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Numerical Methods For Engineers_S. C. Chapra And R. P. Canale.pdf

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The document is a textbook companion for 'Numerical Methods for Engineers' by S.C. Chapra and R.P. Canale, created by Meghana Sundaresan, which includes Scilab codes and examples related to numerical methods applicable in engineering. It details various topics such as mathematical modeling, optimization, numerical differentiation, and integration methods, organized with specific codes for reference to examples in the main textbook. The Scilab codes are available for download on a specified website as part of the textbook companion project.

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Scilab Textbook Companion for Numerical Methods For Engineers by S. C. Chapra And R. P. Canale1 Created by Meghana Sundaresan Numerical Methods Chemical Engineering VNIT College Teacher Kailas L. Wasewar Cross-Checked by July 31, 2019 1Funded by a grant from the National Mission on Education through ICT, http://spoken-tutorial.org/NMEICT-Intro. This Textbook Companion and Scilab codes written in it can be downloaded from the ”Textbook Companion Project” section at the website http://scilab.in
Book Description Title: Numerical Methods For Engineers Author: S. C. Chapra And R. P. Canale Publisher: McGraw Hill, New York Edition: 5 Year: 2006 ISBN: 0071244298 1
Scilab numbering policy used in this document and the relation to the above book. Exa Example (Solved example) Eqn Equation (Particular equation of the above book) AP Appendix to Example(Scilab Code that is an Appednix to a particular Example of the above book) For example, Exa 3.51 means solved example 3.51 of this book. Sec 2.3 means a scilab code whose theory is explained in Section 2.3 of the book. 2
Contents List of Scilab Codes 4 1 Mathematical Modelling and Engineering Problem Solving 6 2 Programming and Software 8 3 Approximations and Round off Errors 10 4 Truncation Errors and the Taylor Series 15 5 Bracketing Methods 25 6 Open Methods 34 7 Roots of Polynomials 44 9 Gauss Elimination 51 10 LU Decomposition and matrix inverse 61 11 Special Matrices and gauss seidel 65 13 One dimensional unconstrained optimization 69 14 Multidimensional Unconstrainted Optimization 72 3
15 Constrained Optimization 76 17 Least squares regression 82 18 Interpolation 96 19 Fourier Approximation 108 21 Newton Cotes Integration Formulas 116 22 Integration of equations 129 23 Numerical differentiation 135 25 Runga Kutta methods 139 26 Stiffness and multistep methods 156 27 Boundary Value and Eigen Value problems 165 29 Finite Difference Elliptic Equations 180 30 Finite Difference Parabolic Equations 185 31 Finite Element Method 191 4
List of Scilab Codes Exa 1.1 Analytical Solution to Falling Parachutist Prob- lem . . . . . . . . . . . . . . . . . . . . . . . 6 Exa 1.2 Numerical Solution to Falling Parachutist Prob- lem . . . . . . . . . . . . . . . . . . . . . . . 6 Exa 2.1 roots of quadratic . . . . . . . . . . . . . . . 8 Exa 3.1 Calculations of Errors . . . . . . . . . . . . 10 Exa 3.2 Iterative error estimation . . . . . . . . . . . 11 Exa 3.3 Range of Integers . . . . . . . . . . . . . . . 11 Exa 3.4 Floating Point Numbers . . . . . . . . . . . 12 Exa 3.5 Machine Epsilon . . . . . . . . . . . . . . . 12 Exa 3.6 Interdependent Computations . . . . . . . . 12 Exa 3.7 Subtractive Cancellation . . . . . . . . . . . 13 Exa 3.8 Infinite Series Evaluation . . . . . . . . . . . 13 Exa 4.1 Polynomial Taylor Series . . . . . . . . . . . 15 Exa 4.2 Taylor Series Expansion . . . . . . . . . . . 17 Exa 4.3 Effect of Nonlinearity and Stepsize on Taylor expansion . . . . . . . . . . . . . . . . . . . 20 Exa 4.4 Finite divided difference approximation of deriva- tives . . . . . . . . . . . . . . . . . . . . . . 21 Exa 4.5 Error propagation in function of single variable 22 Exa 4.6 Error propagation in multivariable function 22 Exa 4.7 Condition Number . . . . . . . . . . . . . . 24 Exa 5.1 Graphical Approach . . . . . . . . . . . . . 25 Exa 5.2 Computer Graphics to Locate Roots . . . . 27 Exa 5.3 Bisection . . . . . . . . . . . . . . . . . . . 28 Exa 5.4 Error Estimates for Bisection . . . . . . . . 29 Exa 5.5 False Position . . . . . . . . . . . . . . . . . 30 Exa 5.6 Bracketing and False Position Methods . . . 31 5
Exa 6.1 simple fixed point iteration . . . . . . . . . 34 Exa 6.2 The Two curve graphical method . . . . . . 34 Exa 6.3 Newton Raphson Method . . . . . . . . . . 36 Exa 6.4 Error analysis of Newton Raphson method . 37 Exa 6.5 slowly converging function with Newton Raph- son method . . . . . . . . . . . . . . . . . . 38 Exa 6.6 The secant method . . . . . . . . . . . . . . 38 Exa 6.7 secant and false position techniques . . . . . 39 Exa 6.8 Modified secant method . . . . . . . . . . . 40 Exa 6.9 modified newton raphson method . . . . . . 41 Exa 6.10 fixed point iteration for nonlinear system . . 42 Exa 6.11 Newton Raphson for a nonlinear Problem . 42 Exa 7.1 Polynomial Deflation . . . . . . . . . . . . . 44 Exa 7.2 Mullers Method . . . . . . . . . . . . . . . . 45 Exa 7.3 Bairstows Method . . . . . . . . . . . . . . 46 Exa 7.4 Locate single root . . . . . . . . . . . . . . . 47 Exa 7.5 Solving nonlinear system . . . . . . . . . . . 48 Exa 7.6 Root Location . . . . . . . . . . . . . . . . . 49 Exa 7.7 Roots of Polynomials . . . . . . . . . . . . . 49 Exa 7.8 Root Location . . . . . . . . . . . . . . . . . 49 Exa 9.1 Graphical Method for two Equations . . . . 51 Exa 9.2 Deterinants . . . . . . . . . . . . . . . . . . 53 Exa 9.3 Cramers Rule . . . . . . . . . . . . . . . . . 53 Exa 9.4 Elimination of Unknowns . . . . . . . . . . 54 Exa 9.5 Naive Gauss Elimination . . . . . . . . . . . 55 Exa 9.6 ill conditioned systems . . . . . . . . . . . . 56 Exa 9.7 Effect of Scale on Determinant . . . . . . . 56 Exa 9.8 Scaling . . . . . . . . . . . . . . . . . . . . . 57 Exa 9.9 Partial Pivoting . . . . . . . . . . . . . . . . 57 Exa 9.10 Effect of scaling on Pivoting and round off . 58 Exa 9.11 Solution of Linear Algebraic Equations . . . 59 Exa 9.12 Gauss Jordan method . . . . . . . . . . . . 59 Exa 10.1 LU decomposition with gauss elimination . . 61 Exa 10.2 The substitution steps . . . . . . . . . . . . 62 Exa 10.3 Matrix inversion . . . . . . . . . . . . . . . 62 Exa 10.4 Matrix condition evaluation . . . . . . . . . 63 Exa 11.1 Tridiagonal solution with Thomas algorithm 65 Exa 11.2 Cholesky Decomposition . . . . . . . . . . . 66 6
Exa 11.3 Gauss Seidel method . . . . . . . . . . . . . 66 Exa 11.4 Linear systems . . . . . . . . . . . . . . . . 67 Exa 11.5 Manipulate linear algebraic equations . . . . 68 Exa 11.6 Analyze and solve Hilbert matrix . . . . . . 68 Exa 13.1 Golden section method . . . . . . . . . . . . 69 Exa 13.2 Quadratic interpolation . . . . . . . . . . . 70 Exa 13.3 Newtons method . . . . . . . . . . . . . . . 71 Exa 14.1 Random Search Method . . . . . . . . . . . 72 Exa 14.2 Path of Steepest Descent . . . . . . . . . . . 73 Exa 14.3 1 D function along Gradient . . . . . . . . . 73 Exa 14.4 Optimal Steepest Descent . . . . . . . . . . 74 Exa 15.1 Setting up LP problem . . . . . . . . . . . . 76 Exa 15.2 Graphical Solution . . . . . . . . . . . . . . 78 Exa 15.3 Linear Programming Problem . . . . . . . . 78 Exa 15.4 Nonlinear constrained optimization . . . . . 79 Exa 15.5 One dimensional Optimization . . . . . . . . 80 Exa 15.6 Multidimensional Optimization . . . . . . . 80 Exa 15.7 Locate Single Optimum . . . . . . . . . . . 81 Exa 17.1 Linear regression . . . . . . . . . . . . . . . 82 Exa 17.2 Estimation of errors for the linear least square fit . . . . . . . . . . . . . . . . . . . . . . . 83 Exa 17.3.a linear regression using computer . . . . . . . 84 Exa 17.3.b linear regression using computer . . . . . . . 85 Exa 17.4 Linearization of a power function . . . . . . 87 Exa 17.5 polynomial regression . . . . . . . . . . . . . 88 Exa 17.6 Multiple linear regression . . . . . . . . . . 92 Exa 17.7 Confidence interval for linear regression . . . 93 Exa 17.8 Gauss Newton method . . . . . . . . . . . . 94 Exa 18.1 Linear interpolation . . . . . . . . . . . . . 96 Exa 18.2 Quadratic interpolation . . . . . . . . . . . 97 Exa 18.3 Newtons divided difference Interpolating poly- nomials . . . . . . . . . . . . . . . . . . . . 97 Exa 18.4 Error estimation for Newtons polynomial . . 98 Exa 18.5 Error Estimates for Order of Interpolation . 99 Exa 18.6 Lagrange interpolating polynomials . . . . . 100 Exa 18.7 Lagrange interpolation using computer . . . 100 Exa 18.8 First order splines . . . . . . . . . . . . . . 103 Exa 18.9 Quadratic splines . . . . . . . . . . . . . . . 103 7
Exa 18.10 Cubic splines . . . . . . . . . . . . . . . . . 106 Exa 19.1 Least Square Fit . . . . . . . . . . . . . . . 108 Exa 19.2 Continuous Fourier Series Approximation . . 109 Exa 19.3 Trendline . . . . . . . . . . . . . . . . . . . 110 Exa 19.4 Data Analysis . . . . . . . . . . . . . . . . . 110 Exa 19.5 Curve Fitting . . . . . . . . . . . . . . . . . 112 Exa 19.6 Polynomial Regression . . . . . . . . . . . . 115 Exa 21.1 Single trapezoidal rule . . . . . . . . . . . . 116 Exa 21.2 Multiple trapezoidal rule . . . . . . . . . . . 117 Exa 21.3 Evaluating Integrals . . . . . . . . . . . . . 118 Exa 21.4 Single Simpsons 1 by 3 rule . . . . . . . . . 122 Exa 21.5 Multiple Simpsons 1 by 3 rule . . . . . . . . 123 Exa 21.6 Simpsons 3 by 8 rule . . . . . . . . . . . . . 124 Exa 21.7 Unequal Trapezoidal segments . . . . . . . . 126 Exa 21.8 Simpsons Uneven data . . . . . . . . . . . . 127 Exa 21.9 Average Temperature Determination . . . . 127 Exa 22.1 Error corrections of the trapezoidal rule . . 129 Exa 22.2 Higher order error correction of integral esti- mates . . . . . . . . . . . . . . . . . . . . . 129 Exa 22.3 Two point gauss legendre formulae . . . . . 130 Exa 22.4 Three point gauss legendre method . . . . . 131 Exa 22.5 Applying Gauss Quadrature to the falling Parachutist problem . . . . . . . . . . . . . . . . . . . . 132 Exa 22.6 Evaluation of improper integral . . . . . . . 133 Exa 23.1 High accuracy numerical differentiation for- mulas . . . . . . . . . . . . . . . . . . . . . 135 Exa 23.2 Richardson extrapolation . . . . . . . . . . . 136 Exa 23.3 Differentiating unequally spaced data . . . . 137 Exa 23.4 Integration and Differentiation . . . . . . . . 137 Exa 23.5 Integrate a function . . . . . . . . . . . . . . 138 Exa 25.1 Eulers method . . . . . . . . . . . . . . . . 139 Exa 25.2 Taylor series estimate for error by eulers method 140 Exa 25.3 Effect of reduced step size on Eulers method 140 Exa 25.4 Solving ODEs . . . . . . . . . . . . . . . . . 141 Exa 25.5 Heuns method . . . . . . . . . . . . . . . . . 142 Exa 25.6 Comparison of various second order RK 4 method 143 Exa 25.7 Classical fourth order RK method . . . . . . 144 Exa 25.8 Comparison of Runga Kutta methods . . . . 145 8
Exa 25.9 Solving systems of ODE using Eulers method 147 Exa 25.10 Solving systems of ODE using RK 4 method 147 Exa 25.11 Solving systems of ODEs . . . . . . . . . . . 148 Exa 25.12 Adaptive fourth order RK method . . . . . 150 Exa 25.13 Runga kutta fehlberg method . . . . . . . . 153 Exa 25.14 Adaptive Fourth order RK scheme . . . . . 154 Exa 26.1 Explicit and Implicit Euler . . . . . . . . . . 156 Exa 26.2 Non self starting Heun method . . . . . . . 157 Exa 26.3 Estimate of per step truncation error . . . . 160 Exa 26.4 Effect of modifier on Predictor Corrector re- sults . . . . . . . . . . . . . . . . . . . . . . 161 Exa 26.5 Milnes Method . . . . . . . . . . . . . . . . 161 Exa 26.6 Fourth order Adams method . . . . . . . . . 162 Exa 26.7 stability of Milnes and Fourth order Adams method . . . . . . . . . . . . . . . . . . . . 163 Exa 27.1 The shooting method . . . . . . . . . . . . . 165 Exa 27.2 The shooting method for non linear problems 166 Exa 27.3 Finite Difference Approximation . . . . . . . 167 Exa 27.4 Mass Spring System . . . . . . . . . . . . . 168 Exa 27.5 Axially Loaded column . . . . . . . . . . . . 168 Exa 27.6 Polynomial Method . . . . . . . . . . . . . . 169 Exa 27.7 Power Method Highest Eigenvalue . . . . . 171 Exa 27.8 Power Method Lowest Eigenvalue . . . . . . 171 Exa 27.9 Eigenvalues and ODEs . . . . . . . . . . . . 172 Exa 27.10.a Stiff ODEs . . . . . . . . . . . . . . . . . . . 173 Exa 27.10.b Stiff ODEs . . . . . . . . . . . . . . . . . . . 177 Exa 27.11 Solving ODEs . . . . . . . . . . . . . . . . . 178 Exa 29.1 Temperature of a heated plate with fixed bound- ary conditions . . . . . . . . . . . . . . . . . 180 Exa 29.2 Flux distribution for a heated plate . . . . . 181 Exa 29.3 Heated plate with a insulated edge . . . . . 182 Exa 29.4 Heated plate with an irregular boundary . . 183 Exa 30.1 Explicit solution of a one dimensional heat conduction equation . . . . . . . . . . . . . 185 Exa 30.2 Simple implicit solution of a heat conduction equation . . . . . . . . . . . . . . . . . . . . 186 Exa 30.3 Crank Nicolson solution to the heat conduc- tion equation . . . . . . . . . . . . . . . . . 187 9
Exa 30.4 Comparison of Analytical and Numerical so- lution . . . . . . . . . . . . . . . . . . . . . 188 Exa 30.5 ADI Method . . . . . . . . . . . . . . . . . 188 Exa 31.1 Analytical Solution for Heated Rod . . . . . 191 Exa 31.2 Element Equation for Heated Rod . . . . . . 193 Exa 31.3 Temperature of a heated plate . . . . . . . . 193 10
List of Figures 5.1 Graphical Approach . . . . . . . . . . . . . . . . . . . . . . 26 5.2 Computer Graphics to Locate Roots . . . . . . . . . . . . . 27 6.1 The Two curve graphical method . . . . . . . . . . . . . . . 35 9.1 Graphical Method for two Equations . . . . . . . . . . . . . 52 15.1 Graphical Solution . . . . . . . . . . . . . . . . . . . . . . . 77 17.1 linear regression using computer . . . . . . . . . . . . . . . . 85 17.2 linear regression using computer . . . . . . . . . . . . . . . . 86 17.3 Linearization of a power function . . . . . . . . . . . . . . . 88 17.4 Linearization of a power function . . . . . . . . . . . . . . . 89 17.5 polynomial regression . . . . . . . . . . . . . . . . . . . . . . 90 18.1 First order splines . . . . . . . . . . . . . . . . . . . . . . . . 102 18.2 Quadratic splines . . . . . . . . . . . . . . . . . . . . . . . . 104 18.3 Cubic splines . . . . . . . . . . . . . . . . . . . . . . . . . . 106 19.1 Trendline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 19.2 Curve Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . 113 19.3 Curve Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . 114 25.1 Solving ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . 141 25.2 Solving systems of ODEs . . . . . . . . . . . . . . . . . . . . 150 11
25.3 Solving systems of ODEs . . . . . . . . . . . . . . . . . . . . 151 25.4 Adaptive Fourth order RK scheme . . . . . . . . . . . . . . 154 26.1 Explicit and Implicit Euler . . . . . . . . . . . . . . . . . . . 158 26.2 Explicit and Implicit Euler . . . . . . . . . . . . . . . . . . . 159 27.1 The shooting method for non linear problems . . . . . . . . 166 27.2 Eigenvalues and ODEs . . . . . . . . . . . . . . . . . . . . . 174 27.3 Eigenvalues and ODEs . . . . . . . . . . . . . . . . . . . . . 175 27.4 Stiff ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 27.5 Stiff ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 31.1 Analytical Solution for Heated Rod . . . . . . . . . . . . . . 192 12
Chapter 1 Mathematical Modelling and Engineering Problem Solving Scilab code Exa 1.1 Analytical Solution to Falling Parachutist Problem 1 clc; 2 clear; 3 g=9.8; //m/ s ˆ2; a c c e l e r a t i o n due to g r a v i t y 4 m=68.1; // kg 5 c=12.5; // kg/ sec ; drag c o e f f i c i e n t 6 count =1; 7 for i=0:2:12 8 v(count)=g*m*(1-exp(-c*i/m))/c; 9 disp(v(count),”v (m/ s )=”,i,”Time ( s )=”) 10 count=count +1; 11 end 12 disp(g*m/c,”v (m/ s )=”,” i n f i n i t y ”,”Time ( s )=”) Scilab code Exa 1.2 Numerical Solution to Falling Parachutist Problem 1 clc; 13
2 clear; 3 g=9.8; //m/ s ˆ2; a c c e l e r a t i o n due to g r a v i t y 4 m=68.1; // kg 5 c=12.5; // kg/ sec ; drag c o e f f i c i e n t 6 count =2; 7 v(1) =0; 8 disp(v(1),”v (m/ s )=” ,0,”Time ( s )=”) 9 for i=2:2:12 10 v(count)=v(count -1)+(g-c*v(count -1)/m)*(2); 11 disp(v(count),”v (m/ s )=”,i,”Time ( s )=”) 12 count=count +1; 13 end 14 disp(g*m/c,”v (m/ s )=”,” i n f i n i t y ”,”Time ( s )=”) 14
Chapter 2 Programming and Software Scilab code Exa 2.1 roots of quadratic 1 clc; 2 clear; 3 response =1; 4 while response ==1 5 a=input(” Input the value of a : ”) 6 b=input(” Input the value of b : ”) 7 c=input(” Input the value of c : ”) 8 if a==0 then 9 if b~=0 then 10 r1=-c/b; 11 disp(r1 ,”The root : ”) 12 else disp(” T r i v i a l S o l u t i o n . ”) 13 end 14 else 15 discr=b^2-4*a*c; 16 if discr >=0 then 17 r1=(-b+sqrt(discr))/(2*a); 18 r2=(-b-sqrt(discr))/(2*a); 19 disp(r2 ,”and”,r1 ,”The r o o t s are : ”) 20 else 21 r1=-b/(2*a); 15
22 r2=r1; 23 i1=sqrt(abs(discr))/(2*a); 24 i2=-i1; 25 disp(r2+i2*sqrt (-1),r1+i1*sqrt (-1),”The r o o t s are : ”) 26 end 27 end 28 response=input(”Do you want to continue ( p r e s s 1 f o r yes and 2 f o r no ) ?”) 29 if response =2 then exit; 30 end 31 end 32 end 16
Chapter 3 Approximations and Round off Errors Scilab code Exa 3.1 Calculations of Errors 1 clc; 2 clear; 3 lbm =9999; //cm , measured length of bridge 4 lrm =9; //cm , measured length of r i v e t 5 lbt =10000; //cm , true length of bridge 6 lrt =10; //cm , true length of r i v e t 7 // c a l c u l a t i n g true e r r o r below ; 8 Etb=lbt -lbm;//cm , true e r r o r in bridge 9 Etr=lrt -lrm;//cm , true e r r o r in r i v e t 10 // c a l c u l a t i n g percent r e l a t i v e e r r o r below 11 etb=Etb *100/ lbt;// percent r e l a t i v e e r r o r f o r bridge 12 etr=Etb *100/ lrt;// percent r e l a t i v e e r r o r f o r r i v e t 13 disp(”a . The true e r r o r i s ”) 14 disp(” f o r the bridge ”,”cm”,Etb) 15 disp(” f o r the r i v e t ”,”cm”,Etr) 16 disp(”b . The percent r e l a t i v e e r r o r i s ”) 17 disp(” f o r the bridge ”,”%”,etb) 18 disp(” f o r the r i v e t ”,”%”,etr) 17
Scilab code Exa 3.2 Iterative error estimation 1 clc; 2 clear; 3 n=3; //number of s i g n i f i c a n t f i g u r e s 4 es =0.5*(10^(2 -n));// percent , s p e c i f i e d e r r o r c r i t e r i o n 5 x=0.5; 6 f(1) =1; // f i r s t estimate f=e ˆx = 1 7 ft =1.648721; // true value of e ˆ0.5= f 8 et(1)=(ft -f(1))*100/ ft; 9 ea(1) =100; 10 i=2; 11 while ea(i-1) >=es 12 f(i)=f(i-1)+(x^(i-1))/( factorial(i-1)); 13 et(i)=(ft -f(i))*100/ ft; 14 ea(i)=(f(i)-f(i-1))*100/f(i); 15 i=i+1; 16 end 17 for j=1:i-1 18 disp(ea(j),” Approximate estimate of e r r o r (%)=”,et (j),”True % r e l a t i v e e r r o r=”,f(j),” Result=”,j, ” term number=”) 19 disp(” −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 20 end Scilab code Exa 3.3 Range of Integers 1 n=16; // no of b i t s 2 num =0; 18
3 for i=0:(n-2) 4 num=num +(1*(2^i)); 5 end 6 disp ((-1* num),” to ”,num ,”Thus a 16− b i t computer word can s t o r e decimal i n t e g e r s ranging from ”) Scilab code Exa 3.4 Floating Point Numbers 1 clc; 2 clear; 3 n=7; // no . of b i t s 4 // the maximum value of exponents i s given by 5 Max =1*(2^1) +1*(2^0); 6 // mantissa i s found by 7 mantissa =1*(2^ -1) +0*(2^ -3) +0*(2^ -3); 8 num=mantissa *(2^( Max*-1));// s m a l l e s t p o s s i b l e p o s i t i v e number f o r t h i s system 9 disp(num ,”The s m a l l e s t p o s s i b l e p o s i t i v e number f o r t h i s system i s ”) Scilab code Exa 3.5 Machine Epsilon 1 clc; 2 clear; 3 b=2; // base 4 t=3; //number of mantissa b i t s 5 E=2^(1 -t);// e p s i l o n 6 disp(E,” value of e p s i l o n=”) Scilab code Exa 3.6 Interdependent Computations 19
1 clc; 2 clear; 3 num=input(” Input a number : ”) 4 sum =0; 5 for i=1:100000 6 sum=sum+num; 7 end 8 disp(sum ,”The number summed up 100 ,000 times i s=”) Scilab code Exa 3.7 Subtractive Cancellation 1 clc; 2 clear; 3 a=1; 4 b=3000.001; 5 c=3; 6 // the r o o t s of the qu ad ra ti c equation x ˆ2+3000.001∗ x +3=0 are found as 7 D=(b^2) -4*a*c; 8 x1=(-b+(D^0.5))/(2*a); 9 x2=(-b-(D^0.5))/(2*a); 10 disp(x2 ,and ,x1 ,”The r o o t s of the qu ad ra ti c equation ( x ˆ2) +(3000.001∗ x )+3=0 are = ”) Scilab code Exa 3.8 Infinite Series Evaluation 1 clc; 2 clear; 3 function y=f(x) 4 y=exp(x) 5 endfunction 6 sum =1; 7 test =0; 20
8 i=0; 9 term =1; 10 x=input(” Input value of x : ”) 11 while sum~= test 12 disp(sum ,”sum : ”,term ,” term : ”,i,” i : ”) 13 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−”) 14 i=i+1; 15 term=term*x/i; 16 test=sum; 17 sum=sum+term; 18 end 19 disp(f(x),” Exact Value ”) 21
Chapter 4 Truncation Errors and the Taylor Series Scilab code Exa 4.1 Polynomial Taylor Series 1 clc; 2 clear; 3 function y=f(x) 4 y= -0.1*x^4 -0.15*x^3 -0.5*x^2 -0.25*x+1.2; 5 endfunction 6 xi=0; 7 xf=1; 8 h=xf -xi; 9 fi=f(xi);// f u n c t i o n value at xi 10 ffa=f(xf);// a c t u a l f u n c t i o n value at xf 11 12 // f o r n=0, i . e , zero order approximation 13 ff=fi; 14 Et(1)=ffa -ff;// t r u n c a t i o n e r r o r at x=1 15 disp(fi ,”The value of f at x=0 : ”) 16 disp(ff ,”The value of f at x=1 due to zero order approximation : ”) 17 disp(Et(1),” Truncation e r r o r : ”) 18 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− 22
”) 19 20 // f o r n=1, i . e , f i r s t order approximation 21 deff( ’ y=f1 ( x ) ’ , ’ y=d e r i v a t i v e ( f , x , order =4) ’ ) 22 f1i=f1(xi);// value of f i r s t d e r i v a t i v e of f u n c t i o n at xi 23 f1f=fi+f1i*h;// value of f i r s t d e r i v a t i v e of f u n c t i o n at xf 24 Et(2)=ffa -f1f;// t r u n c a t i o n e r r o r at x=1 25 disp(f1i ,”The value of f i r s t d e r i v a t i v e of f at x=0 : ”) 26 disp(f1f ,”The value of f at x=1 due to f i r s t order approximation : ”) 27 disp(Et(2),” Truncation e r r o r : ”) 28 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 29 30 31 // f o r n=2, i . e , second order approximation 32 deff( ’ y=f2 ( x ) ’ , ’ y=d e r i v a t i v e ( f1 , x , order =4) ’ ) 33 f2i=f2(xi);// value of second d e r i v a t i v e of f u n c t i o n at xi 34 f2f=f1f+f2i*(h^2)/factorial (2);// value of second d e r i v a t i v e of f u n c t i o n at xf 35 Et(3)=ffa -f2f;// t r u n c a t i o n e r r o r at x=1 36 disp(f2i ,”The value of second d e r i v a t i v e of f at x=0 : ”) 37 disp(f2f ,”The value of f at x=1 due to second order approximation : ”) 38 disp(Et(3),” Truncation e r r o r : ”) 39 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 40 41 // f o r n=3, i . e , t h i r d order approximation 42 deff( ’ y=f3 ( x ) ’ , ’ y=d e r i v a t i v e ( f2 , x , order =4) ’ ) 43 f3i=f3(xi);// value of t h i r d d e r i v a t i v e of f u n c t i o n at xi 44 f3f=f2f+f3i*(h^3)/factorial (3);// value of t h i r d 23
d e r i v a t i v e of f u n c t i o n at xf 45 Et(4)=ffa -f3f;// t r u n c a t i o n e r r o r at x=1 46 disp(f3i ,”The value of t h i r d d e r i v a t i v e of f at x=0 : ”) 47 disp(f3f ,”The value of f at x=1 due to t h i r d order approximation : ”) 48 disp(Et(4),” Truncation e r r o r : ”) 49 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 50 51 // f o r n=4, i . e , f o u r t h order approximation 52 deff( ’ y=f4 ( x ) ’ , ’ y=d e r i v a t i v e ( f3 , x , order =4) ’ ) 53 f4i=f4(xi);// value of f o u r t h d e r i v a t i v e of f u n c t i o n at xi 54 f4f=f3f+f4i*(h^4)/factorial (4);// value of f o u r t h d e r i v a t i v e of f u n c t i o n at xf 55 Et(5)=ffa -f4f;// t r u n c a t i o n e r r o r at x=1 56 disp(f4i ,”The value of f o u r t h d e r i v a t i v e of f at x=0 : ”) 57 disp(f4f ,”The value of f at x=1 due to f o u r t h order approximation : ”) 58 disp(Et(5),” Truncation e r r o r : ”) 59 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) Scilab code Exa 4.2 Taylor Series Expansion 1 clc; 2 clear; 3 function y=f(x) 4 y=cos(x) 5 endfunction 6 xi=%pi /4; 7 xf=%pi /3; 8 h=xf -xi; 24
9 fi=f(xi);// f u n c t i o n value at xi 10 ffa=f(xf);// a c t u a l f u n c t i o n value at xf 11 12 // f o r n=0, i . e , zero order approximation 13 ff=fi; 14 et(1)=(ffa -ff)*100/ ffa;// percent r e l a t i v e e r r o r at x =1 15 disp(ff ,”The value of f at x=1 due to zero order approximation : ”) 16 disp(et(1),”% r e l a t i v e e r r o r : ”) 17 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 18 19 // f o r n=1, i . e , f i r s t order approximation 20 deff( ’ y=f1 ( x ) ’ , ’ y=d e r i v a t i v e ( f , x , order =4) ’ ) 21 f1i=f1(xi);// value of f i r s t d e r i v a t i v e of f u n c t i o n at xi 22 f1f=fi+f1i*h;// value of f i r s t d e r i v a t i v e of f u n c t i o n at xf 23 et(2)=(ffa -f1f)*100/ ffa;//% r e l a t i v e e r r o r at x=1 24 disp(f1f ,”The value of f at x=1 due to f i r s t order approximation : ”) 25 disp(et(2),”% r e l a t i v e e r r o r : ”) 26 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 27 28 29 // f o r n=2, i . e , second order approximation 30 deff( ’ y=f2 ( x ) ’ , ’ y=d e r i v a t i v e ( f1 , x , order =4) ’ ) 31 f2i=f2(xi);// value of second d e r i v a t i v e of f u n c t i o n at xi 32 f2f=f1f+f2i*(h^2)/factorial (2);// value of second d e r i v a t i v e of f u n c t i o n at xf 33 et(3)=(ffa -f2f)*100/ ffa;//% r e l a t i v e e r r o r at x=1 34 disp(f2f ,”The value of f at x=1 due to second order approximation : ”) 35 disp(et(3),”% r e l a t i v e e r r o r : ”) 36 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− 25
”) 37 38 39 // f o r n=3, i . e , t h i r d order approximation 40 deff( ’ y=f3 ( x ) ’ , ’ y=d e r i v a t i v e ( f2 , x , order =4) ’ ) 41 f3i=f3(xi);// value of t h i r d d e r i v a t i v e of f u n c t i o n at xi 42 f3f=f2f+f3i*(h^3)/factorial (3);// value of t h i r d d e r i v a t i v e of f u n c t i o n at xf 43 et(4)=(ffa -f3f)*100/ ffa;//% r e l a t i v e e r r o r at x=1 44 disp(f3f ,”The value of f at x=1 due to t h i r d order approximation : ”) 45 disp(et(4),”% r e l a t i v e e r r o r : ”) 46 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 47 48 49 // f o r n=4, i . e , f o u r t h order approximation 50 deff( ’ y=f4 ( x ) ’ , ’ y=d e r i v a t i v e ( f3 , x , order =4) ’ ) 51 f4i=f4(xi);// value of f o u r t h d e r i v a t i v e of f u n c t i o n at xi 52 f4f=f3f+f4i*(h^4)/factorial (4);// value of f o u r t h d e r i v a t i v e of f u n c t i o n at xf 53 et(5)=(ffa -f4f)*100/ ffa;//% r e l a t i v e e r r o r at x=1 54 disp(f4f ,”The value of f at x=1 due to f o u r t h order approximation : ”) 55 disp(et(5),”% r e l a t i v e e r r o r : ”) 56 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 57 58 59 // f o r n=5, i . e , f i f t h order approximation 60 f5i=(f4 (1.1* xi)-f4 (0.9* xi))/(2*0.1);// value of f i f t h d e r i v a t i v e of f u n c t i o n at xi ( c e n t r a l d i f f e r e n c e method ) 61 f5f=f4f+f5i*(h^5)/factorial (5);// value of f i f t h d e r i v a t i v e of f u n c t i o n at xf 62 et(6)=(ffa -f5f)*100/ ffa;//% r e l a t i v e e r r o r at x=1 26
63 disp(f5f ,”The value of f at x=1 due to f i f t h order approximation : ”) 64 disp(et(6),”% r e l a t i v e e r r o r : ”) 65 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 66 67 68 // f o r n=6, i . e , s i x t h order approximation 69 deff( ’ y=f6 ( x ) ’ , ’ y=d e r i v a t i v e ( f5 , x , order =4) ’ ) 70 f6i=(f4 (1.1* xi) -2*f4(xi)+f4 (0.9* xi))/(0.1^2);// value of s i x t h d e r i v a t i v e of f u n c t i o n at xi ( c e n t r a l d i f f e r e n c e method ) 71 f6f=f5f+f6i*(h^6)/factorial (6);// value of s i x t h d e r i v a t i v e of f u n c t i o n at xf 72 et(6)=(ffa -f6f)*100/ ffa;//% r e l a t i v e e r r o r at x=1 73 disp(f6f ,”The value of f at x=1 due to s i x t h order approximation : ”) 74 disp(et(6),”% r e l a t i v e e r r o r : ”) 75 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) Scilab code Exa 4.3 Effect of Nonlinearity and Stepsize on Taylor expansion 1 clc; 2 clear; 3 m=input(” Input value of m: ”) 4 h=input(” Input value of h : ”) 5 function y=f(x) 6 y=x^m 7 endfunction 8 x1=1; 9 x2=x1+h; 10 fx1=f(x1); 11 fx2=fx1+m*(fx1^(m-1))*h; 12 if m==1 then 27
13 R=0; 14 else if m==2 then 15 R=2*(h^2)/factorial (2); 16 end 17 if m==3 then 18 R=(6*( x1)*(h^2)/factorial (2))+(6*(h^3)/ factorial (3)); 19 end 20 if m==4 then 21 R=(12*( x1^2)*(h^2)/factorial (2))+(24*( x1)*(h ^3)/factorial (3))+(24*(h^4)/factorial (4)) ; 22 end 23 end 24 disp(R,” Remainder : ”,fx2 ,”The value by f i r s t order approximation : ”) 25 disp(f(x2),”True Value at x2 : ”) Scilab code Exa 4.4 Finite divided difference approximation of derivatives 1 clc; 2 clear; 3 function y=f(x) 4 y= -0.1*(x^4) -0.15*(x^3) -0.5*(x^2) -0.25*(x)+1.2 5 endfunction 6 x=0.5; 7 h=input(” Input h : ”) 8 x1=x-h; 9 x2=x+h; 10 // forward d i f f e r e n c e method 11 fdx1 =(f(x2)-f(x))/h;// d e r i v a t i v e at x 12 et1=abs((fdx1 -derivative(f,x))/derivative(f,x))*100; 13 // backward d i f f e r e n c e method 14 fdx2 =(f(x)-f(x1))/h;// d e r i v a t i v e at x 15 et2=abs((fdx2 -derivative(f,x))/derivative(f,x))*100; 28
16 // c e n t r a l d i f f e r e n c e method 17 fdx3 =(f(x2)-f(x1))/(2*h);// d e r i v a t i v e at x 18 et3=abs((fdx3 -derivative(f,x))/derivative(f,x))*100; 19 disp(h,”For h=”) 20 disp(et1 ,”and percent e r r o r=”,fdx1 ,” D e r i v a t i v e at x by forward d i f f e r e n c e method=”) 21 disp(et2 ,”and percent e r r o r=”,fdx2 ,” D e r i v a t i v e at x by backward d i f f e r e n c e method=”) 22 disp(et3 ,”and percent e r r o r=”,fdx3 ,” D e r i v a t i v e at x by c e n t r a l d i f f e r e n c e method=”) Scilab code Exa 4.5 Error propagation in function of single variable 1 clc; 2 clear; 3 function y=f(x) 4 y=x^3 5 endfunction 6 x=2.5; 7 delta =0.01; 8 deltafx=abs(derivative(f,x))*delta; 9 fx=f(x); 10 disp(fx+deltafx ,”and”,fx -deltafx ,” true value i s between ”) Scilab code Exa 4.6 Error propagation in multivariable function 1 clc; 2 clear; 3 function y=f(F,L,E,I) 4 y=(F*(L^4))/(8*E*I) 5 endfunction 6 Fbar =50; // lb / f t 29
7 Lbar =30; // f t 8 Ebar =1.5*(10^8);// lb / f t ˆ2 9 Ibar =0.06; // f t ˆ4 10 deltaF =2; // lb / f t 11 deltaL =0.1; // f t 12 deltaE =0.01*(10^8);// lb / f t ˆ2 13 deltaI =0.0006; // f t ˆ4 14 ybar =( Fbar *( Lbar ^4))/(8* Ebar*Ibar); 15 function y=f1(F) 16 y=(F*( Lbar ^4))/(8* Ebar*Ibar) 17 endfunction 18 function y=f2(L) 19 y=( Fbar*(L^4))/(8* Ebar*Ibar) 20 endfunction 21 function y=f3(E) 22 y=( Fbar*( Lbar ^4))/(8*E*Ibar) 23 endfunction 24 function y=f4(I) 25 y=( Fbar*( Lbar ^4))/(8* Ebar*I) 26 endfunction 27 28 deltay=abs(derivative(f1 ,Fbar))*deltaF+abs( derivative(f2 ,Lbar))*deltaL+abs(derivative(f3 , Ebar))*deltaE+abs(derivative(f4 ,Ibar))*deltaI; 29 30 disp(ybar+deltay ,”and”,ybar -deltay ,”The value of y i s between : ”) 31 ymin =((Fbar -deltaF)*((Lbar -deltaL)^4))/(8*( Ebar+ deltaE)*( Ibar+deltaI)); 32 ymax =(( Fbar+deltaF)*(( Lbar+deltaL)^4))/(8*( Ebar - deltaE)*(Ibar -deltaI)); 33 disp(ymin ,”ymin i s c a l c u l a t e d at lower extremes of F , L , E, I v a l u e s as =”) 34 disp(ymax ,”ymax i s c a l c u l a t e d at higher extremes of F , L , E, I v a l u e s as =”) 30
Scilab code Exa 4.7 Condition Number 1 clc; 2 clear; 3 function y=f(x) 4 y=tan(x) 5 endfunction 6 x1bar =(%pi /2) +0.1*( %pi /2); 7 x2bar =(%pi /2) +0.01*( %pi /2); 8 // computing c o n d i t i o n number f o r x1bar 9 condnum1=x1bar*derivative(f,x1bar)/f(x1bar); 10 disp(condnum1 ,” i s : ”,x1bar ,”The c o n d i t i o n number of f u n c t i o n f o r x=”) 11 if abs(condnum1) >1 then disp(x1bar ,” Function i s i l l − c o n d i t i o n e d f o r x=”) 12 end 13 // computing c o n d i t i o n number f o r x2bar 14 condnum2=x2bar*derivative(f,x2bar)/f(x2bar); 15 disp(condnum2 ,” i s : ”,x2bar ,”The c o n d i t i o n number of f u n c t i o n f o r x=”) 16 if abs(condnum2) >1 then disp(x2bar ,” Function i s i l l − c o n d i t i o n e d f o r x=”) 17 end 31
Chapter 5 Bracketing Methods Scilab code Exa 5.1 Graphical Approach 1 clc; 2 clear; 3 m=68.1; // kg 4 v=40; //m/ s 5 t=10; // s 6 g=9.8; //m/ s ˆ2 7 function y=f(c) 8 y=g*m*(1-exp(-c*t/m))/c - v; 9 endfunction 10 disp(”For v a r i o u s v a l u e s of c and f ( c ) i s found as : ” ) 11 i=0; 12 for c=4:4:20 13 i=i+1; 14 bracket =[c f(c)]; 15 disp(bracket) 16 fc(i)=f(c); 17 end 18 c=[4 8 12 16 20] 32
Figure 5.1: Graphical Approach 33
Figure 5.2: Computer Graphics to Locate Roots 19 plot2d(c,fc) 20 xtitle( ’ f ( c ) vs c ’ , ’ c ’ , ’ f ( c ) (m/ s ) ’ ) Scilab code Exa 5.2 Computer Graphics to Locate Roots 1 clc; 2 clear; 3 function y=f(x) 4 y=sin (10*x)+cos (3*x); 5 endfunction 6 count =1; 34
7 for i=1:0.05:5 8 val(count)=i; 9 func(count)=f(i); 10 count=count +1; 11 end 12 plot2d(val ,func) 13 xtitle(”x vs f ( x ) ”, ’ x ’ , ’ f ( x ) ’ ) Scilab code Exa 5.3 Bisection 1 clc; 2 clear; 3 m=68.1; // kg 4 v=40; //m/ s 5 t=10; // s 6 g=9.8; //m/ s ˆ2 7 function y=f(c) 8 y=g*m*(1-exp(-c*t/m))/c - v; 9 endfunction 10 x1=12; 11 x2=16; 12 xt =14.7802; // true value 13 e=input(” e n t e r the t o l e r a b l e true percent e r r o r=”) 14 xr=(x1+x2)/2; 15 etemp=abs(xr -xt)/xt *100; // e r r o r 16 while etemp >e 17 if f(x1)*f(xr) >0 then 18 x1=xr; 19 xr=(x1+x2)/2; 20 etemp=abs(xr -xt)/xt *100; 21 end 22 if f(x1)*f(xr) <0 then 23 x2=xr; 24 xr=(x1+x2)/2; 25 etemp=abs(xr -xt)/xt *100; 35
26 end 27 if f(x1)*f(xr)==0 then break; 28 end 29 end 30 31 disp(xr ,”The r e s u l t i s=”) Scilab code Exa 5.4 Error Estimates for Bisection 1 clc; 2 clear; 3 m=68.1; // kg 4 v=40; //m/ s 5 t=10; // s 6 g=9.8; //m/ s ˆ2 7 function y=f(c) 8 y=g*m*(1-exp(-c*t/m))/c - v; 9 endfunction 10 x1=12; 11 x2=16; 12 xt =14.7802; // true value 13 e=input(” e n t e r the t o l e r a b l e approximate e r r o r=”) 14 xr=(x1+x2)/2; 15 i=1; 16 et=abs(xr -xt)/xt *100; // e r r o r 17 disp(i,” I t e r a t i o n : ”) 18 disp(x1 ,” xl : ”) 19 disp(x2 ,”xu : ”) 20 disp(xr ,” xr : ”) 21 disp(et ,” et (%) : ”) 22 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−”) 23 etemp =100; 24 while etemp >e 25 if f(x1)*f(xr) >0 then 26 x1=xr; 36
27 xr=(x1+x2)/2; 28 etemp=abs(xr -x1)/xr *100; 29 et=abs(xr -xt)/xt *100; 30 end 31 if f(x1)*f(xr) <0 then 32 x2=xr; 33 xr=(x1+x2)/2; 34 etemp=abs(xr -x2)/xr *100; 35 et=abs(xr -xt)/xt *100; 36 end 37 if f(x1)*f(xr)==0 then break; 38 end 39 i=i+1; 40 disp(i,” I t e r a t i o n : ”) 41 disp(x1 ,” xl : ”) 42 disp(x2 ,”xu : ”) 43 disp(xr ,” xr : ”) 44 disp(et ,” et (%) : ”) 45 disp(etemp ,” ea (%) ”) 46 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−”) 47 end 48 49 disp(xr ,”The r e s u l t i s=”) Scilab code Exa 5.5 False Position 1 clc; 2 clear; 3 m=68.1; // kg 4 v=40; //m/ s 5 t=10; // s 6 g=9.8; //m/ s ˆ2 7 function y=f(c) 8 y=g*m*(1-exp(-c*t/m))/c - v; 9 endfunction 37
10 x1 =12; 11 x2 =16; 12 xt =14.7802; // true value 13 e=input(” e n t e r the t o l e r a b l e true percent e r r o r=”) 14 xr=x1 -(f(x1)*(x2 -x1))/(f(x2)-f(x1)); 15 etemp=abs(xr -xt)/xt *100; // e r r o r 16 while etemp >e 17 if f(x1)*f(xr) >0 then 18 x1=xr; 19 xr=x1 -(f(x1)*(x2 -x1))/(f(x2)-f(x1)); 20 etemp=abs(xr -xt)/xt *100; 21 end 22 if f(x1)*f(xr) <0 then 23 x2=xr; 24 xr=x1 -(f(x1)*(x2 -x1))/(f(x2)-f(x1)); 25 etemp=abs(xr -xt)/xt *100; 26 end 27 if f(x1)*f(xr)==0 then break; 28 end 29 end 30 disp(xr ,”The r e s u l t i s=”) Scilab code Exa 5.6 Bracketing and False Position Methods 1 clc; 2 clear; 3 function y=f(x) 4 y=x^10 - 1; 5 endfunction 6 x1=0; 7 x2 =1.3; 8 xt=1; 9 10 // using b i s e c t i o n method 11 disp(”BISECTION METHOD: ”) 38
12 xr=(x1+x2)/2; 13 et=abs(xr -xt)/xt *100; // e r r o r 14 disp(1,” I t e r a t i o n : ”) 15 disp(x1 ,” xl : ”) 16 disp(x2 ,”xu : ”) 17 disp(xr ,” xr : ”) 18 disp(et ,” et (%) : ”) 19 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−”) 20 for i=2:5 21 if f(x1)*f(xr) >0 then 22 x1=xr; 23 xr=(x1+x2)/2; 24 ea=abs(xr -x1)/xr *100; 25 et=abs(xr -xt)/xt *100; 26 else if f(x1)*f(xr) <0 then 27 x2=xr; 28 xr=(x1+x2)/2; 29 ea=abs(xr -x2)/xr *100; 30 et=abs(xr -xt)/xt *100; 31 end 32 end 33 if f(x1)*f(xr)==0 then break; 34 end 35 disp(i,” I t e r a t i o n : ”) 36 disp(x1 ,” xl : ”) 37 disp(x2 ,”xu : ”) 38 disp(xr ,” xr : ”) 39 disp(et ,” et (%) : ”) 40 disp(ea ,” ea (%) ”) 41 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−”) 42 end 43 44 // using f a l s e p o s i t i o n method 45 disp(”FALSE POSITION METHOD: ”) 46 x1=0; 47 x2 =1.3; 48 xt=1; 49 xr=x1 -(f(x1)*(x2 -x1))/(f(x2)-f(x1));; 39
50 et=abs(xr -xt)/xt *100; // e r r o r 51 disp(1,” I t e r a t i o n : ”) 52 disp(x1 ,” xl : ”) 53 disp(x2 ,”xu : ”) 54 disp(xr ,” xr : ”) 55 disp(et ,” et (%) : ”) 56 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−”) 57 for i=2:5 58 if f(x1)*f(xr) >0 then 59 x1=xr; 60 xr=x1 -(f(x1)*(x2 -x1))/(f(x2)-f(x1)); 61 ea=abs(xr -x1)/xr *100; 62 et=abs(xr -xt)/xt *100; 63 64 else if f(x1)*f(xr) <0 then 65 x2=xr; 66 xr=x1 -(f(x1)*(x2 -x1))/(f(x2)-f(x1)); 67 ea=abs(xr -x2)/xr *100; 68 et=abs(xr -xt)/xt *100; 69 end 70 end 71 if f(x1)*f(xr)==0 then break; 72 end 73 disp(i,” I t e r a t i o n : ”) 74 disp(x1 ,” xl : ”) 75 disp(x2 ,”xu : ”) 76 disp(xr ,” xr : ”) 77 disp(et ,” et (%) : ”) 78 disp(ea ,” ea (%) ”) 79 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−”) 80 end 40
Chapter 6 Open Methods Scilab code Exa 6.1 simple fixed point iteration 1 // c l c ( ) 2 // f ( x ) = exp(−x ) − x ; 3 // using simple f i x e d point i t e r a t i o n , Xi+1 = exp(−Xi ) 4 x = 0; // i n i t i a l guess 5 for i = 1:11 6 if i == 1 then 7 y(i) = x; 8 else 9 y(i) = exp(-y(i-1)); 10 e(i) = (y(i) - y(i-1)) * 100 / y(i); 11 end 12 end 13 disp(y,”x = ”) 14 disp(”%”,e,” e = ”) Scilab code Exa 6.2 The Two curve graphical method 41
Figure 6.1: The Two curve graphical method 42
1 // c l c ( ) 2 // y1 = x 3 // y2 = exp(−x ) 4 for i = 1:6 5 if i == 1 then 6 x(i) = 0; 7 else x(i) = x(i-1) + 0.2; 8 end 9 y1(i) = x(i); 10 y2(i) = exp(-x(i)); 11 end 12 disp(x,”x = ”) 13 disp(y1 ,”y1 = ”) 14 disp(y2 ,”y2 = ”) 15 plot(x,y1); 16 plot(x,y2); 17 xtitle(” f ( x ) vs x”,”x”,” f ( x ) ”) 18 // from the graph , we get 19 x7 = 5.7; 20 disp(x7 , ” answer using two curve g r a p h i c a l method = ”); Scilab code Exa 6.3 Newton Raphson Method 1 // c l c ( ) 2 // f ( x ) = exp(−x )−x 3 // f ’ ( x ) = −exp(−x )−1 4 for i = 1:5 5 if i == 1 then 6 x(i) = 0; 7 else 8 x(i) = x(i-1) - (exp(-x(i-1))-x(i-1))/(-exp (-x(i-1)) -1); 9 et(i) = (x(i) - x(i-1)) * 100 / x(i); 10 x(i-1) = x(i); 43
11 end 12 end 13 disp(x) 14 disp(et) Scilab code Exa 6.4 Error analysis of Newton Raphson method 1 // c l c ( ) 2 // f ( x ) = exp(−x ) − x 3 // f ’ ( x ) = −exp(−x ) − 1 4 // f ”( x ) = exp(−x ) 5 xr = 0.56714329; 6 //E( t i +1) = −f ”( x ) ∗ E( t i ) / 2 ∗ f ’ ( x ) 7 Et0 = 0.56714329; 8 Et1 = -exp(-xr)* ((Et0)^2) / (2 * (-exp(-xr) - 1)); 9 disp(” which i s c l o s e to the true e r r o r of 0.06714329 ”,Et1 ,”Et1 = ”) 10 Et1true = 0.06714329; 11 Et2 = -exp(-xr)* (( Et1true)^2) / (2 * (-exp(-xr) - 1)); 12 disp(” which i s c l o s e to the true e r r o r of 0.0008323 ” ,Et2 ,”Et2 = ”) 13 Et2true = 0.0008323; 14 Et3 = -exp(-xr)* (( Et2true)^2) / (2 * (-exp(-xr) - 1)); 15 disp(” which i s c l o s e to the true e r r o r ”,Et3 ,”Et3 = ” ) 16 Et4 = -exp(-xr)* ((Et3)^2) / (2 * (-exp(-xr) - 1)); 17 disp(” which i s c l o s e to the true e r r o r ”,Et4 ,”Et4 = ” ) 18 disp(”Thus i t i l l u s t r a t r e s that the e r r o r of newton raphson method f o r t h i s case i s p r o p o r t i o n a l ( by a f a c t o r of 0.18095) to the square of the e r r o r of the p r e v i o u s i t e r a t i o n ”) 44
Scilab code Exa 6.5 slowly converging function with Newton Raphson method 1 // c l c ( ) 2 z = 0.5; 3 // f ( x ) = xˆ10 − 1 4 // f ’ ( x ) = 10∗xˆ9 5 for i = 1:40 6 if i==1 then 7 y(i) = z; 8 else 9 y(i) = y(i-1) - (y(i-1) ^10 - 1) /(10*y(i-1) ^9) 10 end 11 end 12 disp(y) 13 disp(”Thus , a f t e r the f i r s t poor p r e d i c t i o n , the technique i s converging on to the true root of 1 but at a very slow r a t e ”) Scilab code Exa 6.6 The secant method 1 // c l c ( ) 2 funcprot (0) 3 // f ( x ) = exp(−x )−x 4 for i = 1:5 5 if i==1 then 6 x(i) = 0; 7 else 8 if i==2 then 9 x(i) = 1; 10 else 45
11 x(i) =x(i-1) - (exp(-x(i-1))-x(i-1))*(x(i-2) - x(i-1))/(( exp(-x(i-2))-x(i-2)) -(exp(-x (i-1))-x(i-1))) 12 er(i) = (0.56714329 - x(i)) * 100 / 0.56714329; 13 end 14 end 15 end 16 disp(x(1:5) ,”x =”) 17 disp(er (3:5) ,” et =”) Scilab code Exa 6.7 secant and false position techniques 1 // c l c ( ) 2 funcprot (0) 3 // f ( x ) = l og ( x ) 4 disp(” secant method”) 5 for i = 1:4 6 if i==1 then 7 x(i) = 0.5; 8 else 9 if i==2 then 10 x(i) = 5; 11 else 12 x(i) =x(i-1) - log(x(i-1)) * (x(i-2) - x(i -1))/(log(x(i-2)) - log(x(i-1))); 13 end 14 end 15 end 16 disp(x(1:4) ,”x =”) 17 disp(” thus , secant method i s d i v e r g e n t ”) 18 disp(”Now, False p o s i t i o n method”) 19 xl = 0.5; 20 xu = 5; 21 for i = 1:3 46
22 m = log(xl); 23 n = log(xu); 24 xr = xu - n*(xl - xu)/(m - n); 25 disp(xr ,” xr = ”) 26 w = log(xr); 27 if m*w < 0 then 28 xu = xr; 29 else 30 xl = xr; 31 end 32 end 33 34 disp(” thus , f a l s e p o s i t i o n method i s convergent ”) Scilab code Exa 6.8 Modified secant method 1 // c l c ( ) 2 del = 0.01; 3 z = 0.56714329 4 x1 = 1; 5 // f ( x ) = exp(−x ) − x 6 disp(x1 ,”x1 = ”) 7 for i = 1:4 8 if i == 1 then 9 x(i) = 1 10 else 11 w = x(i-1); 12 m = exp(-x(i-1)) - x(i-1); 13 x(i-1) = x(i-1) *(1+ del); 14 n = exp(-x(i-1)) - x(i-1); 15 x(i) = w - (x(i-1)- w) * m/(n-m); 16 em = (z - x(i))*100/z; 17 disp(x(i),”x = ”) 18 disp(”%”,em ,” e r r o r = ”) 19 end 47
20 end Scilab code Exa 6.9 modified newton raphson method 1 // c l c ( ) 2 // f ( x ) = xˆ3 − 5∗xˆ2 + 7∗x −3 3 // f ’ ( x ) = 3∗xˆ2 − 10∗x + 7 4 disp(” standard Newton Raphson method”) 5 for i = 1:7 6 if i == 1 then 7 x(i) = 0; 8 else 9 x(i) = x(i-1) - ((x(i-1))^3 - 5*(x(i-1))^2 + 7*x(i-1) -3)/(3*(x(i-1))^2 - 10*(x(i -1)) + 7); 10 et(i) = (1 - x(i)) * 100 / 1; 11 disp(x(i),”x = ”) 12 disp(”%”,et(i),” e r r o r = ”) 13 x(i-1) = x(i); 14 end 15 end 16 disp(” Modified Newton Raphson method”) 17 // f ”( x ) = 6∗x − 10 18 for i = 1:4 19 if i == 1 then 20 x(i) = 0; 21 else 22 x(i) = x(i-1) - ((x(i-1))^3 - 5*(x(i-1))^2 + 7*x(i-1) -3)*((3*(x(i-1))^2 - 10*(x(i -1)) + 7))/((3*(x(i-1))^2 - 10*(x(i-1)) + 7)^2 - ((x(i-1))^3 - 5*(x(i-1))^2 + 7* x(i-1) -3) * (6*x(i-1) - 10)); 23 et(i) = (1 - x(i)) * 100 / 1; 24 disp(x(i),”x = ”) 25 disp(”%”,et(i),” e r r o r = ”) 48
26 x(i-1) = x(i); 27 end 28 end Scilab code Exa 6.10 fixed point iteration for nonlinear system 1 // c l c ( ) 2 //u ( x , y ) = xˆ2 + x∗y − 10 3 //v ( x , y ) = y + 3∗x∗yˆ2 −57 4 for i = 1:4 5 if i == 1 then 6 x(i) = 1.5; 7 y(i) = 3.5; 8 else 9 x(i) = sqrt (10 - (x(i-1))*(y(i-1))); 10 y(i) = sqrt ((57 - y(i-1))/(3*x(i))); 11 disp(x(i),”x =”) 12 disp(y(i),”y =”) 13 end 14 end 15 disp(”Thus the approaching to the true value 0 f x = 2 and y = 3”) Scilab code Exa 6.11 Newton Raphson for a nonlinear Problem 1 clc; 2 clear; 3 function z=u(x,y) 4 z=x^2+x*y-10 5 endfunction 6 function z=v(x,y) 7 z=y+3*x*y^2-57 8 endfunction 49
9 x=1.5; 10 y=3.5; 11 e=[100 100]; 12 while e(1) >0.0001 & e(2) >0.0001 13 J=[2*x+y x; 3*y^2 1+6*x*y]; 14 deter=determ(J); 15 u1=u(x,y); 16 v1=v(x,y); 17 x=x-((u1*J(2,2)-v1*J(1,2))/deter); 18 y=y-((v1*J(1,1)-u1*J(2,1))/deter); 19 e(1)=abs(2-x); 20 e(2)=abs(3-y); 21 end 22 bracket =[x y]; 23 disp(bracket) 50
Chapter 7 Roots of Polynomials Scilab code Exa 7.1 Polynomial Deflation 1 clc; 2 clear; 3 function y=f(x) 4 y=(x-4)*(x+6) 5 endfunction 6 n=2; 7 a(1) =-24; 8 a(2) =2; 9 a(3) =1; 10 t=4; 11 r=a(3); 12 a(3) =0; 13 for i=(n):-1:1 14 s=a(i); 15 a(i)=r; 16 r=s+r*t; 17 end 18 disp(”The quptient i s a (1 )+a (2 ) ∗x where : ”) 19 disp(a(1),”a (1 )=”) 20 disp(a(2),”a (2 )=”) 21 disp(r,” remainder=”) 51
Scilab code Exa 7.2 Mullers Method 1 clc; 2 clear; 3 function y=f(x) 4 y=x^3 - 13*x - 12 5 endfunction 6 7 x1t=-3; 8 x2t=-1; 9 x3t =4; 10 x0 =4.5; 11 x1 =5.5; 12 x2=5; 13 disp(0,” i t e r a t i o n : ”) 14 disp(x2 ,” xr : ”) 15 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−” ) 16 for i=1:4 17 18 h0=x1 -x0; 19 h1=x2 -x1; 20 d0=(f(x1)-f(x0))/(x1 -x0); 21 d1=(f(x2)-f(x1))/(x2 -x1); 22 a=(d1 -d0)/(h1+h0); 23 b=a*h1+d1; 24 c=f(x2); 25 d=(b^2 - 4*a*c)^0.5; 26 if abs(b+d)>abs(b-d) then x3=x2+(( -2*c)/(b+d)); 27 else x3=x2+(( -2*c)/(b-d)); end 28 ea=abs(x3 -x2)*100/ x3; 29 x0=x1; 30 x1=x2; 31 x2=x3; 52
32 disp(i,” i t e r a t i o n : ”) 33 disp(x2 ,” xr : ”) 34 disp(ea ,” ea (%) : ”) 35 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−” ) 36 end Scilab code Exa 7.3 Bairstows Method 1 clc; 2 clear; 3 function y=f(x) 4 y=x^5 -3.5*x^4+2.75*x^3+2.125*x^2 -3.875*x+1.25 5 endfunction 6 r=-1; 7 s=-1; 8 es=1; //% 9 n=6; 10 count =1; 11 ear =100; 12 eas =100; 13 a=[1.25 -3.875 2.125 2.75 -3.5 1]; 14 while (ear >es) and (eas >es) 15 16 b(n)=a(n); 17 b(n-1)=a(n-1)+r*b(n); 18 for i=n-2: -1:1 19 b(i)=a(i)+r*b(i+1)+s*b(i+2); 20 end 21 c(n)=b(n); 22 c(n-1)=b(n-1)+r*c(n); 23 for i=(n-2) :-1:2 24 c(i)=b(i)+r*c(i+1)+s*c(i+2); 25 end 26 // c ( 3) ∗ dr+c ( 4) ∗ ds=−b (2 ) 53
27 // c ( 2) ∗ dr+c ( 3) ∗ ds=−b (1 ) 28 ds=((-b(1))+(b(2)*c(2)/c(3)))/(c(3) -(c(4)*c(2 )/c(3))); 29 dr=(-b(2)-c(4)*ds)/c(3); 30 r=r+dr; 31 s=s+ds; 32 ear=abs(dr/r)*100; 33 eas=abs(ds/s)*100; 34 disp(count ,” I t e r a t i o n : ”) 35 disp(dr ,” d e l a t a r : ”) 36 disp(ds ,” d e l a t a s : ”) 37 disp(r,” r : ”) 38 disp(s,” s : ”) 39 disp(ear ,” Error in r : ”) 40 disp(eas ,” Error in s : ”) 41 disp(” −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 42 count=count +1; 43 end 44 x1=(r+(r^2 + 4*s)^0.5) /2; 45 x2=(r-(r^2 + 4*s)^0.5) /2; 46 bracket =[x1 x2]; 47 disp(bracket ,”The r o o t s are : ”) 48 disp(”xˆ3 − 4∗xˆ2 + 5.25∗ x − 2. 5 ”,”The q u o t i e n t i s : ” ) 49 disp(” −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) Scilab code Exa 7.4 Locate single root 1 clc; 2 clear; 3 function y=f(x) 54
4 y=x-cos(x) 5 endfunction 6 x1=0; 7 if f(x1) <0 then 8 x2=x1 +0.001; 9 while f(x2) <0 10 x2=x2 +0.001; 11 end 12 elseif x2=x1 +0.001; 13 while f(x2) >0 14 x2=x2 +0.001; 15 end 16 else disp(x1 ,”The root i s=”) 17 end 18 x=x2 -(x2 -x1)*f(x2)/(f(x2)-f(x1)); 19 disp(x,”The root i s=”) Scilab code Exa 7.5 Solving nonlinear system 1 clc; 2 clear; 3 function z=u(x,y) 4 z=x^2+x*y-10 5 endfunction 6 function z=v(x,y) 7 z=y+3*x*y^2-57 8 endfunction 9 x=1; 10 y=3.5; 11 while u(x,y)~=v(x,y) 12 x=x+1; 13 y=y -0.5; 14 end 15 disp(x,”x=”) 16 disp(y,”y=”) 55
Scilab code Exa 7.6 Root Location 1 clc; 2 clear; 3 x=poly(0, ’ s ’ ); 4 p=x^10 -1; 5 disp(”The r o o t s of the polynomial are : ”) 6 disp(roots(p)) Scilab code Exa 7.7 Roots of Polynomials 1 clc; 2 clear; 3 x=poly(0, ’ s ’ ); 4 p=x^5 - 3.5*x^4 +2.75*x^3 +2.125*x^2 - 3.875*x + 1.25; 5 disp(”The r o o t s of the polynomial are : ”) 6 disp(roots(p)) Scilab code Exa 7.8 Root Location 1 clc; 2 clear; 3 function y=f(x) 4 y=x-cos(x) 5 endfunction 6 x1=0; 7 if f(x1) <0 then 8 x2=x1 +0.00001; 56
9 while f(x2) <0 10 x2=x2 +0.00001; 11 end 12 elseif x2=x1 +0.00001; 13 while f(x2) >0 14 x2=x2 +0.00001; 15 end 16 else disp(x1 ,”The root i s=”) 17 end 18 x=x2 -(x2 -x1)*f(x2)/(f(x2)-f(x1)); 19 disp(x,”The root i s=”) 57
Chapter 9 Gauss Elimination Scilab code Exa 9.1 Graphical Method for two Equations 1 clc; 2 clear; 3 // the equations are : 4 // 3∗ x1 + 2∗ x2=18 and −x1 + 2∗ x2=2 5 6 // equation 1 becomes , 7 // x2=−(3/2)∗x1 + 9 8 // equation 2 becomes , 9 // x2=−(1/2)∗x1 + 1 10 11 // p l o t t i n g equation 1 12 for x1 =1:6 13 x2(x1)= -(3/2)*x1 + 9; 14 end 15 x1=[1 2 3 4 5 6]; 16 // p l o t t i n g equation 2 17 for x3 =1:6 18 x4(x3)=(1/2)*x3 + 1; 19 end 58
Figure 9.1: Graphical Method for two Equations 59
20 x3=[1 2 3 4 5 6]; 21 plot(x1 ,x2) 22 plot(x3 ,x4) 23 xtitle(”x2 vs x1”,”x1”,”x2”) 24 // the l i n e s meet at x1=4 amd x2=3 25 disp(3,”x2=”,”and” ,4,”x1=”,”The l i n e s meet at=”) Scilab code Exa 9.2 Deterinants 1 clc; 2 clear; 3 // For f i g 9 . 1 4 a=[3 2;-1 2]; 5 disp(determ(a),”The value of determinant f o r system re pe sen te d in f i g 9.1 =”) 6 // For f i g 9 . 2 ( a ) 7 a=[ -0.5 1; -0.5 1]; 8 disp(determ(a),”The value of determinant f o r system re pe sen te d in f i g 9.2 ( a ) =”) 9 // For f i g 9 . 2 ( b ) 10 a=[ -0.5 1;-1 2]; 11 disp(determ(a),”The value of determinant f o r system re pe sen te d in f i g 9.2 ( b ) =”) 12 // For f i g 9 . 2 ( c ) 13 a=[ -0.5 1; -2.3/5 1]; 14 disp(determ(a),”The value of determinant f o r system re pe sen te d in f i g 9.2 ( c ) =”) Scilab code Exa 9.3 Cramers Rule 1 clc; 2 clear; 3 // the matrix or the system 60
4 b1= -0.01; 5 b2 =0.67; 6 b3= -0.44; 7 a=[0.3 0.52 1;0.5 1 1.9;0.1 0.3 0.5]; 8 a1=[a(2,2) a(2,3);a(3,2) a(3,3)]; 9 A1=determ(a1); 10 a2=[a(2,1) a(2,3);a(3,1) a(3,3)]; 11 A2=determ(a2); 12 a3=[a(2,1) a(2,2);a(3,1) a(3,2)]; 13 A3=determ(a3); 14 D=a(1,1)*A1 -a(1,2)*A2+a(1,3)*A3; 15 p=[b1 0.52 1;b2 1 1.9; b3 0.3 0.5]; 16 q=[0.3 b1 1;0.5 b2 1.9;0.1 b3 0.5]; 17 r=[0.3 0.52 b1 ;0.5 1 b2 ;0.1 0.3 b3]; 18 x1=det(p)/D; 19 x2=det(q)/D; 20 x3=det(r)/D; 21 disp(”The v a l u e s are : ”) 22 disp(x1 ,”x1=”) 23 disp(x2 ,”x2=”) 24 disp(x3 ,”x3=”) Scilab code Exa 9.4 Elimination of Unknowns 1 clc; 2 clear; 3 // the equations are : 4 // 3∗ x1+2∗x2=18 5 //−x1+2∗x2=2 6 a11 =3; 7 a12 =2; 8 b1=18; 9 a21=-1; 10 a22 =2; 11 b2=2; 61
12 x1=(b1*a22 -a12*b2)/(a11*a22 -a12*a21); 13 x2=(b2*a11 -a21*b1)/(a11*a22 -a12*a21); 14 disp(x1 ,”x1=”) 15 disp(x2 ,”x2=”) Scilab code Exa 9.5 Naive Gauss Elimination 1 clc; 2 clear; 3 n=3; 4 b(1) =7.85; 5 b(2) = -19.3; 6 b(3) =71.4; 7 a=[3 -0.1 -0.2;0.1 7 -0.3;0.3 -0.2 10]; 8 for k=1:n-1 9 for i=k+1:n 10 fact=a(i,k)/a(k,k); 11 for j=k+1:n 12 a(i,j)=a(i,j)-fact*(a(k,j)); 13 end 14 b(i)=b(i)-fact*b(k); 15 end 16 end 17 x(n)=b(n)/a(n,n); 18 for i=n-1: -1:1 19 s=b(i); 20 for j=i+1:n 21 s=s-a(i,j)*x(j) 22 end 23 x(i)=b(i)/a(i,i); 24 end 25 disp(x(1),”x1=”) 26 disp(x(2),”x2=”) 27 disp(x(3),”x3=”) 62
Scilab code Exa 9.6 ill conditioned systems 1 clc; 2 clear; 3 a11 =1; 4 a12 =2; 5 b1=10; 6 a21 =1.1; 7 a22 =2; 8 b2 =10.4; 9 x1=(b1*a22 -a12*b2)/(a11*a22 -a12*a21); 10 x2=(b2*a11 -a21*b1)/(a11*a22 -a12*a21); 11 disp(”For the o r i g i n a l system : ”) 12 disp(x1 ,”x1=”) 13 disp(x2 ,”x2=”) 14 a21 =1.05; 15 x1=(b1*a22 -a12*b2)/(a11*a22 -a12*a21); 16 x2=(b2*a11 -a21*b1)/(a11*a22 -a12*a21); 17 disp(”For the new system : ”) 18 disp(x1 ,”x1=”) 19 disp(x2 ,”x2=”) Scilab code Exa 9.7 Effect of Scale on Determinant 1 clc; 2 clear; 3 // part a 4 a=[3 2;-1 2]; 5 b1=18; 6 b2=2; 7 disp(determ(a),”The determinant f o r part ( a )=”) 8 // part b 63
9 a=[1 2;1.1 2]; 10 b1 =10; 11 b2 =10.4; 12 disp(determ(a),”The determinant f o r part ( b )=”) 13 // part c 14 a1=a*10; 15 b1 =100; 16 b2 =104; 17 disp(determ(a1),”The determinant f o r part ( c )=”) Scilab code Exa 9.8 Scaling 1 clc; 2 clear; 3 // part a 4 a=[1 0.667; -0.5 1]; 5 b1=6; 6 b2=1; 7 disp(determ(a),”The determinant f o r part ( a )=”) 8 // part b 9 a=[0.5 1;0.55 1]; 10 b1=5; 11 b2 =5.2; 12 disp(determ(a),”The determinant f o r part ( b )=”) 13 // part c 14 b1=5; 15 b2 =5.2; 16 disp(determ(a),”The determinant f o r part ( c )=”) Scilab code Exa 9.9 Partial Pivoting 1 // c l c ( ) 2 // 0.0003∗ x1 + 3∗ x2 = 2.0001 64
3 // 1∗ x1 + 1∗ x2 = 1 4 a11 = 0.000; 5 // m u l t i p l y i n g f i r s t equation by 1/0.0003 , we get , x1 + 10000∗ x2 = 6667 6 x2 = (6667 -1) /(10000 -1); 7 x1 = 6667 - 10000* x2; 8 disp(x2 ,”x2 = ”) 9 disp(x1 ,”x1 = ”) 10 disp(”The e r r o r v a r i e s depending on the no . of s i g n i f i c a n t f i g u r e s used ”) Scilab code Exa 9.10 Effect of scaling on Pivoting and round off 1 // c l c ( ) 2 // 2∗ x1 + 10000∗ x2 = 10000 3 // x1 + x2 = 2 4 x1 = 1; 5 x2 = 1; 6 disp(” without s c a l i n g , applying forward e l i m i n a t i o n ” ) 7 // x1 i s too small and can be n e g l e c t e d 8 x21 = 10000/10000; 9 x11 = 0; 10 e1 = (x1 - x11)*100/ x1; 11 disp(x21 ,”x2 = ”) 12 disp(x11 ,”x1 = ”) 13 disp(e1 ,” e r r o r f o r x1 = ”) 14 disp(” with s c a l i n g ”) 15 // 0.00002∗ x1 + x2 = 1 16 //now x1 i s n e g l e c t e d because of the co e f f i c i e n t 17 x22 = 1; 18 x12 = 2 - x1; 19 disp(x12 ,”x1 = ”) 20 disp(x22 ,”x2 = ”) 21 // using o r i g i n a l co e f f i c i e n t 65
22 // x1 can be n e g l e c t e d 23 disp(” pivot and r e t a i n i n g o r i g i n a l c o e f f i c i e n t s ”) 24 x22 = 10000/10000; 25 x12 = 2 - x1; 26 disp(x12 ,”x1 = ”) 27 disp(x22 ,”x2 = ”) Scilab code Exa 9.11 Solution of Linear Algebraic Equations 1 clc; 2 clear; 3 a=[70 1 0;60 -1 1;40 0 -1]; 4 b=[636;518;307]; 5 x=abs(linsolve(a,b)); 6 disp(”m/ s ˆ2”,x(1,1),”a=”) 7 disp(”N”,x(2,1),”T=”) 8 disp(”N”,x(3,1),”R=”) Scilab code Exa 9.12 Gauss Jordan method 1 // c l c ( ) 2 // 3∗ x1 − 0.1∗ x2 − 0.2∗ x3 = 7.85 3 // 0.1∗ x1 + 7∗ x2 − 0.3∗ x3 = −19.3 4 // 0.3∗ x1 − 0.2∗ x2 + 10∗ x3 = 71.4 5 // t h i s can be w r i t t e n in matrix form as 6 A = [3 , -0.1 , -0.2 ,7.85;0.1 ,7 , -0.3 , -19.3;0.3 , -0.2 ,10 ,71.4]; 7 disp(A,” Equation in matrix form can be w r i t t e n as ”) 8 X = A(1,:) / det(A(1,1)); 9 Y = A(2,:) - 0.1*X; 10 Z = A(3,:) - 0.3*X; 11 Y = Y/det(Y(1,2)); 66
12 X = X - Y * det(X(1,2)); 13 Z = Z - Y * det(Z(1,2)); 14 Z = Z/det(Z(1,3)); 15 X = X - Z*det(X(1,3)); 16 Y = Y - Z*det(Y(1,3)); 17 A = [X;Y;Z]; 18 disp(A,” f i n a l matrix = ”) 19 disp(det(A(1,4)),”x1 = ”) 20 disp(det(A(2,4)),”x2 = ”) 21 disp(det(A(3,4)),”x3 = ”) 67
Chapter 10 LU Decomposition and matrix inverse Scilab code Exa 10.1 LU decomposition with gauss elimination 1 // c l c ( ) 2 A = [3 , -0.1 , -0.2;0.1 ,7 , -0.3;0.3 , -0.2 ,10]; 3 U = A; 4 disp(A,”A =”) 5 m = det(U(1,1)); 6 n = det(U(2,1)); 7 a = n/m; 8 U(2,:) = U(2,:) - U(1,:) / (m/n); 9 n = det(U(3,1)); 10 b = n/m; 11 U(3,:) = U(3,:) - U(1,:) / (m/n); 12 m = det(U(2,2)); 13 n = det(U(3,2)); 14 c = n/m; 15 U(3,:) = U(3,:) - U(2,:) / (m/n); 16 disp(U,”U = ”) 17 L = [1,0,0;a,1,0;b,c ,1]; 18 disp(L,”L c a l c u l a t e d based on gauss e l i m i n a t i o n method = ”) 68
Scilab code Exa 10.2 The substitution steps 1 // c l c ( ) 2 A = [3 , -0.1 , -0.2;0.1 ,7 , -0.3;0.3 , -0.2 ,10]; 3 B = [7.85; -19.3;71.4]; 4 X = inv(A) * B; 5 disp(X,”X = ”) Scilab code Exa 10.3 Matrix inversion 1 // c l c ( ) 2 A = [3 , -0.1 , -0.2;0.1 ,7 , -0.3;0.3 , -0.2 ,10]; 3 //B = inv (A) 4 L = [1 ,0 ,0;0.033333 ,1 ,0;0.1 , -0.02713 ,1]; 5 U = [3 , -0.1 , -0.2;0 ,7.0033 , -0.293333;0 ,0 ,10.012]; 6 for i =1:3 7 if i==1 then 8 m = [1;0;0]; 9 else 10 if i==2 then 11 m = [0;1;0]; 12 else 13 m = [0;0;1]; 14 end 15 end 16 d = inv(L) * m; 17 x = inv(U) * d; 18 B(:,i) = x 19 end 20 disp(B) 69
Scilab code Exa 10.4 Matrix condition evaluation 1 // c l c ( ) 2 A = [1 ,1/2 ,1/3;1/2 ,1/3 ,1/4;1/3 ,1/4 ,1/5]; 3 n = det(A(2,1)); 4 A(2,:) = A(2,:)/n; 5 n = det(A(3,1)); 6 A(3,:) = A(3,:)/n; 7 B = inv(A); 8 disp(A,”A = ”) 9 for j = 1:3 10 a = 0; 11 for i = 1:3 12 m(i) = det(A(j,i)); 13 su(j) = a + m(i); 14 a = su(j); 15 end 16 end 17 if su(1) < su(2) then 18 if su(2) < su(3) then 19 z = su(3); 20 else 21 z = su(2); 22 end 23 else 24 if su(1) < su(3) then 25 z = su(3); 26 else 27 z = su(1); 28 end 29 end 30 for j = 1:3 31 a = 0; 32 for i = 1:3 70
33 m(i) = det(B(j,i)); 34 sm(j) = a + abs(m(i)); 35 a = sm(j); 36 end 37 end 38 if sm(1) < sm(2) then 39 if sm(2) < sm(3) then 40 y = sm(3); 41 else 42 y = sm(2); 43 end 44 else 45 if sm(1) < sm(3) then 46 y = sm(3); 47 else 48 y = sm(1); 49 end 50 end 51 C = z*y; 52 disp(C,” Condition number f o r the matrix =”) 71
Chapter 11 Special Matrices and gauss seidel Scilab code Exa 11.1 Tridiagonal solution with Thomas algorithm 1 // c l c ( ) 2 A = [2.04 , -1 ,0 ,0; -1 ,2.04 , -1 ,0;0 ,-1,2.04 ,-1;0 ,0, -1 ,2.04]; 3 B = [40.8;0.8;0.8;200.8]; 4 g = det(A(1,2)); 5 f1 =det(A(1,1)); 6 e2 = det(A(2,1))/f1; 7 f2 = det(A(1,1)) - e2 * det(A(2,1)); 8 e3 = det(A(2,1))/f2; 9 f3 = det(A(1,1)) - e3 * det(A(2,1)); 10 e4 = det(A(2,1))/f3; 11 f4 = det(A(1,1)) - e4 * det(A(2,1)); 12 M = [f1 ,g,0,0;e2 ,f2 ,g,0;0,e3 ,f3 ,g;0,0,e4 ,f4]; 13 L = [1,0,0,0;det(M(2,1)) ,1,0,0;0,det(M(3,2)) ,1,0;0,0,det(M(4,3)) ,1]; 14 U = [det(M(1,1)),g,0,0;0,det(M(2,2)),g,0;0,0,det(M (3,3)),g;0,0,0,det(M(4,4))]; 15 r1 = det(B(1,1)); 72
16 r2 = det(B(2,1)) - e2*det(B(1,1)); 17 r3 = det(B(3,1)) - e3*r2; 18 r4= det(B(4,1)) - e4*r3; 19 N = [r1;r2;r3;r4]; 20 T4 = r4/det(U(4,4)); 21 T3 = (r3 - g*T4)/det(U(3,3)); 22 T2 = (r2 - g*T3)/det(U(2,2)); 23 T1 = (r1 - g*T2)/det(U(1,1)); 24 disp(T1 ,”T1 = ”) 25 disp(T2 ,”T2 = ”) 26 disp(T3 ,”T3 = ”) 27 disp(T4 ,”T4 = ”) Scilab code Exa 11.2 Cholesky Decomposition 1 // c l c ( ) 2 A = [6 ,15 ,55;15 ,55 ,225;55 ,225 ,979]; 3 sl = 0 4 l11 = (det(A(1,1)))^(1/2); 5 // f o r second row 6 l21 = (det(A(2,1)))/l11; 7 l22 = (det(A(2,2)) - l21 ^2) ^(0.5); 8 // f o r t h i r d row 9 l31 = (det(A(3,1)))/l11; 10 l32 = (det(A(3,2)) - l21*l31)/l22; 11 l33 = (det(A(3,3)) - l31^2 - l32 ^2) ^(0.5); 12 L = [l11 ,0,0;l21 ,l22 ,0;l31 ,l32 ,l33]; 13 disp(L,”L = ”) Scilab code Exa 11.3 Gauss Seidel method 1 // c l c ( ) 2 //3x − 0. 1 y − 0. 2 z = 7.85 73
3 // 0. 1 x + 7y − 0. 3 z = −19.3 4 // 0. 3 x − 0. 2 y + 10 z = 71.4 5 Y = 0; 6 Z = 0; 7 for i = 1:2 8 x(i) = (7.85 +0.1*Y+0.2*Z)/3; 9 X = x(i); 10 y(i) = ( -19.3 - 0.1*X +0.3*Z)/7; 11 Y = y(i); 12 z(i) = (71.4 - 0.3*X+0.2*Y)/10; 13 Z = z(i); 14 if i==2 then 15 ex = (x(i) - x(i-1))*100/x(i); 16 ey = (y(i) - y(i-1))*100/y(i); 17 ez = (z(i) - z(i-1))*100/z(i); 18 end 19 end 20 disp(x(1:2) ,”x through two i t e r a t i o n s =”) 21 disp(y(1:2) ,”y through two i t e r a t i o n s =”) 22 disp(z(1:2) ,” z through two i t e r a t i o n s =”) 23 disp(”%”,ex ,” e r r o r of x = ”) 24 disp(”%”,ey ,” e r r o r of y = ”) 25 disp(”%”,ez ,” e r r o r of z = ”) Scilab code Exa 11.4 Linear systems 1 // c l c ( ) 2 A = [1 ,0.5 ,1/3;1 ,2/3 ,1/2;1 ,3/4 ,3/5]; 3 B = [1.833333;2.166667;2.35]; 4 U = inv(A); 5 X = U*B; 6 x = det(X(1,1)); 7 y = det(X(2,1)); 8 z = det(X(3,1)); 9 disp(x,”x = ”) 74
10 disp(y,”y = ”) 11 disp(z,” z = ”) Scilab code Exa 11.5 Manipulate linear algebraic equations 1 // c l c ( ) 2 A = [1 ,0.5 ,1/3;1 ,2/3 ,1/2;1 ,3/4 ,3/5]; 3 B = [1.833333;2.166667;2.35]; 4 U = inv(A); 5 X = U*B; 6 x = det(X(1,1)); 7 y = det(X(2,1)); 8 z = det(X(3,1)); 9 disp(x,”x = ”) 10 disp(y,”y = ”) 11 disp(z,” z = ”) Scilab code Exa 11.6 Analyze and solve Hilbert matrix 1 // c l c ( ) 2 A = [1 ,0.5 ,1/3;1/2 ,1/3 ,1/4;1/3 ,1/4 ,1/5]; 3 B = [1.833333;1.083333;0.783333]; 4 U = inv(A); 5 X = U*B; 6 x = det(X(1,1)); 7 y = det(X(2,1)); 8 z = det(X(3,1)); 9 disp(x,”x = ”) 10 disp(y,”y = ”) 11 disp(z,” z = ”) 75
Chapter 13 One dimensional unconstrained optimization Scilab code Exa 13.1 Golden section method 1 // c l c ( ) 2 // f ( x ) = 2 sin x − x ˆ2/10 3 xl(1) = 0; 4 xu(1) = 4; 5 for i = 1:10 6 d(i) = ((5) ^(0.5) - 1)*(xu(i) - xl(i))/2; 7 x1(i) = xl(i) + d(i); 8 x2(i) = xu(i) - d(i); 9 m(i) = 2*sin(x1(i)) - (x1(i)^2) /10; 10 n(i) = 2*sin(x2(i)) - (x2(i)^2) /10; 11 if n(i) > m(i) then 12 xu(i+1) = x1(i); 13 xl(i+1) = xl(i); 14 else 15 xl(i+1) = x2(i); 16 xu(i+1) = xu(i); 17 end 18 end 19 disp(xl ,” xl = ”) 76
20 disp(x2 ,”x2 = ”) 21 disp(x1 ,”x1 = ”) 22 disp(xu ,”xu = ”) Scilab code Exa 13.2 Quadratic interpolation 1 // c l c ( ) 2 // f ( x ) = 2 sin x − x ˆ2/10 3 x0(1) = 0; 4 x1(1) = 1 5 x2(1) = 4; 6 for i = 1:6 7 m(i) = 2*sin(x0(i)) - (x0(i)^2) /10; 8 n(i) = 2*sin(x1(i)) - (x1(i)^2) /10; 9 r(i) = 2*sin(x2(i)) - (x2(i)^2) /10; 10 x3(i) = ((m(i)*(x1(i) ^ 2 -x2(i) ^ 2)) + (n(i)*( x2(i) ^ 2 -x0(i) ^ 2)) + (r(i)*(x0(i) ^ 2 -x1 (i) ^ 2)))/((2*m(i)*(x1(i) -x2(i)))+(2*n(i)*( x2(i) -x0(i)))+(2*r(i)*(x0(i) -x1(i)))); 11 s(i) = 2*sin(x3(i)) - (x3(i)^2) /10; 12 if x1(i) > x3(i) then 13 if n(i)<s(i) then 14 x0(i+1) = x0(i); 15 x1(i+1) = x3(i); 16 x2(i+1) = x1(i); 17 else 18 x0(i+1) = x1(i); 19 x1(i+1) = x3(i); 20 x2(i+1) = x2(i); 21 end 22 else 23 if n(i)>s(i) then 24 x0(i+1) = x0(i); 25 x1(i+1) = x3(i); 26 x2(i+1) = x1(i); 77
27 else 28 x0(i+1) = x1(i); 29 x1(i+1) = x3(i); 30 x2(i+1) = x2(i); 31 end 32 end 33 end 34 disp(x0 (1:6) ,”x0 = ”) 35 disp(x1 (1:6) ,”x1 = ”) 36 disp(x3 (1:6) ,”x3 = ”) 37 disp(x2 (1:6) ,”x2 = ”) Scilab code Exa 13.3 Newtons method 1 // c l c ( ) 2 // f ( x ) = 2 sin x − x ˆ2/10 3 x(1) = 2.5; 4 // f ’ ( x ) = 2 cosx − x/5 5 // f ”( x ) = −2sin x − 1/5 6 for i = 2:10 7 x(i) = x(i-1) - (2* cos(x(i-1)) - x(i-1) /5) /(-2* sin (x(i-1)) - 1/5); 8 end 9 disp(x,”x = ”) 78
Chapter 14 Multidimensional Unconstrainted Optimization Scilab code Exa 14.1 Random Search Method 1 clc; 2 clear; 3 function z=f(x,y) 4 z=y-x -(2*(x^2)) -(2*x*y) -(y^2); 5 endfunction 6 x1=-2; 7 x2=2; 8 y1=1; 9 y2=3; 10 fmax = -1*10^( -15); 11 n=10000; 12 for j=1:n 13 r=rand (1,2); 14 x=x1+(x2 -x1)*r(1,1); 15 y=y1+(y2 -y1)*r(1,2); 16 fn=f(x,y); 17 if fn >fmax then 18 fmax=fn; 19 xmax=x; 79
20 ymax=y; 21 end 22 if modulo(j ,1000) ==0 then 23 24 disp(j,” I t e r a t i o n : ”) 25 disp(x,”x : ”) 26 disp(y,”y : ”) 27 disp(fn ,” f u n c t i o n value : ”) 28 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 29 end 30 end Scilab code Exa 14.2 Path of Steepest Descent 1 clc; 2 clear; 3 function z=f(x,y) 4 z=x*y*y 5 endfunction 6 p1=[2 2]; 7 elevation=f(p1(1),p1(2)); 8 dfx=p1(1)*p1(1); 9 dfy =2*p1(1)*p1(2); 10 theta=atan(dfy/dfx); 11 slope =(dfx^2 + dfy ^2) ^0.5; 12 disp(elevation ,” Elevation : ”) 13 disp(theta ,” Theta : ”) 14 disp(slope ,” s l o p e : ”) Scilab code Exa 14.3 1 D function along Gradient 1 clc; 80
2 clear; 3 function z=f(x,y) 4 z=2*x*y + 2*x - x^2 - 2*y^2 5 endfunction 6 x=-1; 7 y=1; 8 dfx =2*y+2-2*x; 9 dfy =2*x-4*y; 10 // the f u n c t i o n can thus be expressed along h a x i s as 11 // f ( ( x+dfx ∗h ) ,( y+dfy ∗h ) ) 12 disp(” 180∗hˆ2 + 72∗h − 7”,”The f i n a l equation i s=”) Scilab code Exa 14.4 Optimal Steepest Descent 1 clc; 2 clear; 3 function z=f(x,y) 4 z=2*x*y + 2*x - x^2 - 2*y^2 5 endfunction 6 x=-1; 7 y=1; 8 d2fx =-2; 9 d2fy =-4; 10 d2fxy =2; 11 12 modH=d2fx*d2fy -( d2fxy)^2; 13 14 for i=1:25 15 dfx =2*y+2-2*x; 16 dfy =2*x - 4*y; 17 // the f u n c t i o n can thus be expressed along h a x i s as 18 // f ( ( x+dfx ∗h ) ,( y+dfy ∗h ) ) 19 function d=g(h) 20 d=2*(x+dfx*h)*(y+dfy*h) + 2*(x+dfx*h) - (x+dfx*h )^2 - 2*(y+dfy*h)^2 81
21 endfunction 22 // 2∗ dfx ∗( y+dfy ∗h )+2∗dfy ∗( x+dfx ∗h )+2∗dfx −2∗(x+dfx ∗h ) ∗ dfx −4∗(y+dfy ∗h ) ∗ dfy=g ’ ( h )=0 23 // 2∗ dfx ∗y + 2∗ dfx ∗ dfy ∗h + 2∗ dfy ∗x + 2∗ dfy ∗ dfx ∗h + 2∗ dfx − 2∗x∗ dfx − 2∗ dfx ∗ dfx ∗h − 4∗y∗ dfy − 4∗ dfy ∗ dfy ∗h=0 24 //h (2∗ dfx ∗ dfy+2∗dfy ∗ dfx −2∗dfx ∗ dfx −4∗dfy ∗ dfy )=−(2∗dfx ∗y+2∗dfy ∗x−2∗x∗ dfx −4∗y∗ dfy ) 25 h=(2* dfx*y+2* dfy*x-2*x*dfx -4*y*dfy +2* dfx)/( -1*(2* dfx *dfy +2* dfy*dfx -2* dfx*dfx -4* dfy*dfy)); 26 x=x+dfx*h; 27 y=y+dfy*h; 28 end 29 disp ([x y],”The f i n a l v a l u e s are : ”) 82
Chapter 15 Constrained Optimization Scilab code Exa 15.1 Setting up LP problem 1 clc; 2 clear; 3 regular =[7 10 9 150]; 4 premium =[11 8 6 175]; 5 res_avail =[77 80]; 6 // t o t a l p r o f i t ( to be maximized )=z =150∗x1+175∗x2 7 // t o t a l gas used=7∗x1+11∗x2 ( has to be l e s s than 77 mˆ3/ week ) 8 // s i m i l a r l y other c o n s t r a i n t s are developed 9 disp(” Maximize z=150∗x1+175∗x2”) 10 disp(” s u b j e c t to ”) 11 disp(”7∗ x1+11∗x2<=77 ( Material c o n s t r a i n t ) ”) 12 disp(” 10∗ x1+8∗x2<=80 ( Time c o n s t r a i n t ) ”) 13 disp(”x1<=9 ( Regular s t o r a g e c o n s t r a i n t ) ”) 14 disp(”x2<=6 ( Premium s t o r a g e c o n s t r a i n t ) ”) 15 disp(”x1 , x2>=0 ( P o s i t i v i t y c o n s t r a i n t ) ”) 83
Figure 15.1: Graphical Solution 84
Scilab code Exa 15.2 Graphical Solution 1 clc; 2 clear; 3 for x1 =0:8 4 x21(x1+1) = -(7/11)*x1+7; 5 x22(x1+1) =(80 -10* x1)/8; 6 x23(x1+1) =6; 7 x24(x1+1) = -150*x1 /175; 8 x25(x1+1) =(600 -150* x1)/175; 9 x26(x1+1) =(1400 -150* x1)/175; 10 end 11 x1 =0:8; 12 13 plot(x1 ,x24 , ’ o− ’ ) 14 plot(x1 ,x25 , ’ .− ’ ) 15 plot(x1 ,x26 , ’x− ’ ) 16 h1=legend ([ ’Z=0 ’ ; ’Z=600 ’ ; ’Z=1400 ’ ]) 17 plot(x1 ,x21); 18 plot(x1 ,x22); 19 plot(x1 ,x23); 20 xtitle( ’ x2 vs x1 ’ , ’ x1 ( tonnes ) ’ , ’ x2 ( tonnes ) ’ ) Scilab code Exa 15.3 Linear Programming Problem 1 clc; 2 clear; 3 x1 =[4.888889 3.888889]; 4 x2=[7 11]; 5 x3=[10 8]; 6 x4 =[150 175]; 7 x5=[77 80 9 6]; 8 profit =[x1(1)*x4(1) x1(2)*x4(2)]; 9 total =[x1(1)*x3(1)+x1(2)*x3(2) x1(1)*x3(1)+x1(2)*x3 (2) x1(1) x1(2) profit (1)+profit (2)]; 85
10 e=1000; 11 12 while e>total (5) 13 if total (1) <=x5(1) then 14 if total (2) <=x5(2) then 15 if total (3) <=x5(3) then 16 if total (4) <=x5(4) then 17 l=1; 18 end 19 end 20 end 21 end 22 if l==1 then 23 x1(1)=x1(1) +4.888889; 24 x1(2)=x1(2) +3.888889; 25 profit =[x1(1)*x4(1) x1(2)*x4(2)]; 26 total (5)=profit (1)+profit (2); 27 end 28 end 29 disp(total (5),”The maximized p r o f i t i s=”) Scilab code Exa 15.4 Nonlinear constrained optimization 1 clc; 2 clear; 3 Mt =2000; // kg 4 g=9.8; //m/ s ˆ2 5 c0 =200; //$ 6 c1=56; //$/m 7 c2 =0.1; //$/mˆ2 8 vc=20; //m/ s 9 kc=3; // kg /( s ∗mˆ2) 10 z0 =500; //m 11 t=27; 12 r=2.943652; 86
13 n=6; 14 A=2* %pi*r*r; 15 l=(2^0.5)*r; 16 c=3*A; 17 m=Mt/n; 18 function y=f(t) 19 y=(z0+g*m*m/(c*c)*(1-exp(-c*t/m)))*c/(g*m); 20 endfunction 21 22 while abs(f(t)-t) >0.00001 23 t=t+0.00001; 24 end 25 v=g*m*(1-exp(-c*t/m))/c; 26 disp(v,”The f i n a l value of v e l o c i t y=”) 27 disp(n,”The f i n a l no . of load p a r c e l s=”) 28 disp(”m”,r,”The chute r a d i u s=”) 29 disp ((c0+c1*l+c2*A*A)*n,”The minimum c o s t ( $ )=”) Scilab code Exa 15.5 One dimensional Optimization 1 clc; 2 clear; 3 function y=fx(x) 4 y=-(2* sin(x))+x^2/10 5 endfunction 6 x=fminsearch(fx ,0) 7 disp(” After maximization : ”) 8 disp(x,”x=”) 9 disp(fx(x),” f ( x )=”) Scilab code Exa 15.6 Multidimensional Optimization 1 clc; 87
2 clear; 3 function f=fx(x) 4 f=-(2*x(1)*x(2) +2*x(1)-x(1)^2-2*x(2) ^2) 5 endfunction 6 x=fminsearch(fx ,[-1 1]) 7 disp(” After maximization : ”) 8 disp(x,”x=”) 9 disp(fx(x),” f ( x )=”) Scilab code Exa 15.7 Locate Single Optimum 1 clc; 2 clear; 3 function y=fx(x) 4 y=-(2* sin(x)-x^2/10) 5 endfunction 6 x=fminsearch(fx ,0) 7 disp(” After maximization : ”) 8 disp(x,”x=”) 9 disp(fx(x),” f ( x )=”) 88
Chapter 17 Least squares regression Scilab code Exa 17.1 Linear regression 1 // c l c ( ) 2 x = [1,2,3,4,5,6,7]; 3 y = [0.5 ,2.5 ,2 ,4 ,3.5 ,6 ,5.5]; 4 n = 7; 5 s = 0; 6 xsq = 0; 7 xsum = 0; 8 ysum = 0; 9 for i = 1:7 10 s = s + (det(x(1,i)))*(det(y(1,i))); 11 xsq = xsq + (det(x(1,i))^2); 12 xsum = xsum + det(x(1,i)); 13 ysum = ysum + det(y(1,i)); 14 end 15 disp(s,”sum of product of x and y =”) 16 disp(xsq ,”sum of square of x = ”) 17 disp(xsum ,”sum of a l l the x = ”) 18 disp(ysum ,”sum of a l l the y = ”) 19 a = xsum/n; 20 b = ysum/n; 21 a1 = (n*s - xsum*ysum)/(n*xsq -xsum ^2); 89
22 a0 = b - a*a1; 23 disp(a1 ,”a1 = ”) 24 disp(a0 ,”a0 = ”) 25 disp(”The equation of the l i n e obtained i s y = a0 + a1∗x”) Scilab code Exa 17.2 Estimation of errors for the linear least square fit 1 // c l c ( ) 2 x = [1,2,3,4,5,6,7]; 3 y = [0.5 ,2.5 ,2 ,4 ,3.5 ,6 ,5.5]; 4 n = 7; 5 s = 0; 6 ssum = 0; 7 xsq = 0; 8 xsum = 0; 9 ysum = 0; 10 msum = 0; 11 for i = 1:7 12 s = s + (det(x(1,i)))*(det(y(1,i))); 13 xsq = xsq + (det(x(1,i))^2); 14 xsum = xsum + det(x(1,i)); 15 ysum = ysum + det(y(1,i)); 16 end 17 a = xsum/n; 18 b = ysum/n; 19 a1 = (n*s - xsum*ysum)/(n*xsq -xsum ^2); 20 a0 = b - a*a1; 21 for i = 1:7 22 m(i) = (det(y(1,i)) - ysum /7) ^2; 23 msum = msum +m(i); 24 si(i) = (det(y(1,i)) - a0 - a1*det(x(1,i)))^2; 25 ssum = ssum + si(i); 26 end 27 disp(ysum ,”sum of a l l y =”) 90
28 disp(m,” ( yi − yavg ) ˆ2 = ”) 29 disp(msum ,” t o t a l ( yi − yavg ) ˆ2 = ”) 30 disp(si ,” ( yi − a0 − a1∗x ) ˆ2 = ”) 31 disp(ssum ,” t o t a l ( yi − a0 − a1∗x ) ˆ2 = ”) 32 sy = (msum / (n-1))^(0.5); 33 sxy = (ssum /(n-2))^(0.5); 34 disp(sy ,” sy = ”) 35 disp(sxy ,” sxy = ”) 36 r2 = (msum - ssum)/( msum); 37 r = r2 ^0.5; 38 disp(r,” r = ”) 39 disp(”The r e s u l t i n d i c a t e that 86.8 percent of the o r i g i n a l u n c e r t a i n t y has been e xp la in ed by l i n e a r model ”) Scilab code Exa 17.3.a linear regression using computer 1 // c l c ( ) 2 s = [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]; 3 v = [10 ,16.3 ,23 ,27.5 ,31 ,35.6 ,39 ,41.5 ,42.9 ,45 ,46 ,45.5 ,46 ,49 ,50]; 4 g = 9.8 //m/ s ˆ2 5 m = 68.1; // kg 6 c = 12.5 // kg/ s 7 for i = 1:15 8 v1(i) = g*m*(1 - exp(-c*s(i)/m))/c; 9 v2(i) = g*m*s(i)/(c*(3.75+s(i))); 10 end 11 disp(s,” time = ”) 12 disp(v,” measured v =”) 13 disp(v1 ,” using equation ( 1 . 1 0 ) v1 = ”) 14 disp(v2 ,” using equation ( ( 1 7 . 3 ) ) v2 = ”) 15 plot2d(v,v1); 16 xtitle( ’ v vs v1 ’ , ’ v ’ , ’ v1 ’ ); 91
Figure 17.1: linear regression using computer Scilab code Exa 17.3.b linear regression using computer 1 // c l c ( ) 2 s = [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]; 3 v = [10 ,16.3 ,23 ,27.5 ,31 ,35.6 ,39 ,41.5 ,42.9 ,45 ,46 ,45.5 ,46 ,49 ,50]; 4 g = 9.8 //m/ s ˆ2 92
Figure 17.2: linear regression using computer 93
5 m = 68.1; // kg 6 c = 12.5 // kg/ s 7 for i = 1:15 8 v1(i) = g*m*(1 - exp(-c*s(i)/m))/c; 9 v2(i) = g*m*s(i)/(c*(3.75+s(i))); 10 end 11 disp(s,” time = ”) 12 disp(v,” measured v =”) 13 disp(v1 ,” using equation ( 1 . 1 0 ) v1 = ”) 14 disp(v2 ,” using equation ( ( 1 7 . 3 ) ) v2 = ”) 15 plot2d(v,v2); 16 xtitle( ’ v vs v2 ’ , ’ v ’ , ’ v2 ’ ); Scilab code Exa 17.4 Linearization of a power function 1 // c l c ( ) 2 //y = a∗xˆb 3 a1 = -0.3000; 4 a = 10^( a1); 5 b = 1.75; 6 disp(a) 7 for i=1:5 8 x(i) = i; 9 y(i) = a*x(i)^b; 10 m(i) = log10(x(i)); 11 n(i) = log10(y(i)); 12 end 13 disp(x(1:5) ,”x = ”) 14 disp(y(1:5) ,”y = ”) 15 disp(m(1:5) ,”m = ”) 16 disp(n(1:5) ,”n = ”) 17 plot2d(x(1:5) ,y(1:5)) 18 zoom_rect ([0 ,0 ,7 ,7]) 19 xtitle( ’ y vs x ’ , ’ x ’ , ’ y ’ ) 20 plot2d(m(1:5) ,n(1:5)) 94
Figure 17.3: Linearization of a power function 21 zoom_rect ([0,-1,1,1]) 22 xtitle( ’ l og y vs lo g x ’ , ’ l og x ’ , ’ log y ’ ) Scilab code Exa 17.5 polynomial regression 1 // c l c ( ) 2 x = [0,1,2,3,4,5]; 95
Figure 17.4: Linearization of a power function 96
Figure 17.5: polynomial regression 97
3 y = [2.1 ,7.7 ,13.6 ,27.2 ,40.9 ,61.1]; 4 sumy = 0; 5 sumx = 0; 6 m = 2; 7 n = 6; 8 s = 1.12; 9 xsqsum = 0; 10 xcsum = 0; 11 x4sum = 0; 12 xysum = 0; 13 x2ysum = 0; 14 rsum = 0; 15 usum = 0; 16 for i=1:6 17 sumy = sumy+y(i); 18 sumx = sumx+x(i); 19 r(i) = (y(i) - s/n)^2; 20 xsqsum = xsqsum + x(i)^2; 21 xcsum = xcsum +x(i)^3; 22 x4sum = x4sum + x(i)^4; 23 xysum = xysum + x(i)*y(i); 24 x2ysum = x2ysum + y(i)*x(i)^2; 25 rsum = r(i) + rsum; 26 end 27 disp(sumy ,”sum y =”) 28 disp(sumx ,”sum x =”) 29 xavg = sumx/n; 30 yavg = sumy/n; 31 disp(xavg ,” xavg = ”) 32 disp(yavg ,” yavg = ”) 33 disp(xsqsum ,”sum xˆ2 =”) 34 disp(xcsum ,”sum xˆ3 =”) 35 disp(x4sum ,”sum xˆ4 =”) 36 disp(xysum ,”sum x∗y =”) 37 disp(x2ysum ,”sum xˆ2 ∗ y =”) 38 J = [n,sumx ,xsqsum;sumx ,xsqsum ,xcsum;xsqsum ,xcsum , x4sum ]; 39 I = [sumy;xysum;x2ysum ]; 98
40 X = inv(J) * I; 41 a0 = det(X(1,1)); 42 a1 = det(X(2,1)); 43 a2 = det(X(3,1)); 44 for i=1:6 45 u(i) = (y(i) - a0 - a1*x(i) - a2*x(i)^2) ^2; 46 usum = usum + u(i); 47 end 48 disp(r,” ( yi − yavg ) ˆ2 = ”) 49 disp(u,” ( yi − a0 − a1∗x − a2∗x ˆ2) ˆ2 = ”) 50 plot(x,y); 51 xtitle( ’ x vs y ’ , ’ x ’ , ’ y ’ ); 52 syx = (usum /(n-3))^0,5; 53 disp(syx ,”The standard e r r o r of the estimate based on r e g r e s s i o n polynomial =”) 54 R2 = (rsum - usum)/( rsum); 55 disp(”%”,R2*100,” Percentage of o r i g i n a l u n c e r t a i n t y that has been e xp la in ed by the model = ”) Scilab code Exa 17.6 Multiple linear regression 1 // c l c ( ) 2 x1 = [0,2,2.5,1,4,7]; 3 x2 = [0,1,2,3,6,2]; 4 x1sum = 0; 5 x2sum = 0; 6 ysum = 0; 7 x12sum = 0; 8 x22sum = 0; 9 x1ysum = 0; 10 x2ysum = 0; 11 x1x2sum = 0; 12 n = 6; 13 for i=1:6 14 y(i) = 5 + 4*x1(i) - 3*x2(i); 99
15 x12(i) = x1(i)^2; 16 x22(i) = x2(i)^2; 17 x1x2(i) = x1(i) * x2(i); 18 x1y(i) = x1(i) * y(i); 19 x2y(i) = x2(i) * y(i); 20 x1sum = x1sum + x1(i); 21 x2sum = x2sum + x2(i); 22 ysum = ysum + y(i); 23 x1ysum = x1ysum + x1y(i); 24 x2ysum = x2ysum + x2y(i); 25 x1x2sum = x1x2sum + x1x2(i); 26 x12sum = x12sum + x12(i); 27 x22sum = x22sum + x22(i); 28 end 29 X = [n,x1sum ,x2sum;x1sum ,x12sum ,x1x2sum;x2sum , x1x2sum ,x22sum ]; 30 Y = [ysum;x1ysum;x2ysum ]; 31 Z = inv(X)*Y; 32 a0 = det(Z(1,1)); 33 a1 = det(Z(2,1)); 34 a2 = det(Z(3,1)); 35 disp(a0 ,”a0 = ”) 36 disp(a1 ,”a1 = ”) 37 disp(a2 ,”a2 = ”) 38 disp(”Thus , y = a0 + a1∗x1 + a2∗x2”) Scilab code Exa 17.7 Confidence interval for linear regression 1 // c l c ( ) 2 //y = −0.859 + 1.032∗ x 3 Z = [1 ,10;1 ,16.3;1 ,23;1 ,27.5;1 ,31;1 ,35.6;1 ,39;1 ,41.5;1 ,42.9;1 ,45;1 ,46; 4 for i = 1:15 5 Y(i) = 9.8*68.1*(1 - exp ( -12.5*i/68.1))/12.5; 100
6 end 7 M = Z’; 8 R = M*Z; 9 S = M*Y; 10 P = inv(R); 11 X = inv(R)*S; 12 a0 = det(X(1,1)); 13 a1 = det(X(2,1)); 14 disp(a0 ,”a0 = ”) 15 disp(a1 ,”a1 = ”) 16 sxy = 0.863403; 17 sa0 = (det(P(1,1)) * sxy ^2) ^0.5; 18 sa1 = (det(P(2,2)) * sxy ^2) ^0.5; 19 disp(sa0 ,” standard e r r o r of co e f f i c i e n t a0 = ”) 20 disp(sa1 ,” standard e r r o r of co e f f i c i e n t a1 = ”) 21 TINV = 2.160368; 22 a0 = [a0 - TINV *(sa0),a0 + TINV *(sa0)]; 23 a1 = [a1 - TINV *(sa1),a1 + TINV *(sa1)]; 24 disp(a0 ,” i n t e r v a l of a0 = ”) 25 disp(a1 ,” i n t e r v a l of a1 = ”) Scilab code Exa 17.8 Gauss Newton method 1 // c l c ( ) 2 x = [0.25 ,0.75 ,1.25 ,1.75 ,2.25]; 3 y = [0.28 ,0.57 ,0.68 ,0.74 ,0.79]; 4 a0 = 1; 5 a1 = 1; 6 sr = 0.0248; 7 for i = 1:5 8 pda0(i) = 1 - exp(-a1 * x(i)); 9 pda1(i) = a0 * x(i)*exp(-a1*x(i)); 10 end 11 Z0 = [pda0 (1),pda1 (1);pda0 (2),pda1 (2);pda0 (3),pda1 (3);pda0 (4),pda1 (4);pda0 (5),pda1 (5)] 101
12 disp(Z0 ,”Z0 = ”) 13 R = Z0 ’*Z0; 14 S = inv(R); 15 for i = 1:5 16 y1(i) = a0 * (1-exp(-a1*x(i))); 17 D(i) = y(i) - y1(i); 18 end 19 disp(D,”D = ”) 20 M = Z0 ’*D; 21 X = S *M; 22 disp(X,”X = ”) 23 a0 = a0 + det(X(1,1)); 24 a1 = a1 + det(X(2,1)); 25 disp(a0 ,”The value of a0 a f t e r 1 s t i t e r a t i o n = ”) 26 disp(a1 ,”The value of a1 a f t e r 1 s t i t e r a t i o n = ”) 102
Chapter 18 Interpolation Scilab code Exa 18.1 Linear interpolation 1 // c l c ( ) 2 // f1 ( x ) = f0 ( x ) +( f ( x1 ) − f ( x0 ) ∗( x − x0 ) / ( x1 − x0 ) 3 x = 2; 4 x0 = 1; 5 x1 = 6; 6 m = 1.791759; 7 n = 0; 8 r = log (2); 9 f = 0 + (m - n) * (x - x0) / (x1 - x0); 10 disp(f,” value of ln2 f o r i n t e r p o l a t i o n r e g i o n 1 to 6 =”) 11 e = (r - f) * 100/r; 12 disp(”%”,e,” e r r o r by i n t e r p o l a t i o n f o r i n t e r v a l [ 1 , 6 ] =”) 13 x2 = 4; 14 p = 1.386294; 15 f = 0 + (p - n) * (x - x0) / (x2 - x0); 16 disp(f,” value of ln2 f o r i n t e r p o l a t i o n r e g i o n 1 to 6 =”) 17 e = (r - f) * 100/r; 18 disp(”%”,e,” e r r o r by i n t e r p o l a t i o n f o r i n t e r v a l [ 1 , 6 ] 103
=”) Scilab code Exa 18.2 Quadratic interpolation 1 // c l c ( ) 2 x = 2; 3 x0 = 1; 4 m = 0; 5 x1 = 4; 6 n = 1.386294; 7 x2 = 6; 8 p = 1.791759; 9 b0 = m; 10 b1 = (n - m)/(x1 - x0); 11 b2 = ((p - n)/(x2 - x1) - (n - m)/(x1 - x0))/(x2 - x0); 12 disp(b0 ,”b0 = ”) 13 disp(b1 ,”b1 = ”) 14 disp(b2 ,”b2 = ”) 15 f = b0 + b1*(x - x0) + b2*(x - x0)*(x - x1); 16 disp(f,” f (2) = ”) 17 r = log (2); 18 e = (r -f)*100/r; 19 disp(”%”,e,” e r r o r = ”) Scilab code Exa 18.3 Newtons divided difference Interpolating polynomials 1 // c l c ( ) 2 x = 2; 3 x0 = 1; 4 m = 0; 5 x1 = 4; 6 n = 1.386294; 104
7 x3 = 5; 8 p = 1.609438; 9 x2 = 6; 10 o = 1.791759; 11 f01 = (m - n)/(x0 - x1); 12 f12 = (n - o)/(x1 - x2); 13 f23 = (p - o)/(x3 - x2); 14 f210 = (f12 - f01)/(x2 - x0); 15 f321 = (f23 - f12)/(x3 - x1); 16 f0123 = (f321 - f210) / (x3 - x0); 17 b0 = m; 18 b1 = f01; 19 b2 = f210; 20 b3 = f0123; 21 disp(b0 ,”b0 = ”) 22 disp(b1 ,”b1 = ”) 23 disp(b2 ,”b2 = ”) 24 disp(b3 ,”b3 = ”) 25 f = b0 + b1*(x - x0) + b2*(x - x0)*(x - x1) + b3*(x - x0)*(x - x1)*(x - x2); 26 disp(f,” f (2) = ”) 27 r = log (2); 28 e = (r -f)*100/r; 29 disp(”%”,e,” e r r o r = ”) Scilab code Exa 18.4 Error estimation for Newtons polynomial 1 // c l c ( ) 2 x = 2; 3 x0 = 1; 4 m = 0; 5 x1 = 4; 6 n = 1.386294; 7 x3 = 5; 8 p = 1.609438; 105
9 x2 = 6; 10 o = 1.791759; 11 f01 = (m - n)/(x0 - x1); 12 f12 = (n - o)/(x1 - x2); 13 f23 = (p - o)/(x3 - x2); 14 f210 = (f12 - f01)/(x2 - x0); 15 f321 = (f23 - f12)/(x3 - x1); 16 f0123 = (f321 - f210) / (x3 - x0); 17 b0 = m; 18 b1 = f01; 19 b2 = f210; 20 b3 = f0123; 21 R2 = b3 * (x - x0) * (x - x1)*(x-x2); 22 disp(R2 ,” e r r o r R2 = ”) Scilab code Exa 18.5 Error Estimates for Order of Interpolation 1 clc; 2 clear; 3 x=[1 4 6 5 3 1.5 2.5 3.5]; 4 y=[0 1.3862944 1.7917595 1.6094379 1.0986123 0.4054641 0.9162907 1.2527630]; 5 n=8; 6 for i=1:n 7 fdd(i,1)=y(i); 8 end 9 for j=2:n 10 for i=1:n-j+1 11 fdd(i,j)=(fdd(i+1,j-1)-fdd(i,j-1))/(x(i+j-1) -x(i)); 12 end 13 end 14 xterm =1; 15 yint (1)=fdd (1,1); 16 106
17 for order =2:n 18 xterm=xterm *(2-x(order -1)); 19 yint2=yint(order -1)+fdd(1,order)*xterm; 20 Ea(order -1)=yint2 -yint(order -1); 21 yint(order)=yint2; 22 23 end 24 disp(yint ,”F( x )=”) 25 disp(Ea ,”Ea=”) Scilab code Exa 18.6 Lagrange interpolating polynomials 1 // c l c ( ) 2 x = 2; 3 x0 = 1; 4 m = 0; 5 x1 = 4; 6 n = 1.386294; 7 x2 = 6; 8 p = 1.791759; 9 f1 = (x - x1)*m/((x0 - x)) + (x- x0) * n/(x1 - x0); 10 disp(f1 ,” f i r s t order polynomial f1 (2 ) = ”) 11 f2 = (x - x1)*(x - x2)*m/((x0 - x1)*(x0 - x2)) + (x - x0)*(x - x2)*n/((x1 -x0)*(x1 -x2)) + (x - x0)*(x - x1)*p/((x2 - x0)*(x2 - x1)); 12 disp(f2 ,” second order polynomial f2 ( 2) = ”) Scilab code Exa 18.7 Lagrange interpolation using computer 1 // c l c ( ) 2 z = 10; 3 x =[1 ,3 ,5 ,7 ,13]; 4 v = [800 ,2310 ,3090 ,3940 ,4755]; 107
5 f1 = (z - x(5)) * v(4) / (x(4) - x(5)) + (z - x(4)) * v(5) / (x(5) - x(4)); 6 f2 = (z - x(4))*(z - x(5)) *v(3) /((x(3) - x(4))*(x (3) - x(5)))+(z - x(3))*(z - x(5)) *v(4) /((x(4) - x(3))*(x(4) - x(5)))+(z - x(4))*(z - x(3)) *v(5) /((x(5) - x(4))*(x(5) - x(3))); 7 f3 = (z - x(3))*(z - x(4))*(z - x(5)) *v(2) /((x(2) - x(4))*(x(2) - x(5))*(x(2) - x(3)))+(z - x(4))*(z - x(2))*(z - x(5)) *v(3) /((x(3) - x(2))*(x(3) - x(5))*(x(3) - x(4)))+(z - x(2))*(z - x(3))*(z - x (5)) *v(4) /((x(4) - x(3))*(x(4) - x(2))*(x(4) - x (5)))+ (z - x(3))*(z - x(4))*(z - x(2)) *v(5) /((x (5) - x(4))*(x(5) - x(2))*(x(5) - x(3))); 8 f4 = (z - x(2))*(z - x(3))*(z - x(4))*(z - x(5)) *v (1) /((x(1) - x(2))*(x(1) - x(4))*(x(1) - x(5))*(x (1) - x(3)))+ (z - x(1))*(z - x(3))*(z - x(4))*(z - x(5)) *v(2) /((x(2) - x(1))*(x(2) - x(4))*(x(2) - x(5))*(x(2) - x(3)))+(z - x(1))*(z - x(4))*(z - x(2))*(z - x(5)) *v(3) /((x(3) - x(1))*(x(3) - x (2))*(x(3) - x(5))*(x(3) - x(4)))+(z - x(1))*(z - x(2))*(z - x(3))*(z - x(5)) *v(4) /((x(4) - x(1)) *(x(4) - x(3))*(x(4) - x(2))*(x(4) - x(5)))+ (z - x(1))*(z - x(3))*(z - x(4))*(z - x(2)) *v(5) /((x (5) - x(1))*(x(5) - x(4))*(x(5) - x(2))*(x(5) - x (3))); 9 disp(f1 ,” V e l o c i t y at 10 sec by f i r s t order i n t e r p o l a t i o n = ”) 10 disp(f2 ,” V e l o c i t y at 10 sec by second order i n t e r p o l a t i o n = ”) 11 disp(f3 ,” V e l o c i t y at 10 sec by t h i r d order i n t e r p o l a t i o n = ”) 12 disp(f4 ,” V e l o c i t y at 10 sec by f o u r t h order i n t e r p o l a t i o n = ”) 108
Figure 18.1: First order splines 109
Scilab code Exa 18.8 First order splines 1 // c l c ( ) 2 x = [3 ,4.5 ,7 ,9]; 3 fx = [2.5 ,1 ,2.5 ,0.5]; 4 m1 = (fx(2) - fx(1))/(x(2) - x(1)); 5 m2 = (fx(3) - fx(2))/(x(3) - x(2)); 6 m3 = (fx(4) - fx(3))/(x(4) - x(3)); 7 x1 = [3 ,4.5]; 8 x2 = [4.5 ,7]; 9 x3 = [7 ,9]; 10 plot2d(x1 ,m1*x1 +5.5); 11 plot2d(x2 ,m2*x2 -1.7); 12 plot2d(x3 ,m3*x3 +9.5); 13 xtitle(” f ( x ) vs x”,”x”,” f ( x ) ”) 14 r = 5 15 z = m2*r -1.7; 16 disp(z,”The value at x = 5 i s ”) Scilab code Exa 18.9 Quadratic splines 1 // c l c ( ) 2 x = [3 ,4.5 ,7 ,9]; 3 fx = [2.5 ,1 ,2.5 ,0.5]; 4 p = 4; 5 n = 3; 6 uk = n*(p-1); 7 c = 2*n - 2; 8 // 20.25∗ a1 + 4.5∗ b1 + c1 = 1 9 // 20.25∗ a2 + 4.5∗ b2 + c2 = 1 10 // 49∗ a2 + 7∗b2 + c2 = 2.5 11 // 49∗ a3 + 7∗b3 + c3 = 2.5 12 //9a1 + 3b1 + c1 = 2.5 110
Figure 18.2: Quadratic splines 111
13 // 81∗ a3 + 9∗b3 + c3 = 0.5 14 // 9∗ a1 + b1 = 9∗ a2 + b2 15 // 14 a2 + b2 = 14 ∗a3 + b3 16 a1 = 0; 17 // thus we have above 9 equations and 9 unknowns a1 , a2 , a3 , b1 , b2 , b3 , c1 , c2 , c3 18 // thus we get 19 A = [20.25,4.5,1,0,0,0,0,0,0;0,0,0,20.25,4.5,1,0,0,0;0,0,0,49,7,1,0,0, 20 disp(A,”A = ”) 21 B = [1;1;2.5;2.5;2.5;0.5;0;0;0]; 22 disp(B,”B =”) 23 X = inv(A)*B; 24 a1 = det(X(1,1)); 25 b1 = det(X(2,1)); 26 c1 = det(X(3,1)); 27 a2 = det(X(4,1)); 28 b2 = det(X(5,1)); 29 c2 = det(X(6,1)); 30 a3 = det(X(7,1)); 31 b3 = det(X(8,1)); 32 c3 = det(X(9,1)); 33 disp(a1 ,”a1 = ”) 34 disp(b1 ,”b1 = ”) 35 disp(c1 ,” c1 = ”) 36 disp(a2 ,”a2 = ”) 37 disp(b2 ,”b2 = ”) 38 disp(c2 ,” c2 = ”) 39 disp(a3 ,”a3 = ”) 40 disp(b3 ,”b3 = ”) 41 disp(c3 ,” c3 = ”) 42 // thus , f1 ( x ) = −x + 5.5 3 < x < 4. 5 43 // f2 ( x ) = 0.64∗ xˆ2 −6.76∗x + 18.46 4.5 < x < 7 44 // f3 ( x ) = −1.6∗xˆ2 + 24.6∗ x − 91.3 7 < x < 9 112
Figure 18.3: Cubic splines 45 x1 = 3:0.1:4.5; 46 x2 = 4.5:0.1:7; 47 x3 = 7:0.1:9; 48 plot2d(x1 ,-x1 + 5.5); 49 plot2d(x2 ,0.64* x2^2 -6.76*x2+ 18.46); 50 plot2d(x3 , -1.6*x3^2 + 24.6* x3 - 91.3); 51 xtitle(” f ( x ) vs x”,”x”,” f ( x ) ”) 52 x = 5; 53 fx = 0.64*x^2 -6.76*x + 18.46; 54 disp(fx ,”The value at x = 5 i s ”) 113
Scilab code Exa 18.10 Cubic splines 1 // c l c ( ) 2 x = [3 ,4.5 ,7 ,9]; 3 fx = [2.5 ,1 ,2.5 ,0.5]; 4 //we get the f o l l o w i n g equations f o r cubic s p l i n e s 5 // 8∗ f ” ( 4 . 5 ) + 2.5∗ f ”(7) = 9.6 6 // 2.5∗ f ” ( 4 . 5 ) + 9∗ f ”(7) = −9.6 7 // above two equations g i v e 8 m = 1.67909; // (m = f ” ( 4 . 5 ) ) 9 n = -1.53308; // ( n = f ”(7) ) 10 // t h i s v a l u e s can be used to y i e l d the f i n a l equation 11 // f1 ( x ) = 0.186566 ∗ ( x − 3) ˆ3 + 1 . 6 6 6 6 7 ∗ ( 4 . 5 − x ) + 0.246894∗( x − 3) 12 // in s i m i l a r manner other equations can be found too 13 // f2 ( x ) = 0.111939(7 − x ) ˆ3 − 0.102205∗( x − 4 . 5 ) ˆ3 − 0.299621 ∗ (7 − x ) + 1.638783 ∗ ( x − 4 . 5 ) 14 // f3 ( x ) = −0.127757∗(9 − x ) ˆ3 + 1.761027 ∗(9 − x ) + 0 . 2 5 ∗ ( x − 7) 15 x1 = 3:0.1:4.5; 16 x2 = 4.5:0.1:7; 17 x3 = 7:0.1:9; 18 plot2d(x1 ,0.186566 * (x1 - 3)^3 + 1.66667*(4.5 - x1) + 0.246894*( x1 - 3)); 19 plot2d(x2 ,0.111939*(7 - x2)^3 - 0.102205*( x2 - 4.5) ^3 - 0.299621 * (7 - x2) + 1.638783 * (x2 - 4.5)) ; 20 plot2d(x3 , -0.127757*(9 - x3)^3 + 1.761027 *(9 - x3) + 0.25*( x3 - 7)); 21 xtitle(” f ( x ) vs x”,”x”,” f ( x ) ”) 22 x = 5; 23 fx = 0.111939*(7 - x)^3 - 0.102205*(x - 4.5) ^3 - 0.299621 * (7 - x) + 1.638783 * (x - 4.5); 24 disp(fx ,”The value at x = 5 i s ”) 114
Chapter 19 Fourier Approximation Scilab code Exa 19.1 Least Square Fit 1 clc; 2 clear; 3 function y=f(t) 4 y=1.7+ cos (4.189*t+1.0472) 5 endfunction 6 deltat =0.15; 7 t1=0; 8 t2 =1.35; 9 omega =4.189; 10 del=(t2 -t1)/9; 11 for i=1:10 12 t(i)=t1+del*(i-1); 13 end 14 sumy =0; 15 suma =0; 16 sumb =0; 17 for i=1:10 18 y(i)=f(t(i)); 19 a(i)=y(i)*cos(omega*t(i)); 20 b(i)=y(i)*sin(omega*t(i)); 21 sumy=sumy+y(i); 115
22 suma=suma+a(i); 23 sumb=sumb+b(i); 24 end 25 A0=sumy /10; 26 A1=2* suma /10; 27 B1=2* sumb /10; 28 disp(”The l e a s t square f i t i s y=A0+A1∗ cos (w0∗ t )+A2∗ s i n (w0∗ t ) , where ”) 29 disp(A0 ,”A0=”) 30 disp(A1 ,”A1=”) 31 disp(B1 ,”B1=”) 32 theta=atan(-B1/A1); 33 C1=(A1^2 + B1^2) ^0.5; 34 disp(” A l t e r n a t i v e l y , the l e a s t square f i t can be expressed as ”) 35 disp(”y=A0+C1∗ cos (w0∗ t + theta ) , where ”) 36 disp(A0 ,”A0=”) 37 disp(theta ,” Theta=”) 38 disp(C1 ,”C1=”) 39 disp(”Or”) 40 disp(”y=A0+C1∗ s i n (w0∗ t + theta + pi /2) , where ”) 41 disp(A0 ,”A0=”) 42 disp(theta ,” Theta=”) 43 disp(C1 ,”C1=”) Scilab code Exa 19.2 Continuous Fourier Series Approximation 1 clc; 2 clear; 3 a0=0; 4 // f ( t )=−1 f o r −T/2 to −T/4 5 // f ( t )=1 f o r −T/4 to T/4 6 // f ( t )=−1 f o r T/4 to T/2 7 // ak=2/T∗ ( i n t e g r a t i o n of f ( t ) ∗ cos (w0∗ t ) from −T/2 to T/2) 116
8 // ak=2/T∗(( i n t e g r a t i o n of f ( t ) ∗ cos (w0∗ t ) from −T/2 to −T/4) + ( i n t e g r a t i o n of f ( t ) ∗ cos (w0∗ t ) from −T /4 to T/4) + ( i n t e g r a t i o n of f ( t ) ∗ cos (w0∗ t ) from T/4 to T/2) ) 9 // Therefore , 10 // ak=4/(k∗%pi ) f o r k = 1 , 5 , 9 , . . . . . 11 // ak=−4/(k∗%pi ) f o r k = 3 , 7 , 1 1 , . . . . . 12 // ak=0 f o r k=even i n t e g e r s 13 // s i m i l a r l y we f i n d the b ’ s . 14 // a l l the b ’ s=0 15 disp(”The f o u r i e r approximtion i s : ”) 16 disp(” 4/( %pi ) ∗ cos (w) ∗ t ) − 4/(3∗ %pi ) ∗ cos (3∗(w) ∗ t ) + 4/(5∗ %pi ) ∗ cos (5∗(w) ∗ t ) − 4/(7∗ %pi ) ∗ cos (7∗(w) ∗ t ) + . . . . . ”) Scilab code Exa 19.3 Trendline 1 clc; 2 clear; 3 x=0.5:0.5:5.5; 4 for i=1:11 5 y(i)=0.9846* log(x(i))+1.0004; 6 end 7 plot(x,y) 8 xtitle(”y vs x”,”x”,”y”) Scilab code Exa 19.4 Data Analysis 1 clc; 2 clear; 3 s=[0.0002 0.0002 0.0005 0.0005 0.001 0.001]; 117
Figure 19.1: Trendline 118
4 r=[0.2 0.5 0.2 0.5 0.2 0.5]; 5 u=[0.25; 0.5; 0.4; 0.75; 0.5; 1]; 6 logs=log10(s); 7 logr=log10(r); 8 logu=log10(u); 9 for i=1:6 10 m(i,1) =1; 11 m(i,2)=logs(i); 12 m(i,3)=logr(i); 13 end 14 15 a=mlogu; 16 disp (10^a(1),” alpha=”) 17 disp(a(2),” sigma=”) 18 disp(a(3),” rho=”) Scilab code Exa 19.5 Curve Fitting 1 clc; 2 clear; 3 x=0:10; 4 y=sin(x); 5 xi =0:.25:10; 6 // part a 7 yi=interp1(x,y,xi); 8 plot2d(xi ,yi) 9 xtitle(”y vs x ( part a ) ”,”x”,”y”) 10 // part b 11 // f i t t i n g x and y in a f i f t h order polynomial 12 clf(); 13 p=[0.0008 -0.0290 0.3542 -1.6854 2.586 -0.0915]; 14 15 for i=1:41 16 yi(i)=p(1)*(xi(i)^5)+p(2)*(xi(i)^4)+p(3)*(xi(i) ^3)+p(4)*(xi(i)^2)+p(5)*(xi(i))+p(6); 119
Figure 19.2: Curve Fitting 17 end 18 plot2d(xi ,yi); 19 xtitle(”y vs x ( part b ) ”,”x”,”y”) 20 // part c 21 clf(); 22 d=splin(x,y) 23 yi -interp(xi ,x,y,d) 24 plot2d(xi ,yi) 25 xtitle(”y vs x ( part c ) ”,”x”,”y”) 120
Figure 19.3: Curve Fitting 121
Scilab code Exa 19.6 Polynomial Regression 1 clc; 2 clear; 3 x=[0.05 0.12 0.15 0.3 0.45 0.7 0.84 1.05]; 4 y=[0.957 0.851 0.832 0.72 0.583 0.378 0.295 0.156]; 5 sx=sum(x); 6 sxx=sum(x.*x); 7 sx3=sum(x.*x.*x); 8 sx4=sum(x.*x.*x.*x); 9 sx5=sum(x.*x.*x.*x.*x); 10 sx6=sum(x.*x.*x.*x.*x.*x); 11 n=8; 12 sy=sum(y); 13 sxy=sum(x.*y); 14 sx2y=sum(x.*x.*y); 15 sx3y=sum(x.*x.*x.*y); 16 m=[n sx sxx sx3;sx sxx sx3 sx4;sxx sx3 sx4 sx5;sx3 sx4 sx5 sx6]; 17 p=[sy;sxy;sx2y;sx3y ]; 18 a=mp; 19 disp(”The cubic polynomial i s y=a0 + a1∗x + a2∗xˆ2 + a3∗x ˆ3 , where a0 , a1 , a2 and a3 are ”) 20 disp(a) 122
Chapter 21 Newton Cotes Integration Formulas Scilab code Exa 21.1 Single trapezoidal rule 1 clc; 2 clear; 3 function y=f(x) 4 y=(0.2+25*x -200*x^2+675*x^3 -900*x^4+400*x^5) 5 endfunction 6 tval =1.640533; 7 a=0; 8 b=0.8; 9 fa=f(a); 10 fb=f(b); 11 l=(b-a)*((fa+fb)/2); 12 Et=tval -l;// e r r o r 13 et=Et *100/ tval;// percent r e l a t i v e e r r o r 14 15 //by using approximate e r r o r estimate 16 17 // the second d e r i v a t i v e of f 18 function y=g(x) 19 y= -400+4050*x -10800*x^2+8000*x^3 123
20 endfunction 21 f2x=intg (0,0.8,g)/(b-a);// average value of second d e r i v a t i v e 22 Ea = -(1/12) *(f2x)*(b-a)^3; 23 disp(Et ,”The Error Et=”) 24 disp(”%”,et ,”The percent r e l a t i v e e r r o r et=”) 25 disp(Ea ,”The approximate e r r o r estimate without using the true value=”) Scilab code Exa 21.2 Multiple trapezoidal rule 1 clc; 2 clear; 3 function y=f(x) 4 y=(0.2+25*x -200*x^2+675*x^3 -900*x^4+400*x^5) 5 endfunction 6 a=0; 7 b=0.8; 8 tval =1.640533; 9 n=2; 10 h=(b-a)/n; 11 fa=f(a); 12 fb=f(b); 13 fh=f(h); 14 l=(b-a)*(fa+2*fh+fb)/(2*n); 15 Et=tval -l;// e r r o r 16 et=Et *100/ tval;// percent r e l a t i v e e r r o r 17 18 //by using approximate e r r o r estimate 19 20 // the second d e r i v a t i v e of f 21 function y=g(x) 22 y= -400+4050*x -10800*x^2+8000*x^3 23 endfunction 24 f2x=intg (0,0.8,g)/(b-a);// average value of second 124
d e r i v a t i v e 25 Ea = -(1/12) *(f2x)*(b-a)^3/(n^2); 26 disp(Et ,”The Error Et=”) 27 disp(”%”,et ,”The percent r e l a t i v e e r r o r et=”) 28 disp(Ea ,”The approximate e r r o r estimate without using the true value=”) Scilab code Exa 21.3 Evaluating Integrals 1 clc; 2 clear; 3 g=9.8; //m/ s ˆ2; a c c e l e r a t i o n due to g r a v i t y 4 m=68.1; // kg 5 c=12.5; // kg/ sec ; drag c o e f f i c i e n t 6 function v=f(t) 7 v=g*m*(1-exp(-c*t/m))/c 8 endfunction 9 tval =289.43515; //m 10 a=0; 11 b=10; 12 fa=f(a); 13 fb=f(b); 14 for i=10:10:20 15 n=i; 16 h=(b-a)/n; 17 disp(i,”No . of segments=”) 18 disp(h,”Segment s i z e=”) 19 j=a+h; 20 s=0; 21 while j<b 22 s=s+f(j); 23 j=j+h; 24 end 25 l=(b-a)*(fa+2*s+fb)/(2*n); 26 Et=tval -l;// e r r o r 125
27 et=Et *100/ tval;// percent r e l a t i v e e r r o r 28 disp(”m”,l,” Estimated d=”) 29 disp(et ,” et (%) ”) 30 disp(” −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 31 end 32 for i=50:50:100 33 n=i; 34 h=(b-a)/n; 35 disp(i,”No . of segments=”) 36 disp(h,”Segment s i z e=”) 37 j=a+h; 38 s=0; 39 while j<b 40 s=s+f(j); 41 j=j+h; 42 end 43 l=(b-a)*(fa+2*s+fb)/(2*n); 44 Et=tval -l;// e r r o r 45 et=Et *100/ tval;// percent r e l a t i v e e r r o r 46 disp(”m”,l,” Estimated d=”) 47 disp(et ,” et (%) ”) 48 disp(” −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 49 end 50 for i=100:100:200 51 n=i; 52 h=(b-a)/n; 53 disp(i,”No . of segments=”) 54 disp(h,”Segment s i z e=”) 55 j=a+h; 56 s=0; 57 while j<b 58 s=s+f(j); 59 j=j+h; 60 end 126
61 l=(b-a)*(fa+2*s+fb)/(2*n); 62 Et=tval -l;// e r r o r 63 et=Et *100/ tval;// percent r e l a t i v e e r r o r 64 disp(”m”,l,” Estimated d=”) 65 disp(et ,” et (%) ”) 66 disp(” −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 67 end 68 for i=200:300:500 69 n=i; 70 h=(b-a)/n; 71 disp(i,”No . of segments=”) 72 disp(h,”Segment s i z e=”) 73 j=a+h; 74 s=0; 75 while j<b 76 s=s+f(j); 77 j=j+h; 78 end 79 l=(b-a)*(fa+2*s+fb)/(2*n); 80 Et=tval -l;// e r r o r 81 et=Et *100/ tval;// percent r e l a t i v e e r r o r 82 disp(”m”,l,” Estimated d=”) 83 disp(et ,” et (%) ”) 84 disp(” −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 85 end 86 for i=1000:1000:2000 87 n=i; 88 h=(b-a)/n; 89 disp(i,”No . of segments=”) 90 disp(h,”Segment s i z e=”) 91 j=a+h; 92 s=0; 93 while j<b 94 s=s+f(j); 127
95 j=j+h; 96 end 97 l=(b-a)*(fa+2*s+fb)/(2*n); 98 Et=tval -l;// e r r o r 99 et=Et *100/ tval;// percent r e l a t i v e e r r o r 100 disp(”m”,l,” Estimated d=”) 101 disp(et ,” et (%) ”) 102 disp(” −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 103 end 104 for i=2000:3000:5000 105 n=i; 106 h=(b-a)/n; 107 disp(i,”No . of segments=”) 108 disp(h,”Segment s i z e=”) 109 j=a+h; 110 s=0; 111 while j<b 112 s=s+f(j); 113 j=j+h; 114 end 115 l=(b-a)*(fa+2*s+fb)/(2*n); 116 Et=tval -l;// e r r o r 117 et=Et *100/ tval;// percent r e l a t i v e e r r o r 118 disp(”m”,l,” Estimated d=”) 119 disp(et ,” et (%) ”) 120 disp(” −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 121 end 122 for i=5000:5000:10000 123 n=i; 124 h=(b-a)/n; 125 disp(i,”No . of segments=”) 126 disp(h,”Segment s i z e=”) 127 j=a+h; 128 s=0; 128
129 while j<b 130 s=s+f(j); 131 j=j+h; 132 end 133 l=(b-a)*(fa+2*s+fb)/(2*n); 134 Et=tval -l;// e r r o r 135 et=Et *100/ tval;// percent r e l a t i v e e r r o r 136 disp(”m”,l,” Estimated d=”) 137 disp(et ,” et (%) ”) 138 disp(” −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 139 end Scilab code Exa 21.4 Single Simpsons 1 by 3 rule 1 clc; 2 clear; 3 function y=f(x) 4 y=(0.2+25*x -200*x^2+675*x^3 -900*x^4+400*x^5) 5 endfunction 6 a=0; 7 b=0.8; 8 tval =1.640533; 9 n=2; 10 h=(b-a)/n; 11 fa=f(a); 12 fb=f(b); 13 fh=f(h); 14 l=(b-a)*(fa+4*fh+fb)/(3*n); 15 disp(l,” l=”) 16 Et=tval -l;// e r r o r 17 et=Et *100/ tval;// percent r e l a t i v e e r r o r 18 19 //by using approximate e r r o r estimate 129
20 21 // the f o u r t h d e r i v a t i v e of f 22 function y=g(x) 23 y= -21600+48000*x 24 endfunction 25 f4x=intg (0,0.8,g)/(b-a);// average value of f o u r t h d e r i v a t i v e 26 Ea = -(1/2880) *(f4x)*(b-a)^5; 27 disp(Et ,”The Error Et=”) 28 disp(”%”,et ,”The percent r e l a t i v e e r r o r et=”) 29 disp(Ea ,”The approximate e r r o r estimate without using the true value=”) Scilab code Exa 21.5 Multiple Simpsons 1 by 3 rule 1 clc; 2 clear; 3 function y=f(x) 4 y=(0.2+25*x -200*x^2+675*x^3 -900*x^4+400*x^5) 5 endfunction 6 a=0; 7 b=0.8; 8 tval =1.640533; 9 n=4; 10 h=(b-a)/n; 11 fa=f(a); 12 fb=f(b); 13 j=a+h; 14 s=0; 15 count =1; 16 while j<b 17 if (-1)^count ==-1 then 18 s=s+4*f(j); 19 else 20 s=s+2*f(j); 130
21 end 22 count=count +1; 23 j=j+h; 24 end 25 l=(b-a)*(fa+s+fb)/(3*n); 26 disp(l,” l=”) 27 Et=tval -l;// e r r o r 28 et=Et *100/ tval;// percent r e l a t i v e e r r o r 29 30 //by using approximate e r r o r estimate 31 32 // the f o u r t h d e r i v a t i v e of f 33 function y=g(x) 34 y= -21600+48000*x 35 endfunction 36 f4x=intg (0,0.8,g)/(b-a);// average value of f o u r t h d e r i v a t i v e 37 Ea = -(1/(180*4^4))*(f4x)*(b-a)^5; 38 disp(Et ,”The Error Et=”) 39 disp(”%”,et ,”The percent r e l a t i v e e r r o r et=”) 40 disp(Ea ,”The approximate e r r o r estimate without using the true value=”) Scilab code Exa 21.6 Simpsons 3 by 8 rule 1 clc; 2 clear; 3 function y=f(x) 4 y=(0.2+25*x -200*x^2+675*x^3 -900*x^4+400*x^5) 5 endfunction 6 a=0; 7 b=0.8; 8 tval =1.640533; 9 // part a 10 n=3; 131
11 h=(b-a)/n; 12 fa=f(a); 13 fb=f(b); 14 j=a+h; 15 s=0; 16 count =1; 17 while j<b 18 s=s+3*f(j); 19 count=count +1; 20 j=j+h; 21 end 22 l=(b-a)*(fa+s+fb)/(8); 23 disp(” Part A: ”) 24 disp(l,” l=”) 25 Et=tval -l;// e r r o r 26 et=Et *100/ tval;// percent r e l a t i v e e r r o r 27 28 //by using approximate e r r o r estimate 29 30 // the f o u r t h d e r i v a t i v e of f 31 function y=g(x) 32 y= -21600+48000*x 33 endfunction 34 f4x=intg (0,0.8,g)/(b-a);// average value of f o u r t h d e r i v a t i v e 35 Ea = -(1/6480) *(f4x)*(b-a)^5; 36 disp(Et ,”The Error Et=”) 37 disp(”%”,et ,”The percent r e l a t i v e e r r o r et=”) 38 disp(Ea ,”The approximate e r r o r estimate without using the true value=”) 39 40 // part b 41 n=5; 42 h=(b-a)/n; 43 l1=(a+2*h-a)*(fa+4*f(a+h)+f(a+2*h))/6; 44 l2=(a+5*h-a-2*h)*(f(a+2*h)+3*(f(a+3*h)+f(a+4*h))+fb) /8; 45 l=l1+l2; 132
46 disp(” −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 47 disp(” Part B: ”) 48 disp(l,” l=”) 49 Et=tval -l;// e r r o r 50 et=Et *100/ tval;// percent r e l a t i v e e r r o r 51 disp(Et ,”The Error Et=”) 52 disp(”%”,et ,”The percent r e l a t i v e e r r o r et=”) Scilab code Exa 21.7 Unequal Trapezoidal segments 1 clc; 2 clear; 3 function y=f(x) 4 y=(0.2+25*x -200*x^2+675*x^3 -900*x^4+400*x^5) 5 endfunction 6 tval =1.640533; 7 x=[0 0.12 0.22 0.32 0.36 0.4 0.44 0.54 0.64 0.7 0.8]; 8 for i=1:11 9 func(i)=f(x(i)); 10 end 11 l=0; 12 for i=1:10 13 l=l+(x(i+1)-x(i))*( func(i)+func(i+1))/2; 14 end 15 disp(l,” l=”) 16 Et=tval -l;// e r r o r 17 et=Et *100/ tval;// percent r e l a t i v e e r r o r 18 disp(Et ,”The Error Et=”) 19 disp(”%”,et ,”The percent r e l a t i v e e r r o r et=”) 133
Scilab code Exa 21.8 Simpsons Uneven data 1 clc; 2 clear; 3 function y=f(x) 4 y=(0.2+25*x -200*x^2+675*x^3 -900*x^4+400*x^5) 5 endfunction 6 tval =1.640533; 7 x=[0 0.12 0.22 0.32 0.36 0.4 0.44 0.54 0.64 0.7 0.8]; 8 for i=1:11 9 func(i)=f(x(i)); 10 end 11 l1=(x(2)-x(1))*((f(x(1))+f(x(2)))/2); 12 l2=(x(4)-x(2))*(f(x(4))+4*f(x(3))+f(x(2)))/6; 13 l3=(x(7)-x(4))*(f(x(4))+3*(f(x(5))+f(x(6)))+f(x(7))) /8; 14 l4=(x(9)-x(7))*(f(x(7))+4*f(x(8))+f(x(9)))/6; 15 l5=(x(10) -x(9))*((f(x(10))+f(x(9)))/2); 16 l6=(x(11) -x(10))*((f(x(11))+f(x(10)))/2); 17 l=l1+l2+l3+l4+l5+l6; 18 disp(l,” l=”) 19 Et=tval -l;// e r r o r 20 et=Et *100/ tval;// percent r e l a t i v e e r r o r 21 disp(Et ,”The Error Et=”) 22 disp(”%”,et ,”The percent r e l a t i v e e r r o r et=”) Scilab code Exa 21.9 Average Temperature Determination 1 clc; 2 clear; 3 function t=f(x,y) 4 t=2*x*y+2*x-x^2-2*y^2+72 5 endfunction 6 len =8; //m, length 134
7 wid =6; //m, width 8 a=0; 9 b=len; 10 n=2; 11 h=(b-a)/n; 12 a1=0; 13 b1=wid; 14 h1=(b1 -a1)/n; 15 16 fa=f(a,0); 17 fb=f(b,0); 18 fh=f(h,0); 19 lx1=(b-a)*(fa+2*fh+fb)/(2*n); 20 21 fa=f(a,h1); 22 fb=f(b,h1); 23 fh=f(h,h1); 24 lx2=(b-a)*(fa+2*fh+fb)/(2*n); 25 26 fa=f(a,b1); 27 fb=f(b,b1); 28 fh=f(h,b1); 29 lx3=(b-a)*(fa+2*fh+fb)/(2*n); 30 31 l=(b1 -a1)*(lx1 +2* lx2+lx3)/(2*n); 32 33 avg_temp=l/(len*wid); 34 disp(avg_temp ,”The average termperature i s=”) 135
Chapter 22 Integration of equations Scilab code Exa 22.1 Error corrections of the trapezoidal rule 1 // c l c ( ) 2 h = [0.8 ,0.4 ,0.2]; 3 I = [0.1728 ,1.0688 ,1.4848]; 4 E = [89.5 ,34.9 ,9.5]; 5 I1 = 4 * I(2) / 3 - I(1) / 3; 6 t = 1.640533; 7 et1 = t - I1 ; 8 Et1 = et1 * 100/t; 9 disp(”%”,Et1 ,” Error of the improved i n t e g r a l f o r segment 1 and 2 = ”) 10 I2 = 4 * I(3) / 3 - I(2) / 3; 11 et2 = t - I2 ; 12 Et2 = et2 * 100/t; 13 disp(”%”,Et2 ,” Error of the improved i n t e g r a l f o r segment 4 and 2 = ”) Scilab code Exa 22.2 Higher order error correction of integral estimates 136
1 // c l c ( ) 2 I1 = 1.367467; 3 I2 = 1.623467; 4 I = 16 * I2 /15 - I1 / 15; 5 disp(I,” Obtained i n t e g r a l which i s the c o r r e c t answer t i l l the seventh decimal ”) Scilab code Exa 22.3 Two point gauss legendre formulae 1 // c l c ( ) 2 // f ( x ) = 0 .2 + 25∗x − 200∗ xˆ2 + 675∗ xˆ3 − 900∗ xˆ4 + 400∗ xˆ5 3 // f o r using two point gauss l e g e n d r e formulae , the i n t e r v a l s have to be changed to −1 and 1 4 // t h e r e f o r e , x = 0 .4 + 0 .4 ∗ xd 5 // thus the i n t e g r a l i s t r a n s f e r r e d to 6 // ( 0 . 2 + 25∗(0.4+0.4∗ x ) − 200∗(0.4 + 0.4∗ x ) ˆ2 + 675∗(0.4 + 0.4∗ x ) ˆ3 − 900∗(0.4 + 0.4∗ x ) ˆ4 + 400∗(0.4 + 0.4∗ x ) ˆ5) ∗0.4 7 // f o r t h r e e point gauss l e g e n d r e formulae 8 x1 = -(1/3) ^ 0.5; 9 x2 = (1/3) ^ 0.5; 10 I1 = (0.2 + 25*(0.4+0.4* x1) - 200*(0.4 + 0.4*x1)^2 + 675*(0.4 + 0.4* x1)^3 - 900*(0.4 + 0.4* x1)^4 + 400*(0.4 + 0.4* x1)^5) *0.4; 11 I2 = (0.2 + 25*(0.4+0.4* x2) - 200*(0.4 + 0.4*x2)^2 + 675*(0.4 + 0.4* x2)^3 - 900*(0.4 + 0.4* x2)^4 + 400*(0.4 + 0.4* x2)^5) *0.4; 12 I = I1 + I2; 13 disp(I,” I n t e g r a l obtained using gauss l e g e n d r e formulae =”) 14 t = 1.640533; 15 e = (t - I)*100/t; 16 disp(”%”,e,” e r r o r = ”) 137
Scilab code Exa 22.4 Three point gauss legendre method 1 // c l c ( ) 2 // f ( x ) = 0 .2 + 25∗x − 200∗ xˆ2 + 675∗ xˆ3 − 900∗ xˆ4 + 400∗ xˆ5 3 // f o r using t h r e e point gauss l e g e n d r e formulae , the i n t e r v a l s have to be changed to −1 and 1 4 // t h e r e f o r e , x = 0 .4 + 0 .4 ∗ xd 5 // thus the i n t e g r a l i s t r a n s f e r r e d to 6 // ( 0 . 2 + 25∗(0.4+0.4∗ x ) − 200∗(0.4 + 0.4∗ x ) ˆ2 + 675∗(0.4 + 0.4∗ x ) ˆ3 − 900∗(0.4 + 0.4∗ x ) ˆ4 + 400∗(0.4 + 0.4∗ x ) ˆ5) ∗0.4 7 // f o r t h r e e point gauss l e g e n d r e formulae 8 x1 = -0.7745967; 9 x2 = 0; 10 x3 = 0.7745967; 11 c0 = 0.5555556; 12 c1 = 0.8888889; 13 c2 = 0.5555556; 14 I1 = (0.2 + 25*(0.4+0.4* x1) - 200*(0.4 + 0.4*x1)^2 + 675*(0.4 + 0.4* x1)^3 - 900*(0.4 + 0.4* x1)^4 + 400*(0.4 + 0.4* x1)^5) *0.4; 15 I2 = (0.2 + 25*(0.4+0.4* x2) - 200*(0.4 + 0.4*x2)^2 + 675*(0.4 + 0.4* x2)^3 - 900*(0.4 + 0.4* x2)^4 + 400*(0.4 + 0.4* x2)^5) *0.4; 16 I3 = (0.2 + 25*(0.4+0.4* x3) - 200*(0.4 + 0.4*x3)^2 + 675*(0.4 + 0.4* x3)^3 - 900*(0.4 + 0.4* x3)^4 + 400*(0.4 + 0.4* x3)^5) *0.4; 17 I = c0 * I1 + c1 * I2 + c2 * I3; 18 disp(I,” i n t e g r a l obtained using t h r e e point gauss l e g e n d r e formulae = ”) 138
Scilab code Exa 22.5 Applying Gauss Quadrature to the falling Parachutist problem 1 // c l c ( ) 2 // f ( t ) = g∗m∗( i n t (0 ,10 ,(1 − exp(−c∗ t /m) ) ) ) / c 3 // f o r using gauss quadrature method , l i m i t s are changed to −1 to 1 by r e p l c i n g x = 5 + 5∗xd 4 // the new i n t e g r a l obtained i s 5 // (1 − exp(−c ∗(5 + 5∗x ) /m ) ) ∗5 6 g = 9.8; 7 c = 12.5; 8 m = 68.1; 9 // f o r two point method 10 x1 = -(1/3) ^0.5; 11 x2 = (1/3) ^0.5; 12 I1 = g*m*(1 - exp(-c*(5 + 5*x1)/m ))*5 / c; 13 I2 = g*m*(1 - exp(-c*(5 + 5*x2)/m ))*5 / c; 14 I = I1 + I2; 15 disp(I,” i n t e g r a l by two point method = ”) 16 x1 = -0.7745967; 17 x2 = 0; 18 x3 = 0.7745967; 19 c0 = 0.5555556; 20 c1 = 0.8888889; 21 c2 = 0.5555556; 22 I1 = g*m*(1 - exp(-c*(5 + 5*x1)/m ))*5 / c; 23 I2 = g*m*(1 - exp(-c*(5 + 5*x2)/m ))*5 / c; 24 I3 = g*m*(1 - exp(-c*(5 + 5*x3)/m ))*5 / c; 25 I = c0*I1 + c1 * I2 + c2 * I3; 26 disp(I,” i n t e g r a l by t h r e e point method =”) 27 x1 = -0.861136312; 28 x2 = -0.339981044; 29 x3 = 0.339981044; 30 x4 = 0.861136312; 31 c1 = 0.3478548; 32 c2 = 0.6521452; 33 c3 = 0.6521452; 34 c4 = 0.3478548; 35 I1 = g*m*(1 - exp(-c*(5 + 5*x1)/m ))*5 / c; 139
36 I2 = g*m*(1 - exp(-c*(5 + 5*x2)/m ))*5 / c; 37 I3 = g*m*(1 - exp(-c*(5 + 5*x3)/m ))*5 / c; 38 I4 = g*m*(1 - exp(-c*(5 + 5*x4)/m ))*5 / c; 39 I = c1*I1 + c2 * I2 + c3 * I3 + c4 * I4; 40 disp(I,” i n t e g r a l by f o u r point method =”) 41 x1 = -0.906179846; 42 x2 = -0.538469310; 43 x3 = 0; 44 x4 = 0.538469310; 45 x5 = 0.906179846 46 c1 = 0.2369269; 47 c2 = 0.4786287; 48 c3 = 0.5688889; 49 c4 = 0.4786287; 50 c5 = 0.2369269; 51 I1 = g*m*(1 - exp(-c*(5 + 5*x1)/m ))*5 / c; 52 I2 = g*m*(1 - exp(-c*(5 + 5*x2)/m ))*5 / c; 53 I3 = g*m*(1 - exp(-c*(5 + 5*x3)/m ))*5 / c; 54 I4 = g*m*(1 - exp(-c*(5 + 5*x4)/m ))*5 / c; 55 I5 = g*m*(1 - exp(-c*(5 + 5*x5)/m ))*5 / c; 56 I = c1*I1 + c2 * I2 + c3 * I3 + c4 * I4 + c5 * I5; 57 disp(I,” i n t e g r a l by f i v e point method =”) Scilab code Exa 22.6 Evaluation of improper integral 1 // c l c ( ) 2 //N( x ) = ( i n t (− i n f i n i t y , −2 , exp (−(x ˆ2) /2) ) + i n t ( −2 ,1 , exp (−(x ˆ2) /2) ) ) /(2∗ pi ) ˆ0.5 3 // f i r s t i n t e g r a l can be s o l v e d as 4 // i n t (− i n f i n i t y , −2 , exp (−(x ˆ2) /2) ) = i n t ( −0.5 ,0 , exp ( −1/(2∗ t ˆ2) ) / t ˆ2) 5 h = 1/8; 6 // i n t ( −0.5 ,0 , exp ( −1/(2∗ t ˆ2) ) / t ˆ2) = h ∗( f ( x −7/16) + f ( x −5/16) + f ( x −3/16) + f ( x −1/16) ) 7 t1 = -7/16; 140
8 t2 = -5/16; 9 t3 = -3/16; 10 t4 = -1/16; 11 m1 = exp ( -1/(2*t1^2))/t1^2; 12 m2 = exp ( -1/(2*t2^2))/t2^2; 13 m3 = exp ( -1/(2*t3^2))/t3^2; 14 m4 = exp ( -1/(2*t4^2))/t4^2; 15 I1 = h*(m1 + m2 + m3 + m4); 16 disp(I1 ,” the value of f i r s t i n t e g r a l = ”) 17 // simpsons 1/3 rd s u l e i s a p p l i e d f o r the second i n t e g r a l 18 h1 = 0.5; 19 x1 = -2; 20 x2 = -1.5; 21 x3 = -1; 22 x4 = -0.5; 23 x5 = 0; 24 x6 = 0.5; 25 x7 = 1; 26 n1 = exp(-(x1^2) /2); 27 n2 = exp(-(x2^2) /2); 28 n3 = exp(-(x3^2) /2); 29 n4 = exp(-(x4^2) /2); 30 n5 = exp(-(x5^2) /2); 31 n6 = exp(-(x6^2) /2); 32 n7 = exp(-(x7^2) /2); 33 I2 =(1-(-2)) * (n1 + 4 *(n2 + n4 + n6) + 2*(n3 + n5) + n7)/(18); 34 disp(I2 ,”The value of second i n t e g r a l = ”) 35 f = (I1 + I2)/(2 * %pi)^0.5; 36 disp(f,” Therefore the f i n a l r e s u l t can be computed as ”) 37 N = 0.8413; 38 e = (N - f) * 100 / N; 39 disp(”%”,e,” e r r o r = ”) 141
Chapter 23 Numerical differentiation Scilab code Exa 23.1 High accuracy numerical differentiation formulas 1 // c l c ( ) 2 // f ( x ) = −0.1∗xˆ4 − 0.15∗ xˆ3 − 0. 5 ∗ xˆ2 − 0.25 ∗x + 1. 2 3 h = 0.25; 4 t = -0.9125; 5 x = 0:h:1; 6 disp(x,”x = ”) 7 fx = -0.1*x^4 - 0.15*x^3 - 0.5 * x^2 - 0.25 *x + 1.2; 8 disp(fx ,” f ( x ) = ”) 9 fd = (- fx(5) + 4*fx(4) - 3 * fx(3))/(2 * h); 10 efd = (t - fd) * 100 / t; 11 disp(fd ,”by forward d i f f e r e n c e ”) 12 disp(”%”,efd ,” e r r o r in forward d i f f e r e n c e method = ” ) 13 bd = (3 * fx(3) - 4 * fx(2) + fx(1))/ (2*h); 14 ebd = (t - bd) * 100 / t; 15 disp(bd ,”by backward d i f f e r e n c e ”) 16 disp(”%”,ebd ,” e r r o r in backward d i f f e r e n c e method = ”) 17 cdm = (-fx(5) + 8*(fx(4)) -8*fx(2) + fx(1) ) / (12*h 142
); 18 ecdm = (t - cdm) * 100 / t; 19 disp(cdm ,”by c e n t r a l d i f f e r e n c e ”) 20 disp(”%”,ecdm ,” e r r o r in c e n t r a l d i f f e r e n c e method = ”) Scilab code Exa 23.2 Richardson extrapolation 1 // c l c ( ) 2 // f ( x ) = −0.1∗xˆ4 − 0.15∗ xˆ3 − 0. 5 ∗ xˆ2 − 0.25 ∗x + 1. 2 3 h = 0.5; 4 t = -0.9125; 5 x = 0:h:1; 6 disp(x,”x with h = 0 .5 i s ”) 7 fx = -0.1*x^4 - 0.15*x^3 - 0.5 * x^2 - 0.25 *x + 1.2; 8 disp(fx ,” f ( x ) with h = 0. 5 i s ”) 9 cdm = (fx(3) - fx(1))/ 1; 10 ecdm = (t - cdm) * 100 / t; 11 disp(cdm ,”by c e n t r a l d i f f e r e n c e ( h = 0.5 ) ”) 12 disp(”%”,ecdm ,” e r r o r in c e n t r a l d i f f e r e n c e method ( h = 0. 5 ) = ”) 13 h1 = 0.25; 14 x1 = 0:h1:1; 15 disp(x1 ,”x with h = 0.25 i s ”) 16 fx1 = -0.1*x1^4 - 0.15* x1^3 - 0.5 * x1^2 - 0.25 *x1 + 1.2; 17 disp(fx1 ,” fx with h = 0.25 i s ”) 18 cdm1 = (fx1 (4) - fx1 (2))/ (2*h1); 19 ecdm1 = (t - cdm1) * 100 / t; 20 disp(cdm1 ,”by c e n t r a l d i f f e r e n c e ( h = 0.25 ) ”) 21 disp(”%”,ecdm1 ,” e r r o r in c e n t r a l d i f f e r e n c e method ( h = 0.25 ) = ”) 22 D = 4 * cdm1 /3 - cdm / 3; 143
23 disp(D,” improved estimate =”) Scilab code Exa 23.3 Differentiating unequally spaced data 1 // c l c ( ) 2 //q ( z = 0) = −k∗p∗C∗(dT/dz ) /( z = 0) 3 k = 3.5 * 10^ - 7; //mˆ2/ s 4 p = 1800; // kg/mˆ3 5 C = 840; // ( J /( kg .C) ) 6 x = 0; 7 fx0 = 13.5; 8 fx1 = 12; 9 fx2 = 10; 10 x0 = 0; 11 x1 = 1.25; 12 x2 = 3.75; 13 dfx = fx0 *(2*x - x1 - x2)/((x0 - x1)*(x0 - x2)) + fx1 *(2*x - x0 - x2)/((x1 - x0)*(x1 - x2)) + fx2 *(2*x - x1 - x0)/((x2 - x1)*(x2 - x0)); 14 disp(”C/cm”,dfx ,” (dT/dz ) = ”) 15 q = - k * p *C * dfx *100; 16 disp(”W/mˆ2”,q,”q ( z = 0) =”) Scilab code Exa 23.4 Integration and Differentiation 1 clc; 2 clear; 3 function y=f(x) 4 y=0.2+25*x -200*x^2+675*x^3 -900*x^4+400*x^5 5 endfunction 6 a=0; 7 b=0.8; 8 Q=intg (0,0.8,f); 144
9 disp(Q,”Q=”) 10 x=[0 0.12 0.22 0.32 0.36 0.4 0.44 0.54 0.64 0.7 0.8]; 11 y=f(x); 12 integral=inttrap(x,y) 13 disp(integral ,” i n t e r g r a l=”) 14 disp(diff(x),” d i f f ( x )=”) 15 d=diff(y)./ diff(x); 16 disp(d,”d=”) Scilab code Exa 23.5 Integrate a function 1 clc; 2 clear; 3 function y=f(x) 4 y=0.2+25*x -200*x^2+675*x^3 -900*x^4+400*x^5 5 endfunction 6 a=0; 7 b=0.8; 8 Qt =1.640533; 9 Q=intg (0,0.8,f); 10 disp(Q,”Computed=”) 11 disp(abs(Q-Qt)*100/Qt ,” Error estimate=”) 145
Chapter 25 Runga Kutta methods Scilab code Exa 25.1 Eulers method 1 // c l c ( ) 2 //dy/dx = −2∗xˆ3 + 12∗xˆ2 − 20∗x + 8.5 3 // t h e r e f o r e , y = −0.5∗xˆ4 + 4∗xˆ3 − 10∗xˆ2 + 8 .5 + c 4 x1 = 0; 5 y1 = 1; 6 h = 0.5; 7 c =-(-0.5*x1^4 + 4*x1^3 - 10*x1^2 + 8.5* x1 - y1); 8 x = 0:0.5:4; 9 disp(x,”x = ”) 10 y = -0.5*x^4 + 4*x^3 - 10*x^2 + 8.5*x + c; 11 disp(y,” true v a l u e s of y = ”) 12 fxy = -2*x^3 + 12*x^2 - 20*x + 8.5; 13 y2(1) = y(1); 14 e(1) = (y(1) - y2(1)) * 100 / y(1); 15 for i = 2:9 16 y2(i) = y2(i-1) + fxy(i-1)*h; 17 e(i) = (y(i) - y2(i))*100/y(i); 18 end 19 disp(y2 ,”y by e u l e r method =”) 20 disp(e,” e r r o r =”) 146
Scilab code Exa 25.2 Taylor series estimate for error by eulers method 1 // c l c ( ) 2 // f ( x , y ) = dy/dx = −2∗xˆ3 + 12∗xˆ2 − 20∗x + 8.5 3 // f ’ ( x , y ) = −6∗xˆ2 + 24∗x − 20 4 // f ”( x , y ) = −12∗x + 24 5 // f ” ’( x , y ) = −12 6 x = 0; 7 Et2 = (-6*x^2 + 24*x - 20) * 0.5^2 / 2; 8 Et3 = (-12*x + 24) * (0.5) ^3 / 6; 9 Et4 = (-12) *(0.5 ^ 4) / 24; 10 Et = Et2 + Et3 + Et4; 11 disp(Et ,” Total t r u n c a t i o n e r r o r =”) Scilab code Exa 25.3 Effect of reduced step size on Eulers method 1 // c l c ( ) 2 //dy/dx = −2∗xˆ3 + 12∗xˆ2 − 20∗x + 8.5 3 // t h e r e f o r e , y = −0.5∗xˆ4 + 4∗xˆ3 − 10∗xˆ2 + 8 .5 + c 4 x1 = 0; 5 y1 = 1; 6 h = 0.25; 7 c =-(-0.5*x1^4 + 4*x1^3 - 10*x1^2 + 8.5* x1 - y1); 8 x = 0:h:4; 9 disp(x,”x = ”) 10 y = -0.5*x^4 + 4*x^3 - 10*x^2 + 8.5*x + c; 11 disp(y,” true v a l u e s of y = ”) 12 fxy = -2*x^3 + 12*x^2 - 20*x + 8.5; 13 y2(1) = y(1); 14 e(1) = (y(1) - y2(1)) * 100 / y(1); 15 for i = 2:17 16 y2(i) = y2(i-1) + fxy(i-1)*h; 147
Figure 25.1: Solving ODEs 17 e(i) = (y(i) - y2(i))*100/y(i); 18 end 19 disp(y2 ,”y by e u l e r method =”) 20 disp(e,” e r r o r =”) Scilab code Exa 25.4 Solving ODEs 1 clc; 2 clear; 3 m=68.1; 4 g=9.8; 148
5 c=12.5; 6 a=8.3; 7 b=2.2; 8 vmax =46; 9 function yp=f(t,v) 10 yp=g-c*v/m; 11 endfunction 12 v0=0; 13 t=0:15; 14 x=ode(v0 ,0,t,f); 15 disp(x) 16 plot(t,x, ’ .− ’ ) 17 18 function yp=f1(t,v) 19 yp=g-(c/m)*(v+a*(v/vmax)^b) 20 endfunction 21 x1=ode(v0 ,0,t,f1); 22 plot(t,x1) 23 xtitle(” v e l o c i t y vs time ”,” t ( s ) ”,”v (m/ s ) ”) 24 h1=legend ([” Linear ”;” Nonlinear ”]) Scilab code Exa 25.5 Heuns method 1 // c l c ( ) 2 //y ’ = 4∗ exp ( 0 . 8 ∗ x ) − 0.5∗ y 3 //y = 4∗( exp ( 0 . 8 ∗ x ) − exp ( −0.5∗x ) ) /1.3 + 2∗ exp ( −0.5∗ x ) 4 x = 0:1:4; 5 disp(x) 6 x1 = 0; 7 y1 = 2; 8 y2(1) = y1; 9 for i = 1:5 10 y(i) = 4*( exp (0.8*x(i)) - exp ( -0.5*x(i)))/1.3 + 2*exp ( -0.5*x(i)); 149
11 dy(i) = 4*exp (0.8*x(i)) - 0.5* y2(i); 12 y2(i + 1) = y2(i) + dy(i); 13 if i>1 then 14 m(i) = (dy(i) + dy(i-1))/2; 15 y2(i) = y2(i-1) + m(i); 16 dy(i) = 4*exp (0.8*x(i)) - 0.5* y2(i); 17 end 18 e(i) = (y(i) - y2(i)) * 100 / y(i); 19 end 20 disp(y2 (1:5) ,”By heuns method (1 i t e r a t i o n ) ”) 21 disp(”%”,e(1:5) ,” e r r o r = ”) Scilab code Exa 25.6 Comparison of various second order RK 4 method 1 // c l c ( ) 2 // f ’ ( x , y ) = −2∗xˆ3 + 12∗xˆ2 −20∗x + 8.5 3 // f ( x , y ) = −xˆ4 / 2 + 4∗xˆ3 − 10∗xˆ2 + 8.5∗ x + 1 4 h = 0.5; 5 x = 0:h:4; 6 y1 = -x^4 / 2 + 4*x^3 - 10*x^2 + 8.5*x + 1; 7 y(1) = 1; 8 disp(x,”x =”) 9 disp(y1 ,” true value of y =”) 10 for i = 1:8 11 k1(i) = -2*x(i)^3 + 12*x(i)^2 -20*x(i) + 8.5; 12 x1(i) = x(i) + h/2; 13 k2(i) = -2*x1(i)^3 + 12*x1(i)^2 -20*x1(i) + 8.5; 14 y(i+1) = y(i) + k2(i)*h; 15 e(i) = (y1(i) - y(i))*100/ y1(i); 16 end 17 disp(y(1:9) ,”y by midpoint method”) 18 disp(e,” e r r o r = ”) 19 for i = 1:8 20 k1(i) = -2*x(i)^3 + 12*x(i)^2 -20*x(i) + 8.5; 21 x(i) = x(i) + 3*h/4; 150
22 k2(i) = -2*x(i)^3 + 12*x(i)^2 -20*x(i) + 8.5; 23 y(i+1) = y(i) + (k1(i)/3 + 2*k2(i)/3)*h; 24 e(i) = (y1(i) - y(i))*100/ y1(i); 25 end 26 disp(y(1:9) ,”y by second order Ralston RK”) 27 disp(e,” e r r o r = ”) Scilab code Exa 25.7 Classical fourth order RK method 1 // c l c ( ) 2 // f ’ ( x , y ) = −2∗xˆ3 + 12∗xˆ2 −20∗x + 8.5 3 // f ( x , y ) = −xˆ4 / 2 + 4∗xˆ3 − 10∗xˆ2 + 8.5∗ x + 1 4 h = 0.5; 5 x = 0:h:4; 6 y1 = -x^4 / 2 + 4*x^3 - 10*x^2 + 8.5*x + 1; 7 y(1) = 1; 8 for i=1:8 9 k1(i) = -2*x(i)^3 + 12*x(i)^2 -20*x(i) + 8.5; 10 x1(i) = x(i) + h/2; 11 k2(i) = -2*x1(i)^3 + 12*x1(i)^2 -20*x1(i) + 8.5; 12 k3(i) = -2*x1(i)^3 + 12*x1(i)^2 -20*x1(i) + 8.5; 13 x2(i) = x(i) + h; 14 k4(i) = -2*x2(i)^3 + 12*x2(i)^2 -20*x2(i) + 8.5; 15 y(i+1) = y(i) + (k1(i)+2*k2(i)+2*k3(i)+k4(i))*h /6; 16 e(i) = (y1(i) - y(i))*100/ y1(i); 17 end 18 disp(” f ( x , y ) = −2∗xˆ3 + 12∗xˆ2 −20∗x + 8.5 ”) 19 disp(y(1:9) ,”y by f o u r t h order Ralston RK”) 20 disp(” f ( x , y ) = 4∗ exp ( 0 . 8 ∗ x ) − 0.5∗ y”) 21 x = 0:h:0.5; 22 y(1) = 2; 23 k1 = 4*( exp (0.8*x(1))) -0.5*y(1); 24 x1 = x(1) + 0.5*h; 25 y1 = y(1) + 0.5* k1*h; 151
26 k2 = 4*exp (0.8* x1) - 0.5* y1; 27 y2 = y(1) + 0.5* k2*h; 28 k3 = 4*exp (0.8* x1) - 0.5* y2; 29 x1 = x(1) + h; 30 y1 = y(1) + k3*h; 31 k4 = 4*exp (0.8* x1) - 0.5* y1; 32 y(2) = y(1) + (k1+2*k2+2*k3+k4)*h/6; 33 disp(y(1:2) ,”y = ”) Scilab code Exa 25.8 Comparison of Runga Kutta methods 1 // c l c ( ) 2 disp(” f ( x , y ) = 4∗ exp ( 0 . 8 ∗ x ) − 0.5∗ y”) 3 h = 1; 4 x = 0:h:4; 5 y(1) = 2; 6 for i = 1:5 7 k1(i) = 4*( exp (0.8*x(i))) -0.5*y(i); 8 x1 = x(i) + h; 9 y1 = y(i) + k1(i)*h; 10 k2(i) = 4*( exp (0.8* x1)) -0.5*y1; 11 y(i+1) = y(i) + (k1(i)/2 + k2(i)/2)*h; 12 end 13 disp(y(1:5) ,”y ( second order RK method ) = ”) 14 for i = 1:5 15 k1(i) = 4*( exp (0.8*x(i))) -0.5*y(i); 16 x1 = x(i) + 0.5*h; 17 y1 = y(i) + 0.5*h*k1(i); 18 k2(i) = 4*( exp (0.8* x1)) -0.5*y1; 19 x1 = x(i) + h; 20 y1 = y(i) -k1(i)*h + 2*k2(i)*h; 21 k3(i) = 4*( exp (0.8* x1)) -0.5*y1; 22 y(i+1) = y(i) + (k1(i) + 4*k2(i) + k3(i))*h/6; 23 end 24 disp(y(1:5) ,”y ( t h i r d order RK method ) = ”) 152
25 for i = 1:5 26 k1(i) = 4*( exp (0.8*x(i))) -0.5*y(i); 27 x1 = x(i) + 0.5*h; 28 y1 = y(i) + 0.5* k1(i)*h; 29 k2(i) = 4*exp (0.8* x1) - 0.5* y1; 30 y2 = y(i) + 0.5* k2(i)*h; 31 k3(i) = 4*exp (0.8* x1) - 0.5* y2; 32 x1 = x(i) + h; 33 y1 = y(i) + k3(i)*h; 34 k4(i) = 4*exp (0.8* x1) - 0.5* y1; 35 y(i+1) = y(i) + (k1(i)+2*k2(i)+2*k3(i)+k4(i))*h/6; 36 end 37 disp(y(1:5) ,”y ( f o u r t h order RK method ) = ”) 38 for i = 1:5 39 40 k1(i) = 4*( exp (0.8*x(i))) -0.5*y(i); 41 x1 = x(i) + 0.25*h; 42 y1 = y(i) + 0.25* k1(i)*h; 43 k2(i) = 4*exp (0.8* x1) - 0.5* y1; 44 y2 = y(i) + 0.125* k2(i)*h + 0.125* k1(i)*h; 45 k3(i) = 4*exp (0.8* x1) - 0.5* y2; 46 x1 = x(i) + 0.5*h; 47 y1 = y(i) -0.5*k2(i)*h + k3(i)*h; 48 k4(i) = 4*exp (0.8* x1) - 0.5* y1; 49 x1 = x(i) + 0.75*h; 50 y1 = y(i) + 3*k1(i)*h/16 + 9*k4(i)*h/16; 51 k5(i) = 4*exp (0.8* x1) - 0.5* y1; 52 x1 = x(i) + h; 53 y1 = y(i) - 3*k1(i)*h/7 + 2*k2(i)*h/7 + 12*k3(i)*h/7 - 12*k4(i)*h/7 + 8*k5(i)*h/7; 54 k6(i) = 4*exp (0.8* x1) - 0.5* y1; 55 y(i+1) = y(i) + (7*k1(i)+32* k3(i)+12* k4(i) + 32*k5(i ) + 7*k6(i))*h/90; 56 end 57 disp(y(1:5) ,”y ( f i f t h order RK method ) ”) 153
Scilab code Exa 25.9 Solving systems of ODE using Eulers method 1 // c l c ( ) 2 // dy1/dx = −0.5∗ y1 3 // dy2/dx = 4 − 0.3∗ y2 − 0.1∗ y1 4 x1 = 0; 5 h =0.5; 6 y1(1) = 4; 7 y2(1) = 6; 8 x = 0:h:2; 9 for i = 2:5 10 y1(i) = y1(i-1) -0.5*y1(i-1)*h; 11 y2(i) = y2(i-1) + (4 - 0.3* y2(i-1) - 0.1* y1(i-1) )*h; 12 end 13 disp(x,”x = ”) 14 disp(y1 ,”y1 = ”) 15 disp(y2 ,”y2 = ”) Scilab code Exa 25.10 Solving systems of ODE using RK 4 method 1 // c l c ( ) 2 // dy1/dx = −0.5∗ y1 3 // dy2/dx = 4 − 0.3∗ y2 − 0.1∗ y1 4 x1 = 0; 5 h =0.5; 6 y1(1) = 4; 7 y2(1) = 6; 8 x = 0:h:2; 9 for i = 1:5 10 k11(i) = -0.5*y1(i); 11 k12(i) = 4 - 0.3* y2(i) - 0.1* y1(i); 154
12 y11 = y1(i) + k11(i) * h/2; 13 y21 = y2(i) + k12(i) * h/2; 14 k21(i) = -0.5* y11; 15 k22(i) = 4 - 0.3* y21 - 0.1* y11; 16 y11 = y1(i) + k21(i) * h/2; 17 y21 = y2(i) + k22(i) * h/2; 18 k31(i) = -0.5* y11; 19 k32(i) = 4 - 0.3* y21 - 0.1* y11; 20 y11 = y1(i) + k31(i) * h; 21 y21 = y2(i) + k32(i) * h; 22 k41(i) = -0.5* y11; 23 k42(i) = 4 - 0.3* y21 - 0.1* y11; 24 y1(i+1) = y1(i) + ( k11(i) + 2*k21(i) + 2*k31(i) + k41(i) )*h / 6; 25 y2(i+1) = y2(i) + ( k12(i) + 2*k22(i) + 2*k32(i) + k42(i) ) *h/ 6; 26 end 27 disp(” using f o u r t h order RK method”) 28 disp(x,”x =”) 29 disp(y1 (1:5) ,”y1 = ”) 30 disp(y2 (1:5) ,”y2 = ”) Scilab code Exa 25.11 Solving systems of ODEs 1 clc; 2 clear; 3 function yp=f(x,y) 4 yp=[y(2) ; -16.1*y(1)]; 5 endfunction 6 x=0:0.1:4 7 y0 =[0.1 0]; 8 sol=ode(y0 ,0,x,f); 9 count =1; 10 disp(sol) 11 for i=1:2:81 155
12 a(count)=sol(i); 13 b(count)=sol(i+1); 14 count=count +1; 15 end 16 plot(x,a) 17 plot(x,b,”.−”) 18 h1=legend ([”y1 , y3”,”y2 , y4”]) 19 xtitle(”y vs x”,”x”,”y”) 20 function yp=g(x,y) 21 yp=[y(2) ; -16.1* sin(y(1))]; 22 endfunction 23 sol=ode(y0 ,0,x,g); 24 count =1; 25 disp(sol) 26 for i=1:2:81 27 a(count)=sol(i); 28 b(count)=sol(i+1); 29 count=count +1; 30 end 31 plot(x,a) 32 plot(x,b,”.−”) 33 clf(); 34 y0=[%pi/4 0]; 35 sol=ode(y0 ,0,x,f); 36 count =1; 37 disp(sol) 38 for i=1:2:81 39 a(count)=sol(i); 40 b(count)=sol(i+1); 41 count=count +1; 42 end 43 plot(x,a) 44 plot(x,b,”.−”) 45 46 xtitle(”y vs x”,”x”,”y”) 47 sol=ode(y0 ,0,x,g); 48 count =1; 49 disp(sol) 156
Figure 25.2: Solving systems of ODEs 50 for i=1:2:81 51 a(count)=sol(i); 52 b(count)=sol(i+1); 53 count=count +1; 54 end 55 plot(x,a, ’ o− ’ ) 56 plot(x,b,”x−”) 57 h1=legend ([”y1”,”y2”,”y3”,”y4”]) 157
Figure 25.3: Solving systems of ODEs 158
Scilab code Exa 25.12 Adaptive fourth order RK method 1 // c l c ( ) 2 disp(” f ( x , y ) = 4∗ exp ( 0 . 8 ∗ x ) − 0.5∗ y”) 3 // f ’ ( x , y ) = 4∗ exp ( 0 . 8 ∗ x ) − 0.5∗ y 4 x = 0:2:2; 5 y(1) = 2; 6 h =2; 7 t = 14.84392; 8 k1 = 4*exp (0.8*x(1)) - 0.5*y(1); 9 x1 = x(1) + h/2; 10 y1 = y(1) + k1*h/2; 11 k2 = 4*exp (0.8* x1) - 0.5* y1; 12 x1 = x(1) + h/2; 13 y1 = y(1) + k2*h/2; 14 k3 = 4*exp (0.8* x1) - 0.5* y1; 15 x1 = x(1) + h; 16 y1 = y(1) + k3*h; 17 k4 = 4*exp (0.8* x1) - 0.5* y1; 18 y(2) = y(1) + (k1 + 2*k2 + 2*k3 + k4)*h/6; 19 e = (t - y(2))/(t); 20 disp(y(1:2) ,”y by h = 2 i s ”) 21 disp(e,” e r r o r = ”) 22 h = 1; 23 x = 0:h:2; 24 for i=1:3 25 k1(i) = 4*exp (0.8*x(i)) - 0.5*y(i); 26 x1 = x(i) + h/2; 27 y1 = y(i) + k1(i)*h/2; 28 k2(i) = 4*exp (0.8* x1) - 0.5* y1; 29 x1 = x(i) + h/2; 30 y1 = y(i) + k2(i)*h/2; 31 k3(i) = 4*exp (0.8* x1) - 0.5* y1; 32 x1 = x(i) + h; 33 y1 = y(i) + k3(i)*h; 34 k4(i) = 4*exp (0.8* x1) - 0.5* y1; 35 y(i+1) = y(i) + (k1(i) + 2*k2(i) + 2*k3(i) + k4( i))*h/6; 159
36 end 37 e = (t - (y(3)))/t; 38 disp(y(1:3) ,”y by h = 1 i s ”) 39 disp(e,” e r r o r = ”) Scilab code Exa 25.13 Runga kutta fehlberg method 1 // c l c ( ) 2 disp(” f ( x , y ) = 4∗ exp ( 0 . 8 ∗ x ) − 0.5∗ y”) 3 // f ’ ( x , y ) = 4∗ exp ( 0 . 8 ∗ x ) − 0.5∗ y 4 h =2; 5 x = 0:h:2; 6 y(1) = 2; 7 t = 14.84392; 8 k1 = 4*exp (0.8*x(1)) - 0.5*y(1); 9 x1 = x(1) + h/5; 10 y1 = y(1) + k1*h/5; 11 k2 = 4*exp (0.8* x1) - 0.5* y1; 12 x1 = x(1) + 3*h/10; 13 y1 = y(1) + 3*k1*h/40 + 9*k2*h/40; 14 k3 = 4*exp (0.8* x1) - 0.5* y1; 15 x1 = x(1) + 3*h/5; 16 y1 = y(1) + 3*k1*h/10 - 9*k2*h/10 + 6*k3*h/5; 17 k4 = 4*exp (0.8* x1) - 0.5* y1; 18 x1 = x(1) + h; 19 y1 = y(1) -11*k1*h/54 + 5*k2*h/2 - 70*k3*h/27 + 35* k4*h/27; 20 k5 = 4*exp (0.8* x1) - 0.5* y1; 21 x1 = x(1) + 7*h/8; 22 y1 = y(1) + 1631* k1*h/55296 + 175* k2*h/512 + 575* k3* h/13824 + 44275* k4*h/110592 +253* k5*h/4096; 23 k6 = 4*exp (0.8* x1) - 0.5* y1; 24 y1 = y(1) + (37* k1/378 + 250* k3 /621 + 125* k4 /594 + 512* k6 /1771)*h; 25 y2 = y(1) + (2825* k1 /27648 + 18575* k3 /48384 + 13525* 160
Figure 25.4: Adaptive Fourth order RK scheme k4 /55296 + 277* k5 /14336 + k6/4)*h; 26 disp(y1 ,”y ( f o u r t h order p r e d i c t i o n ) = ”) 27 disp(y2 ,”y ( f i f t h order p r e d i c t i o n ) = ”) Scilab code Exa 25.14 Adaptive Fourth order RK scheme 1 clc; 2 clear; 3 function yp=f(x,y) 4 yp =10* exp(-(x-2) ^2/(2*(0.075^2))) -0.6*y 5 endfunction 161
6 x=0:0.1:4 7 y0 =0.5; 8 sol=ode(y0 ,0,x,f); 9 plot(x,sol) 10 xtitle(”y vs x”,”x”,”y”) 162
Chapter 26 Stiffness and multistep methods Scilab code Exa 26.1 Explicit and Implicit Euler 1 clc; 2 clear; 3 function yp=f(t,y) 4 yp= -1000*y+3000 -2000* exp(-t) 5 endfunction 6 y0=0; 7 // e x p l i c i t e u l e r 8 h1 =0.0005; 9 y1(1)=y0; 10 count =2; 11 t=0:0.0001:0.006 12 for i=0:0.0001:0.0059 13 y1(count)=y1(count -1)+f(i,y1(count -1))*h1 14 count=count +1; 15 end 16 plot(t,y1) 17 h2 =0.0015; 18 y2(1)=y0; 19 count =2; 20 t=0:0.0001:0.006 21 for i=0:0.0001:0.0059 163
22 y2(count)=y2(count -1)+f(i,y2(count -1))*h2 23 count=count +1; 24 end 25 plot(t,y2 , ’ .− ’ ) 26 xtitle(”y vs t ”,” t ”,”y”) 27 h=legend ([”h −0.0005 ”,”h=0.0015 ”]) 28 clf(); 29 // i m p l i c i t order 30 h3 =0.05; 31 y3(1)=y0; 32 count =2; 33 t=0:0.01:0.4 34 for j=0:0.01:0.39 35 y3(count)=(y3(count -1) +3000*h3 -2000* h3*exp(-(j +0.01)))/(1+1000* h3) 36 count=count +1; 37 end 38 39 plot(t,y3) 40 xtitle(”y vs t ”,” t ”,”y”) Scilab code Exa 26.2 Non self starting Heun method 1 // c l c ( ) 2 disp(” f ( x , y ) = 4∗ exp ( 0 . 8 ∗ x ) − 0.5∗ y”) 3 // f ’ ( x , y ) = 4∗ exp ( 0 . 8 ∗ x ) − 0.5∗ y 4 h = 1; 5 x=0:h:4; 6 y(1) = 2; 7 x1 = -1; 8 y1 = -0.3929953; 164
Figure 26.1: Explicit and Implicit Euler 165
Figure 26.2: Explicit and Implicit Euler 166
9 y10 = y1 + (4* exp (0.8*x(1)) - 0.5*y(1))*2; 10 y11 = y(1) + (4* exp (0.8*x(1)) - 0.5*y(1) + 4*exp (0.8*x(2)) - 0.5* y10)*h/2; 11 y12 = y(1) + (3 + 4*exp (0.8*x(2)) - 0.5* y11)*h/2; 12 t = 6.360865; 13 y20 = y(1) + (4* exp (0.8*x(2)) - 0.5*t) *2; 14 y21 = t + (4* exp (0.8*x(2)) - 0.5*t + 4*exp (0.8*x(3)) - 0.5* y20)*h/2; 15 disp(y21 ,” the f i r s t c o r r e c t o r y i e l d s y = ”) 16 t = 14.84392 17 e = (t - y21)*100/t; 18 disp(”%”,e,” e r r o r = ”) Scilab code Exa 26.3 Estimate of per step truncation error 1 // c l c ( ) 2 x1 = 1; 3 x2 = 2; 4 y1 = 6.194631; 5 y2 = 14.84392; 6 y10 = 5.607005; 7 y11 = 6.360865; 8 y20 = 13.44346; 9 y21 = 15.30224; 10 Ec1 = -(y11 - y10)/5; 11 disp(Ec1 ,”Ec ( x = 1) = ”) 12 e1 = y1 - y11; 13 disp(e1 ,” true e r r o r ( x = 1) = ”) 14 Ec2 = -(y21 - y20)/5; 15 disp(Ec2 ,”Ec ( x = 2) = ”) 16 e2 = y2 - y21; 17 disp(e2 ,” true e r r o r ( x = 2) = ”) 167
Scilab code Exa 26.4 Effect of modifier on Predictor Corrector results 1 // c l c ( ) 2 // c l c ( ) 3 x0 = 0; 4 x1 = 1; 5 x2 = 2; 6 y1 = 6.194631; 7 y2 = 14.84392; 8 y10 = 5.607005; 9 y11 = 6.360865; 10 y1m = y11 - (y11 - y10)/5; 11 e = (y1 - y1m)*100/ y1; 12 disp(y1m ,”ym”) 13 disp(”%”,e,” e r r o r = ”) 14 y20 =2+(4* exp (0.8* x1) - 0.5* y1m)*2; 15 e2 = (y2 - y20)*100/ y2; 16 disp(y20 ,” y20 = ”) 17 disp(”%”,e2 ,” e r r o r = ”) 18 y2o = y20 + 4* (y11 - y10)/5; 19 e2 = (y2 - y2o)*100/ y2; 20 disp(y2o ,” y20 = ”) 21 disp(”%”,e2 ,” e r r o r = ”) 22 y21 = 15.21178; 23 y23 = y21 - (y21 - y20)/5; 24 disp(y23 ,”y2 = ”) 25 e3 = (y2 - y23)*100/ y2; 26 disp(”%”,e3 ,” e r r o r = ”) Scilab code Exa 26.5 Milnes Method 1 // c l c ( ) 2 disp(” f ( x , y ) = 4∗ exp ( 0 . 8 ∗ x ) − 0.5∗ y”) 3 // f ’ ( x , y ) = 4∗ exp ( 0 . 8 ∗ x ) − 0.5∗ y 4 h = 1; 168
5 x = -3:h:4; 6 y(1) = -4.547302; 7 y(2) = -2.306160; 8 y(3) = -0.3929953; 9 y(4) = 2; 10 y1(4) = 2; 11 for i = 4:7; 12 y(i+1) = y(i-3) + 4*h*(2*(4* exp (0.8*x(i)) - 0.5* y(i)) - 4*exp (0.8*x(i-1)) + 0.5*y(i-1) + 2*(4* exp (0.8*x(i-2)) - 0.5*y(i-2)))/3; 13 y1(i+1) = y(i-1) + h*(4* exp (0.8*x(i-1)) - 0.5*y( i-1) +4 * (4* exp (0.8*x(i)) - 0.5*y(i)) + 4* exp (0.8*x(i+1)) - 0.5*y(i+1))/3; 14 end 15 disp(x(4:8) ,”x = ”) 16 disp(y(4:8) ,”y0 = ”) 17 disp(y1 (4:8) ,” c o r r e c t e d y1 = ”) Scilab code Exa 26.6 Fourth order Adams method 1 // c l c ( ) 2 disp(” f ( x , y ) = 4∗ exp ( 0 . 8 ∗ x ) − 0.5∗ y”) 3 // f ’ ( x , y ) = 4∗ exp ( 0 . 8 ∗ x ) − 0.5∗ y 4 h = 1; 5 x = -3:h:4; 6 y(1) = -4.547302; 7 y(2) = -2.306160; 8 y(3) = -0.3929953; 9 y(4) = 2; 10 m(4) = y(4); 11 for i =4:7 12 y(i+1) = y(i) + h *(55* (4* exp (0.8*x(i)) - 0.5*y (i)) / 24 - 59 * (4* exp (0.8*x(i-1)) - 0.5*y(i -1)) / 24 + 37*(4* exp (0.8*x(i-2)) - 0.5*y(i -2))/24 - 9*(4* exp (0.8*x(i-3)) - 0.5*y(i-3)) 169
/24); 13 m(i+1) = y(i+1) 14 y(i+1) = y(i) + h*(9*(4* exp (0.8*x(i+1)) - 0.5*y( i+1))/24 +19*(4* exp (0.8*x(i)) - 0.5*y(i))/24 - 5*(4* exp (0.8*x(i-1)) - 0.5*y(i-1))/24 + (4* exp (0.8*x(i-2)) - 0.5*y(i-2))/24); 15 end 16 disp(x(4:8) ,”x = ”) 17 disp(m(4:8) ,”y0 = ”) 18 disp(y(4:8) ,”y1 = ”) Scilab code Exa 26.7 stability of Milnes and Fourth order Adams method 1 // c l c ( ) 2 //dy/dx = −y 3 //y = exp(−x ) 4 h = 0.5; 5 x = -1.5:h:10; 6 y(1) = exp(-x(1)); 7 y(2) = exp(-x(2)); 8 y(3) = exp(-x(3)); 9 y(4) = 1; 10 m(4) = y(4); 11 for i = 4:23; 12 y(i+1) = y(i-3) + 4*h*(2*( -y(i)) + y(i-1) + 2*(- y(i-2)))/3; 13 m(i+1) = y(i+1); 14 y(i+1) = y(i-1) + h*(-y(i-1) +4 * (-y(i)) + (-y( i+1)))/3; 15 end 16 disp(x(4:24) ,”x = ”) 17 disp(m(4:24) ,”y0 ( milnes method ) = ”) 18 disp(y(4:24) ,” c o r r e c t e d y1 ( milnes method ) = ”) 19 for i =4:23 20 y(i+1) = y(i) + h *(55* (-y(i)) / 24 - 59 * (-y( 170
i-1)) / 24 + 37*(-y(i-2))/24 - 9*(-y(i-3)) /24); 21 m(i+1) = y(i+1) 22 y(i+1) = y(i) + h*(9*( -y(i+1))/24 +19*( -y(i))/24 - 5*(-y(i-1))/24 + (-y(i-2))/24); 23 end 24 disp(m(4:23) ,”y0 ( f o u r t h order adams method ) = ”) 25 disp(y(4:23) ,”y1 ( f o u r t h order adams method ) = ”) 171
Chapter 27 Boundary Value and Eigen Value problems Scilab code Exa 27.1 The shooting method 1 // c l c ( ) 2 l = 10; //m 3 h = 0.01; //mˆ( −2) 4 Ta = 20; 5 T0 = 40; 6 T10 = 200; 7 //dT/dx = z 8 // dz/dx = h ’ ( T−Ta) 9 // we use z = 10 as i n i t i a l guess and i n t e g r a t i n g above equations simultaneously , s o l v i n g using RK4 method , we get T10 = 168.3797 10 // s i m i l a r l y , with z = 20 , T10 = 285.898 11 z01 = 10; 12 z02 = 20; 13 T10a = 168.3797; 14 T10b = 285.898; 15 z0 = z01 + (z02 - z01)*(T10 - T10a)/( T10b - T10a); 16 disp(z0 ,” z0 = ”) 17 disp(” t h i s value of z can be used to get the c o r r e c t 172
Figure 27.1: The shooting method for non linear problems value ”) Scilab code Exa 27.2 The shooting method for non linear problems 1 // c l c ( ) 2 z(1) = 10.0035; 3 T(1) = 40; 4 Ta = 20; 5 h = 0.5; 6 for i = 1:20 7 k11(i) = z(i); 173
8 k12(i) = 5*10^ -8*(T(i) - Ta)^4; 9 z1 = z(i) + h/2; 10 T1 = T(i) + h/2; 11 k21(i) = z1; 12 k22(i) = 5*10^ -8*( T1 - Ta)^4; 13 z1 = z(i) + h/2; 14 T1 = T(i) + h/2; 15 k31(i) = z1; 16 k32(i) = 5*10^ -8*( T1 - Ta)^4; 17 z1 = z(i) + h; 18 T1 = T(i) + h; 19 k41(i) = z1; 20 k42(i) = 5*10^ -8*( T1 - Ta)^4; 21 T(i+1) = T(i) + ( k11(i) + 2* k21(i) + 2*k31(i) + k41(i))*h/6; 22 z(i+1) = z(i) + ( k12(i) + 2* k22(i) + 2*k32(i) + k42(i))*h/6; 23 end 24 x=0:0.5:10; 25 plot(x,T(1:21)) 26 xtitle(”T vs x”,”x”,”T”) Scilab code Exa 27.3 Finite Difference Approximation 1 clc; 2 clear; 3 h=0.01; 4 delx =2; 5 x=2+h*delx ^2; 6 a=[x -1 0 0;-1 x -1 0;0 -1 x -1;0 0 -1 x]; 7 b=[40.8; 0.8; 0.8; 200.8]; 8 T=linsolve(a,b); 9 disp(”The temperature at the i n t e r i o r nodes : ”) 10 disp(abs(T)) 174
Scilab code Exa 27.4 Mass Spring System 1 clc; 2 clear; 3 m1=40; // kg 4 m2=40; // kg 5 k=200; //N/m 6 sqw=poly(0,” s ”); 7 p=sqw ^2 -20* sqw +75; 8 r=roots(p); 9 f1=(r(1))^0.5; 10 f2=(r(2))^0.5; 11 Tp1 =(2* %pi)/f1; 12 Tp2 =(2* %pi)/f2; 13 // f o r f i r s t mode 14 disp(”For f i r s t mode : ”) 15 disp(Tp1 ,” Period of o s c i l l a t i o n : ”) 16 disp(”A1=−A2”) 17 disp(” −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 18 // f o r f i r s t mode 19 disp(”For second mode : ”) 20 disp(Tp2 ,” Period of o s c i l l a t i o n : ”) 21 disp(”A1=A2”) Scilab code Exa 27.5 Axially Loaded column 1 clc; 2 clear; 3 E=10*10^9; //Pa 4 I=1.25*10^ -5; //mˆ4 175
5 L=3; //m 6 for i=1:8 7 p=i*%pi/L; 8 P=i^2*( %pi)^2*E*I/(L^2*1000); 9 disp(i,”n=”) 10 disp(”mˆ−2”,p,”p=”) 11 disp(”kN”,P,”P=”) 12 disp(” −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 13 end Scilab code Exa 27.6 Polynomial Method 1 clc; 2 clear; 3 E=10*10^9; //Pa 4 I=1.25*10^ -5; //mˆ4 5 L=3; //m 6 true =[1.0472 2.0944 3.1416 4.1888]; 7 // part a 8 h1 =3/2; 9 p=poly(0,” s ”) 10 a=2-h1^2*p^2; 11 x=roots(a); 12 e=abs(abs(x(1))-true (1))*100/ true (1); 13 disp(x,”p=”) 14 disp(e,” e r r o r=”) 15 disp(” −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−” ) 16 // part b 17 h2 =3/3; 18 p=poly(0,” s ”) 19 a=(2-h2^2*p^2)^2 - 1; 176
20 x=roots(a); 21 e(1)=abs(abs(x(1))-true (2))*100/ true (2); 22 e(2)=abs(abs(x(2))-true (1))*100/ true (1); 23 disp(x,”p=”) 24 disp(e,” e r r o r=”) 25 disp(” −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 26 // part c 27 h3 =3/4; 28 p=poly(0,” s ”) 29 a=(2-h3^2*p^2)^3 - 2*(2-h3^2*p^2); 30 x=roots(a); 31 e(1)=abs(abs(x(1))-true (3))*100/ true (3); 32 e(2)=abs(abs(x(2))-true (2))*100/ true (2); 33 e(3)=abs(abs(x(3))-true (1))*100/ true (1); 34 disp(x,”p=”) 35 disp(e,” e r r o r=”) 36 disp(” −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 37 38 39 // part d 40 h4 =3/5; 41 p=poly(0,” s ”) 42 a=(2-h4^2*p^2)^4 - 3*(2-h4^2*p^2)^2 + 1; 43 x=roots(a); 44 e(1)=abs(abs(x(1))-true (4))*100/ true (4); 45 e(2)=abs(abs(x(2))-true (3))*100/ true (3); 46 e(3)=abs(abs(x(3))-true (2))*100/ true (2); 47 e(4)=abs(abs(x(4))-true (1))*100/ true (1); 48 disp(x,”p=”) 49 disp(e,” e r r o r=”) 50 disp(” −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 177
Scilab code Exa 27.7 Power Method Highest Eigenvalue 1 clc; 2 clear; 3 a=[3.556 -1.668 0; -1.778 3.556 -1.778;0 -1.778 3.556]; 4 b=[1.778;0;1.778]; 5 ea =100; 6 count =1; 7 while ea >0.1 8 maxim=b(1); 9 for i=2:3 10 if abs(b(i))>abs(maxim) then 11 maxim=b(i); 12 end 13 end 14 eigen(count)=maxim; 15 b=a*(b./ maxim); 16 if count ==1 then 17 ea =20; 18 count=count +1; 19 20 else 21 ea=abs(eigen(count)-eigen(count -1))*100/ abs( eigen(count)); 22 count=count +1; 23 end 24 end 25 disp(eigen(count -1),”The l a r g e s t e i g e n value ”) Scilab code Exa 27.8 Power Method Lowest Eigenvalue 178
1 clc; 2 clear; 3 a=[3.556 -1.668 0; -1.778 3.556 -1.778;0 -1.778 3.556]; 4 b=[1.778;0;1.778]; 5 ea =100; 6 count =1; 7 ai=inv(a); 8 while ea >4 9 maxim=b(1); 10 for i=2:3 11 if abs(b(i))>abs(maxim) then 12 maxim=b(i); 13 end 14 end 15 eigen(count)=maxim; 16 b=ai*(b./ maxim); 17 if count ==1 then 18 ea=20; 19 count=count +1; 20 21 else 22 ea=abs(eigen(count)-eigen(count -1))*100/ abs( eigen(count)); 23 count=count +1; 24 end 25 end 26 disp ((1/ eigen(count -1))^0.5,”The s m a l l e s t e i g e n value ”) Scilab code Exa 27.9 Eigenvalues and ODEs 1 clc; 2 clear; 3 function yp=predprey(t,y) 179
4 yp =[1.2*y(1) -0.6*y(1)*y(2) ; -0.8*y(2) +0.3*y(1)*y (2)]; 5 endfunction 6 t=0:0.1:20; 7 y0=[2 1]; 8 sol=ode(y0 ,0,t,predprey); 9 count =1; 10 for i=1:2:401 11 x(count)=sol(i); 12 z(count)=sol(i+1); 13 count=count +1; 14 end 15 16 plot(t,x) 17 plot(t,z) 18 xtitle(”y vs t ”, ” t ”,”y”) 19 clf(); 20 plot(x,z) 21 xtitle(” space−space p l o t ( y1 vs y2 ) ”, ”y1”,”y2”) Scilab code Exa 27.10.a Stiff ODEs 1 clc; 2 clear; 3 function yp=vanderpol(t,y) 4 yp=[y(2) ;1*(1 -y(1) ^2)*y(2)-y(1)] 5 endfunction 6 y0=[1 1]; 7 t=0:0.1:20; 180
Figure 27.2: Eigenvalues and ODEs 181
Figure 27.3: Eigenvalues and ODEs 182
Figure 27.4: Stiff ODEs 183
Figure 27.5: Stiff ODEs 8 sol=ode(y0 ,0,t,vanderpol); 9 count =1; 10 for i=1:2:401 11 x(count)=sol(i); 12 count=count +1; 13 end 14 plot(t,x) 15 xtitle(”y vs t ”,” t ”,”y”) Scilab code Exa 27.10.b Stiff ODEs 184
1 clc; 2 clear; 3 clf(); 4 function yp=vanderpol1(t,y) 5 yp=[y(2) ;1000*(1 -y(1) ^2)*y(2)-y(1)] 6 endfunction 7 t=0:3000; 8 yo=[1 1]; 9 sol=ode(yo ,0,t,vanderpol1); 10 count =1; 11 12 for i=1:2:6001 13 z(count)=sol(i)*-1; 14 count=count +1; 15 end 16 17 plot(t,z) 18 xtitle(”y vs t ”,” t ”,”y”) Scilab code Exa 27.11 Solving ODEs 1 clc; 2 clear; 3 function yp=predprey(t,y) 4 yp =[1.2*y(1) -0.6*y(1)*y(2) ; -0.8*y(2) +0.3*y(1)*y (2)]; 5 endfunction 6 t=0:10; 7 y0=[2 1]; 8 sol=ode(y0 ,0,t,predprey); 9 count =0; 10 for i=1:2:22 11 disp(count ,” i s t e p=”) 12 disp(count ,” time=”) 13 disp(sol(i),”y1=”) 185
14 disp(sol(i+1),”y2=”) 15 disp(” −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 16 count=count +1; 17 end 186
Chapter 29 Finite Difference Elliptic Equations Scilab code Exa 29.1 Temperature of a heated plate with fixed boundary conditions 1 // c l c ( ) 2 wf = 1.5; 3 for i = 1:5 4 for j = 1:5 5 T(i,j) = 0; 6 if j == 1 then 7 T(i,j) = 0; //C 8 else 9 if j == 5 then 10 T(i,j) = 100; //C 11 end 12 end 13 if i == 1 then 14 T(i,j) = 75; //C 15 else 16 if i == 5 then 17 T(i,j) = 50; //C 18 end 19 end 187
20 end 21 end 22 e = 100; 23 while e>1 24 for i=1:5 25 for j = 1:5 26 if i>1 & j>1 & i<5 & j<5 then 27 Tn(i,j) = (T(i + 1,j) + T(i-1,j) + T(i,j+1) + T(i,j-1))/4; 28 Tn(i,j) = wf * Tn(i,j) + (1-wf)*T(i,j); 29 if i==2 & j==2 then 30 e = abs((Tn(i,j) - T(i,j)) * 100/ (Tn(i, j))); 31 end 32 T(i,j) = Tn(i,j); 33 end 34 end 35 end 36 end 37 disp(T(2:4 ,2:4) ,” f o r e r r o r < 1 , the temperatures are ”) Scilab code Exa 29.2 Flux distribution for a heated plate 1 // c l c ( ) 2 k = 0.49; // c a l /( s . cm .C) 3 T1 = 33.29755; //C 4 T2 = 75; //C 5 T3 = 63.21152; //C 6 T4 = 0; //C 7 qx = -k * (T1 - T2) / (2*10); 8 qy = -k * (T3 - T4) / (2*10); 9 disp(qx ,”qx = ”) 10 disp(qy ,”qy = ”) 11 q = (qx^2 + qy^2) ^0.5; 188
12 t = (atan(qy/qx)); 13 r = t * 180/( %pi); 14 disp(r,”Thus , the f l u x i s d i r e c e d at the angle ”) Scilab code Exa 29.3 Heated plate with a insulated edge 1 // c l c ( ) 2 // since , i n s u l a t i o n i s done at one end , the g e n e r a l equation becomes ( j =0) , 3 //T( i +1 ,0) + T( i −1 ,0) + 2∗T( i , 1 ) −2∗y ∗(dT/dy ) − 4∗T( i , 0 ) = 0 4 // the d e r i v a t i v e i s zero and we get , T( i +1 ,0) + T( i −1 ,0) + 2∗T( i , 1 ) − 4∗T( i , 0 ) = 0 5 // the simultaneous equations obtained can be w r i t t e n in matrix form as f o l l o w s , 6 A = [4,-1,0,-2,0,0,0,0,0,0,0,0;-1,4,-1,0,-2,0,0,0,0,0,0,0;0,-1,4,0,0,- 7 disp(A,”A = ”) 8 B = [75;0;50;75;0;50;75;0;50;175;100;150]; 9 disp(B,”B = ”) 10 T = inv(A)*B; 11 T10 = det(T(1,1)); 12 T20 = det(T(2,1)); 13 T30 = det(T(3,1)); 14 T11 = det(T(4,1)); 15 T21 = det(T(5,1)); 16 T31 = det(T(6,1)); 17 T12 = det(T(7,1)); 18 T22 = det(T(8,1)); 19 T32 = det(T(9,1)); 20 T13 = det(T(10 ,1)); 21 T23 = det(T(11 ,1)); 22 T33 = det(T(12 ,1)); 23 disp(T10 ,”T10 = ”) 189
24 disp(T20 ,”T20 = ”) 25 disp(T30 ,”T30 = ”) 26 disp(T11 ,”T11 = ”) 27 disp(T21 ,”T21 = ”) 28 disp(T31 ,”T31 = ”) 29 disp(T12 ,”T12 = ”) 30 disp(T22 ,”T22 = ”) 31 disp(T32 ,”T32 = ”) 32 disp(T13 ,”T13 = ”) 33 disp(T23 ,”T23 = ”) 34 disp(T33 ,”T33 = ”) Scilab code Exa 29.4 Heated plate with an irregular boundary 1 // c l c ( ) 2 // f o r i r r e g u l a r boundary , 3 //x = y 4 alpha1 = 0.732; 5 beta1 = 0.732; 6 alpha2 = 1; 7 beta2 = 1; 8 // s u b s t i t u t i n g the above value to obtain simultaneous equation we get the f o l l o w i n g matrix , 9 A = [4,-0.845,0,-0.845,0,0,0,0,0;-1,4,-1,0,-1,0,0,0,0;0,-1,4,0,0,-1,0, 10 disp(A,”A = ”) 11 B = [173.2;75;125;75;0;50;175;100;150]; 12 disp(B,”B = ”) 13 T = inv(A)*B; 14 T11 = det(T(1,1)); 15 T21 = det(T(2,1)); 16 T31 = det(T(3,1)); 17 T12 = det(T(4,1)); 190
18 T22 = det(T(5,1)); 19 T32 = det(T(6,1)); 20 T13 = det(T(7,1)); 21 T23 = det(T(8,1)); 22 T33 = det(T(9,1)); 23 disp(T11 ,”T11 = ”) 24 disp(T21 ,”T21 = ”) 25 disp(T31 ,”T31 = ”) 26 disp(T12 ,”T12 = ”) 27 disp(T22 ,”T22 = ”) 28 disp(T32 ,”T32 = ”) 29 disp(T13 ,”T13 = ”) 30 disp(T23 ,”T23 = ”) 31 disp(T33 ,”T33 = ”) 191
Chapter 30 Finite Difference Parabolic Equations Scilab code Exa 30.1 Explicit solution of a one dimensional heat conduction equati 1 // c l c ( ) 2 l = 10; //cm 3 k1 = 0.49; // c a l /( s . cm .C) 4 x = 2; //cm 5 dt = 0.1; // sec 6 T0 = 100; //C 7 T10 = 50; //C 8 C = 0.2174; // c a l /( g .C) 9 rho = 2.7; //g/cmˆ3 10 k = k1/(C*rho); 11 L = k * dt / x^2; 12 disp(L,”L =”) 13 T(1,1) = 100; 14 T(1,2) = 0; 15 T(1,3) = 0; 16 T(1,4) = 0; 17 T(1,5) = 0; 18 T(1,6) = 50; 19 T(2,1) = 100; 192
20 T(2,6) = 50; 21 for i = 2:3 22 for j = 2:5 23 T(i,j) = T(i-1,j) + L * (T(i-1,j+1) - 2* T(i -1,j) + T(i-1,j-1)); 24 end 25 end 26 disp(T(2,2),”T11 = ”) 27 disp(T(2,3),”T12 = ”) 28 disp(T(2,4),”T13 = ”) 29 disp(T(2,5),”T14 = ”) 30 disp(T(3,2),”T21 = ”) 31 disp(T(3,3),”T22 = ”) 32 disp(T(3,4),”T23 = ”) 33 disp(T(3,5),”T24 = ”) Scilab code Exa 30.2 Simple implicit solution of a heat conduction equation 1 // c l c ( ) 2 l = 10; //cm 3 k1 = 0.49; // c a l /( s . cm .C) 4 x = 2; //cm 5 dt = 0.1; // sec 6 C = 0.2174; // c a l /( g .C) 7 rho = 2.7; //g/cmˆ3 8 k = k1/(C*rho); 9 L = k * dt / x^2; 10 disp(L,”L =”) 11 //now , at t = 0 , 1.04175 ∗T’ 1 + 0.020875 ∗T’ 2 = 0 + 0.020875∗100 12 // s i m i l a r l y g e t t i n g other simultaneous eqautions , we get the f o l l o w i n g matrix 13 A = [1.04175 , -0.020875 ,0 ,0; -0.020875 ,1.04175 , -0.020875 ,0;0 , -0.020875 , 193
14 B = [2.0875;0;0;1.04375]; 15 X = inv(A) * B; 16 T11 = det(X(1,1)); 17 T12 = det(X(2,1)); 18 T13 = det(X(3,1)); 19 T14 = det(X(4,1)); 20 disp(” f o r t = 0”) 21 disp(T11 ,”T11 = ”) 22 disp(T12 ,”T12 = ”) 23 disp(T13 ,”T13 = ”) 24 disp(T14 ,”T14 = ”) 25 // to s o l v e f o r t = 0 . 2 , the r i g h t hand s i d e v e c t o r i s modified to D, 26 D = [4.09215;0.04059;0.02090;2.04069]; 27 Y = inv(A)*D; 28 T21 = det(Y(1,1)); 29 T22 = det(Y(2,1)); 30 T23 = det(Y(3,1)); 31 T24 = det(Y(4,1)); 32 disp(” f o r t = 0 .2 ”) 33 disp(T21 ,”T21 = ”) 34 disp(T22 ,”T22 = ”) 35 disp(T23 ,”T23 = ”) 36 disp(T24 ,”T24 = ”) Scilab code Exa 30.3 Crank Nicolson solution to the heat conduction equation 1 // c l c ( ) 2 // using crank n i c o l s o n method , we get simultaneous equations which can be s i m p l i f i e d to f o l l o w i n g matrics 3 A = [2.04175 , -0.020875 ,0 ,0; -0.020875 ,2.04175 , -0.020875 ,0;0 , -0.020875 , 4 B = [4.175;0;0;2.0875]; 194
5 X = inv(A)*B; 6 disp(” at t = 0. 1 s ”) 7 disp(det(X(1,1)),”T11 = ”) 8 disp(det(X(2,1)),”T12 = ”) 9 disp(det(X(3,1)),”T13 = ”) 10 disp(det(X(4,1)),”T14 = ”) 11 C = [8.1801;0.0841;0.0427;4.0901]; 12 Y = inv(A)*C; 13 disp(” at t = 0. 2 s ”) 14 disp(det(Y(1,1)),”T21 = ”) 15 disp(det(Y(2,1)),”T22 = ”) 16 disp(det(Y(3,1)),”T23 = ”) 17 disp(det(Y(4,1)),”T24 = ”) Scilab code Exa 30.4 Comparison of Analytical and Numerical solution 1 // c l c ( ) 2 x = 2; //cm 3 L = 10; //cm 4 k = 0.835; //cmˆ2/ s 5 t = 10; // s 6 n = 1:10000; 7 T = 324.* (x/L + sum (2.*(( -1)^n).*sin(n.*x/L).*exp(- n^2.* %pi ^2.* k.* t / L^2)/(n.*%pi))); 8 disp(T,”T[ 2 , 1 0 ] a n a l y t i c a l = ”) 9 Tex = 64.97; 10 disp(Tex ,”T[ 2 , 1 0 ] e x p l i c i t = ”) 11 Tim = 64.49; 12 disp(Tim ,”T[ 2 , 1 0 ] i m p l i c i t = ”) 13 Tcn = 64.73; 14 disp(Tcn ,”T[ 2 , 1 0 ] crank−n i c o l s o n = ”) Scilab code Exa 30.5 ADI Method 195
1 // c l c ( ) 2 x = 10; //cm 3 L = 0.0835; 4 t1 = 5; 5 // f o r f i r s t step t = 5 i s a p p l i e d to nodes ( 1 , 1 ) , ( 1 , 2 ) and ( 1 , 3 ) to y i e l d f o l l o w i n g matrices 6 A = [2.167 , -0.0835 ,0; -0.0835 ,2.167 , -0.0835;0 , -0.0835 ,2.167]; 7 B = [6.2625;6.2625;14.6125]; 8 X = inv(A)*B; 9 disp(”At t = 5 s ”) 10 disp(det(X(1,1)),”T11 = ”) 11 disp(det(X(2,1)),”T12 = ”) 12 disp(det(X(3,1)),”T13 = ”) 13 // s i m i l a r l y we get , 14 T21 = 0.1274; 15 T22 = 0.2900; 16 T23 = 4.1291; 17 T31 = 2.0181; 18 T32 = 2.2477; 19 T33 = 6.0256; 20 disp(T21 ,”T21 = ”) 21 disp(T22 ,”T22 = ”) 22 disp(T23 ,”T23 = ”) 23 disp(T31 ,”T31 = ”) 24 disp(T32 ,”T32 = ”) 25 disp(T33 ,”T33 = ”) 26 C = [13.0639;0.2577;8.0619]; 27 Y = inv(A)*C; 28 disp(”At t = 10 s ”) 29 disp(det(Y(1,1)),”T11 = ”) 30 disp(det(Y(2,1)),”T12 = ”) 31 disp(det(Y(3,1)),”T13 = ”) 32 // s i m i l a r l y we get , 33 T21 = 6.1683; 34 T22 = 0.8238; 35 T23 = 4.2359; 196
36 T31 = 13.1120; 37 T32 = 8.3207; 38 T33 = 11.3606; 39 disp(T21 ,”T21 = ”) 40 disp(T22 ,”T22 = ”) 41 disp(T23 ,”T23 = ”) 42 disp(T31 ,”T31 = ”) 43 disp(T32 ,”T32 = ”) 44 disp(T33 ,”T33 = ”) 197
Chapter 31 Finite Element Method Scilab code Exa 31.1 Analytical Solution for Heated Rod 1 clc; 2 clear; 3 //d2T/dx2=−10; equation to be s o l v e d 4 //T(0 , t ) =40; boundary c o n d i t i o n 5 //T(10 , t ) =200; boundary c o n d i t i o n 6 // f ( x ) =10; uniform heat source 7 //we assume a s o l u t i o n T=a∗Xˆ2 + b∗x +c 8 // d i f f e r e n t i a t i n g twice we get d2T/dx2=2∗a 9 a= -10/2; 10 // using f i r s t boundary c o n d i t i o n 11 c=40; 12 // using second boundary condtion 13 b=66; 14 // hence f i n a l s o l u t i o n T=−5∗xˆ2 + 66∗x + 40 15 function T=f(x) 16 T=-5*x^2 + 66*x + 40 17 endfunction 18 count =1; 19 for i=0:0.1:11 198
Figure 31.1: Analytical Solution for Heated Rod 199
20 T(count)=f(i); 21 count=count +1; 22 end 23 x=0:0.1:11 24 plot(x,T) 25 xtitle(” Temperature (T) vs d i s t a n c e ( x ) ”,”x (cm) ”,”T ( u n i t s ) ”) Scilab code Exa 31.2 Element Equation for Heated Rod 1 clc; 2 clear; 3 xf=10; //cm 4 xe =2.5; //cm 5 //T(0 , t ) =40; boundary c o n d i t i o n 6 //T(10 , t ) =200; boundary c o n d i t i o n 7 // f ( x ) =10; uniform heat source 8 function y=f(x) 9 y=10*(xe -x)/xe; 10 endfunction 11 int1=intg(0,xe ,f) 12 function y=g(x) 13 y=10*(x-0)/xe; 14 endfunction 15 int2=intg(0,xe ,g) 16 disp(”The r e s u l t s are : ”) 17 disp(” 0.4∗ T1−0.4∗T2=−(dT/dx ) ∗x1 + c1 ”) 18 disp(int1 ,” where c1=”) 19 disp(”and”) 20 disp(” −0.4∗T1+0.4∗T2=−(dT/dx ) ∗x2 + c2 ”) 21 disp(int2 ,” where c2=”) Scilab code Exa 31.3 Temperature of a heated plate 200
1 // c l c ( ) 2 wf = 01.5; 3 for i = 1:41 4 for j = 1:41 5 T(i,j) = 0; 6 if j == 1 then 7 T(i,j) = 0; //C 8 else 9 if j == 41 then 10 T(i,j) = 100; //C 11 end 12 end 13 if i == 1 then 14 T(i,j) = 75; //C 15 else 16 if i == 41 then 17 T(i,j) = 50; //C 18 end 19 end 20 end 21 end 22 e = 100; 23 while e>1 24 for i=1:41 25 for j = 1:41 26 if i>1 & j>1 & i<41 & j<41 then 27 Tn(i,j) = (T(i + 1,j) + T(i-1,j) + T(i,j+1) + T(i,j-1))/4; 28 Tn(i,j) = wf * Tn(i,j) + (1-wf)*T(i,j); 29 if i==2 & j==2 then 30 e = abs((Tn(i,j) - T(i,j)) * 100/ (Tn(i, j))); 31 end 32 T(i,j) = Tn(i,j); 33 end 34 end 35 end 36 end 201
37 disp(T,” f o r e r r o r < 1 , the temperatures are ”) 202

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Numerical Methods For Engineers_S. C. Chapra And R. P. Canale.pdf

  • 1. Scilab Textbook Companion for Numerical Methods For Engineers by S. C. Chapra And R. P. Canale1 Created by Meghana Sundaresan Numerical Methods Chemical Engineering VNIT College Teacher Kailas L. Wasewar Cross-Checked by July 31, 2019 1Funded by a grant from the National Mission on Education through ICT, http://spoken-tutorial.org/NMEICT-Intro. This Textbook Companion and Scilab codes written in it can be downloaded from the ”Textbook Companion Project” section at the website http://scilab.in
  • 2. Book Description Title: Numerical Methods For Engineers Author: S. C. Chapra And R. P. Canale Publisher: McGraw Hill, New York Edition: 5 Year: 2006 ISBN: 0071244298 1
  • 3. Scilab numbering policy used in this document and the relation to the above book. Exa Example (Solved example) Eqn Equation (Particular equation of the above book) AP Appendix to Example(Scilab Code that is an Appednix to a particular Example of the above book) For example, Exa 3.51 means solved example 3.51 of this book. Sec 2.3 means a scilab code whose theory is explained in Section 2.3 of the book. 2
  • 4. Contents List of Scilab Codes 4 1 Mathematical Modelling and Engineering Problem Solving 6 2 Programming and Software 8 3 Approximations and Round off Errors 10 4 Truncation Errors and the Taylor Series 15 5 Bracketing Methods 25 6 Open Methods 34 7 Roots of Polynomials 44 9 Gauss Elimination 51 10 LU Decomposition and matrix inverse 61 11 Special Matrices and gauss seidel 65 13 One dimensional unconstrained optimization 69 14 Multidimensional Unconstrainted Optimization 72 3
  • 5. 15 Constrained Optimization 76 17 Least squares regression 82 18 Interpolation 96 19 Fourier Approximation 108 21 Newton Cotes Integration Formulas 116 22 Integration of equations 129 23 Numerical differentiation 135 25 Runga Kutta methods 139 26 Stiffness and multistep methods 156 27 Boundary Value and Eigen Value problems 165 29 Finite Difference Elliptic Equations 180 30 Finite Difference Parabolic Equations 185 31 Finite Element Method 191 4
  • 6. List of Scilab Codes Exa 1.1 Analytical Solution to Falling Parachutist Prob- lem . . . . . . . . . . . . . . . . . . . . . . . 6 Exa 1.2 Numerical Solution to Falling Parachutist Prob- lem . . . . . . . . . . . . . . . . . . . . . . . 6 Exa 2.1 roots of quadratic . . . . . . . . . . . . . . . 8 Exa 3.1 Calculations of Errors . . . . . . . . . . . . 10 Exa 3.2 Iterative error estimation . . . . . . . . . . . 11 Exa 3.3 Range of Integers . . . . . . . . . . . . . . . 11 Exa 3.4 Floating Point Numbers . . . . . . . . . . . 12 Exa 3.5 Machine Epsilon . . . . . . . . . . . . . . . 12 Exa 3.6 Interdependent Computations . . . . . . . . 12 Exa 3.7 Subtractive Cancellation . . . . . . . . . . . 13 Exa 3.8 Infinite Series Evaluation . . . . . . . . . . . 13 Exa 4.1 Polynomial Taylor Series . . . . . . . . . . . 15 Exa 4.2 Taylor Series Expansion . . . . . . . . . . . 17 Exa 4.3 Effect of Nonlinearity and Stepsize on Taylor expansion . . . . . . . . . . . . . . . . . . . 20 Exa 4.4 Finite divided difference approximation of deriva- tives . . . . . . . . . . . . . . . . . . . . . . 21 Exa 4.5 Error propagation in function of single variable 22 Exa 4.6 Error propagation in multivariable function 22 Exa 4.7 Condition Number . . . . . . . . . . . . . . 24 Exa 5.1 Graphical Approach . . . . . . . . . . . . . 25 Exa 5.2 Computer Graphics to Locate Roots . . . . 27 Exa 5.3 Bisection . . . . . . . . . . . . . . . . . . . 28 Exa 5.4 Error Estimates for Bisection . . . . . . . . 29 Exa 5.5 False Position . . . . . . . . . . . . . . . . . 30 Exa 5.6 Bracketing and False Position Methods . . . 31 5
  • 7. Exa 6.1 simple fixed point iteration . . . . . . . . . 34 Exa 6.2 The Two curve graphical method . . . . . . 34 Exa 6.3 Newton Raphson Method . . . . . . . . . . 36 Exa 6.4 Error analysis of Newton Raphson method . 37 Exa 6.5 slowly converging function with Newton Raph- son method . . . . . . . . . . . . . . . . . . 38 Exa 6.6 The secant method . . . . . . . . . . . . . . 38 Exa 6.7 secant and false position techniques . . . . . 39 Exa 6.8 Modified secant method . . . . . . . . . . . 40 Exa 6.9 modified newton raphson method . . . . . . 41 Exa 6.10 fixed point iteration for nonlinear system . . 42 Exa 6.11 Newton Raphson for a nonlinear Problem . 42 Exa 7.1 Polynomial Deflation . . . . . . . . . . . . . 44 Exa 7.2 Mullers Method . . . . . . . . . . . . . . . . 45 Exa 7.3 Bairstows Method . . . . . . . . . . . . . . 46 Exa 7.4 Locate single root . . . . . . . . . . . . . . . 47 Exa 7.5 Solving nonlinear system . . . . . . . . . . . 48 Exa 7.6 Root Location . . . . . . . . . . . . . . . . . 49 Exa 7.7 Roots of Polynomials . . . . . . . . . . . . . 49 Exa 7.8 Root Location . . . . . . . . . . . . . . . . . 49 Exa 9.1 Graphical Method for two Equations . . . . 51 Exa 9.2 Deterinants . . . . . . . . . . . . . . . . . . 53 Exa 9.3 Cramers Rule . . . . . . . . . . . . . . . . . 53 Exa 9.4 Elimination of Unknowns . . . . . . . . . . 54 Exa 9.5 Naive Gauss Elimination . . . . . . . . . . . 55 Exa 9.6 ill conditioned systems . . . . . . . . . . . . 56 Exa 9.7 Effect of Scale on Determinant . . . . . . . 56 Exa 9.8 Scaling . . . . . . . . . . . . . . . . . . . . . 57 Exa 9.9 Partial Pivoting . . . . . . . . . . . . . . . . 57 Exa 9.10 Effect of scaling on Pivoting and round off . 58 Exa 9.11 Solution of Linear Algebraic Equations . . . 59 Exa 9.12 Gauss Jordan method . . . . . . . . . . . . 59 Exa 10.1 LU decomposition with gauss elimination . . 61 Exa 10.2 The substitution steps . . . . . . . . . . . . 62 Exa 10.3 Matrix inversion . . . . . . . . . . . . . . . 62 Exa 10.4 Matrix condition evaluation . . . . . . . . . 63 Exa 11.1 Tridiagonal solution with Thomas algorithm 65 Exa 11.2 Cholesky Decomposition . . . . . . . . . . . 66 6
  • 8. Exa 11.3 Gauss Seidel method . . . . . . . . . . . . . 66 Exa 11.4 Linear systems . . . . . . . . . . . . . . . . 67 Exa 11.5 Manipulate linear algebraic equations . . . . 68 Exa 11.6 Analyze and solve Hilbert matrix . . . . . . 68 Exa 13.1 Golden section method . . . . . . . . . . . . 69 Exa 13.2 Quadratic interpolation . . . . . . . . . . . 70 Exa 13.3 Newtons method . . . . . . . . . . . . . . . 71 Exa 14.1 Random Search Method . . . . . . . . . . . 72 Exa 14.2 Path of Steepest Descent . . . . . . . . . . . 73 Exa 14.3 1 D function along Gradient . . . . . . . . . 73 Exa 14.4 Optimal Steepest Descent . . . . . . . . . . 74 Exa 15.1 Setting up LP problem . . . . . . . . . . . . 76 Exa 15.2 Graphical Solution . . . . . . . . . . . . . . 78 Exa 15.3 Linear Programming Problem . . . . . . . . 78 Exa 15.4 Nonlinear constrained optimization . . . . . 79 Exa 15.5 One dimensional Optimization . . . . . . . . 80 Exa 15.6 Multidimensional Optimization . . . . . . . 80 Exa 15.7 Locate Single Optimum . . . . . . . . . . . 81 Exa 17.1 Linear regression . . . . . . . . . . . . . . . 82 Exa 17.2 Estimation of errors for the linear least square fit . . . . . . . . . . . . . . . . . . . . . . . 83 Exa 17.3.a linear regression using computer . . . . . . . 84 Exa 17.3.b linear regression using computer . . . . . . . 85 Exa 17.4 Linearization of a power function . . . . . . 87 Exa 17.5 polynomial regression . . . . . . . . . . . . . 88 Exa 17.6 Multiple linear regression . . . . . . . . . . 92 Exa 17.7 Confidence interval for linear regression . . . 93 Exa 17.8 Gauss Newton method . . . . . . . . . . . . 94 Exa 18.1 Linear interpolation . . . . . . . . . . . . . 96 Exa 18.2 Quadratic interpolation . . . . . . . . . . . 97 Exa 18.3 Newtons divided difference Interpolating poly- nomials . . . . . . . . . . . . . . . . . . . . 97 Exa 18.4 Error estimation for Newtons polynomial . . 98 Exa 18.5 Error Estimates for Order of Interpolation . 99 Exa 18.6 Lagrange interpolating polynomials . . . . . 100 Exa 18.7 Lagrange interpolation using computer . . . 100 Exa 18.8 First order splines . . . . . . . . . . . . . . 103 Exa 18.9 Quadratic splines . . . . . . . . . . . . . . . 103 7
  • 9. Exa 18.10 Cubic splines . . . . . . . . . . . . . . . . . 106 Exa 19.1 Least Square Fit . . . . . . . . . . . . . . . 108 Exa 19.2 Continuous Fourier Series Approximation . . 109 Exa 19.3 Trendline . . . . . . . . . . . . . . . . . . . 110 Exa 19.4 Data Analysis . . . . . . . . . . . . . . . . . 110 Exa 19.5 Curve Fitting . . . . . . . . . . . . . . . . . 112 Exa 19.6 Polynomial Regression . . . . . . . . . . . . 115 Exa 21.1 Single trapezoidal rule . . . . . . . . . . . . 116 Exa 21.2 Multiple trapezoidal rule . . . . . . . . . . . 117 Exa 21.3 Evaluating Integrals . . . . . . . . . . . . . 118 Exa 21.4 Single Simpsons 1 by 3 rule . . . . . . . . . 122 Exa 21.5 Multiple Simpsons 1 by 3 rule . . . . . . . . 123 Exa 21.6 Simpsons 3 by 8 rule . . . . . . . . . . . . . 124 Exa 21.7 Unequal Trapezoidal segments . . . . . . . . 126 Exa 21.8 Simpsons Uneven data . . . . . . . . . . . . 127 Exa 21.9 Average Temperature Determination . . . . 127 Exa 22.1 Error corrections of the trapezoidal rule . . 129 Exa 22.2 Higher order error correction of integral esti- mates . . . . . . . . . . . . . . . . . . . . . 129 Exa 22.3 Two point gauss legendre formulae . . . . . 130 Exa 22.4 Three point gauss legendre method . . . . . 131 Exa 22.5 Applying Gauss Quadrature to the falling Parachutist problem . . . . . . . . . . . . . . . . . . . . 132 Exa 22.6 Evaluation of improper integral . . . . . . . 133 Exa 23.1 High accuracy numerical differentiation for- mulas . . . . . . . . . . . . . . . . . . . . . 135 Exa 23.2 Richardson extrapolation . . . . . . . . . . . 136 Exa 23.3 Differentiating unequally spaced data . . . . 137 Exa 23.4 Integration and Differentiation . . . . . . . . 137 Exa 23.5 Integrate a function . . . . . . . . . . . . . . 138 Exa 25.1 Eulers method . . . . . . . . . . . . . . . . 139 Exa 25.2 Taylor series estimate for error by eulers method 140 Exa 25.3 Effect of reduced step size on Eulers method 140 Exa 25.4 Solving ODEs . . . . . . . . . . . . . . . . . 141 Exa 25.5 Heuns method . . . . . . . . . . . . . . . . . 142 Exa 25.6 Comparison of various second order RK 4 method 143 Exa 25.7 Classical fourth order RK method . . . . . . 144 Exa 25.8 Comparison of Runga Kutta methods . . . . 145 8
  • 10. Exa 25.9 Solving systems of ODE using Eulers method 147 Exa 25.10 Solving systems of ODE using RK 4 method 147 Exa 25.11 Solving systems of ODEs . . . . . . . . . . . 148 Exa 25.12 Adaptive fourth order RK method . . . . . 150 Exa 25.13 Runga kutta fehlberg method . . . . . . . . 153 Exa 25.14 Adaptive Fourth order RK scheme . . . . . 154 Exa 26.1 Explicit and Implicit Euler . . . . . . . . . . 156 Exa 26.2 Non self starting Heun method . . . . . . . 157 Exa 26.3 Estimate of per step truncation error . . . . 160 Exa 26.4 Effect of modifier on Predictor Corrector re- sults . . . . . . . . . . . . . . . . . . . . . . 161 Exa 26.5 Milnes Method . . . . . . . . . . . . . . . . 161 Exa 26.6 Fourth order Adams method . . . . . . . . . 162 Exa 26.7 stability of Milnes and Fourth order Adams method . . . . . . . . . . . . . . . . . . . . 163 Exa 27.1 The shooting method . . . . . . . . . . . . . 165 Exa 27.2 The shooting method for non linear problems 166 Exa 27.3 Finite Difference Approximation . . . . . . . 167 Exa 27.4 Mass Spring System . . . . . . . . . . . . . 168 Exa 27.5 Axially Loaded column . . . . . . . . . . . . 168 Exa 27.6 Polynomial Method . . . . . . . . . . . . . . 169 Exa 27.7 Power Method Highest Eigenvalue . . . . . 171 Exa 27.8 Power Method Lowest Eigenvalue . . . . . . 171 Exa 27.9 Eigenvalues and ODEs . . . . . . . . . . . . 172 Exa 27.10.a Stiff ODEs . . . . . . . . . . . . . . . . . . . 173 Exa 27.10.b Stiff ODEs . . . . . . . . . . . . . . . . . . . 177 Exa 27.11 Solving ODEs . . . . . . . . . . . . . . . . . 178 Exa 29.1 Temperature of a heated plate with fixed bound- ary conditions . . . . . . . . . . . . . . . . . 180 Exa 29.2 Flux distribution for a heated plate . . . . . 181 Exa 29.3 Heated plate with a insulated edge . . . . . 182 Exa 29.4 Heated plate with an irregular boundary . . 183 Exa 30.1 Explicit solution of a one dimensional heat conduction equation . . . . . . . . . . . . . 185 Exa 30.2 Simple implicit solution of a heat conduction equation . . . . . . . . . . . . . . . . . . . . 186 Exa 30.3 Crank Nicolson solution to the heat conduc- tion equation . . . . . . . . . . . . . . . . . 187 9
  • 11. Exa 30.4 Comparison of Analytical and Numerical so- lution . . . . . . . . . . . . . . . . . . . . . 188 Exa 30.5 ADI Method . . . . . . . . . . . . . . . . . 188 Exa 31.1 Analytical Solution for Heated Rod . . . . . 191 Exa 31.2 Element Equation for Heated Rod . . . . . . 193 Exa 31.3 Temperature of a heated plate . . . . . . . . 193 10
  • 12. List of Figures 5.1 Graphical Approach . . . . . . . . . . . . . . . . . . . . . . 26 5.2 Computer Graphics to Locate Roots . . . . . . . . . . . . . 27 6.1 The Two curve graphical method . . . . . . . . . . . . . . . 35 9.1 Graphical Method for two Equations . . . . . . . . . . . . . 52 15.1 Graphical Solution . . . . . . . . . . . . . . . . . . . . . . . 77 17.1 linear regression using computer . . . . . . . . . . . . . . . . 85 17.2 linear regression using computer . . . . . . . . . . . . . . . . 86 17.3 Linearization of a power function . . . . . . . . . . . . . . . 88 17.4 Linearization of a power function . . . . . . . . . . . . . . . 89 17.5 polynomial regression . . . . . . . . . . . . . . . . . . . . . . 90 18.1 First order splines . . . . . . . . . . . . . . . . . . . . . . . . 102 18.2 Quadratic splines . . . . . . . . . . . . . . . . . . . . . . . . 104 18.3 Cubic splines . . . . . . . . . . . . . . . . . . . . . . . . . . 106 19.1 Trendline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 19.2 Curve Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . 113 19.3 Curve Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . 114 25.1 Solving ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . 141 25.2 Solving systems of ODEs . . . . . . . . . . . . . . . . . . . . 150 11
  • 13. 25.3 Solving systems of ODEs . . . . . . . . . . . . . . . . . . . . 151 25.4 Adaptive Fourth order RK scheme . . . . . . . . . . . . . . 154 26.1 Explicit and Implicit Euler . . . . . . . . . . . . . . . . . . . 158 26.2 Explicit and Implicit Euler . . . . . . . . . . . . . . . . . . . 159 27.1 The shooting method for non linear problems . . . . . . . . 166 27.2 Eigenvalues and ODEs . . . . . . . . . . . . . . . . . . . . . 174 27.3 Eigenvalues and ODEs . . . . . . . . . . . . . . . . . . . . . 175 27.4 Stiff ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 27.5 Stiff ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 31.1 Analytical Solution for Heated Rod . . . . . . . . . . . . . . 192 12
  • 14. Chapter 1 Mathematical Modelling and Engineering Problem Solving Scilab code Exa 1.1 Analytical Solution to Falling Parachutist Problem 1 clc; 2 clear; 3 g=9.8; //m/ s ˆ2; a c c e l e r a t i o n due to g r a v i t y 4 m=68.1; // kg 5 c=12.5; // kg/ sec ; drag c o e f f i c i e n t 6 count =1; 7 for i=0:2:12 8 v(count)=g*m*(1-exp(-c*i/m))/c; 9 disp(v(count),”v (m/ s )=”,i,”Time ( s )=”) 10 count=count +1; 11 end 12 disp(g*m/c,”v (m/ s )=”,” i n f i n i t y ”,”Time ( s )=”) Scilab code Exa 1.2 Numerical Solution to Falling Parachutist Problem 1 clc; 13
  • 15. 2 clear; 3 g=9.8; //m/ s ˆ2; a c c e l e r a t i o n due to g r a v i t y 4 m=68.1; // kg 5 c=12.5; // kg/ sec ; drag c o e f f i c i e n t 6 count =2; 7 v(1) =0; 8 disp(v(1),”v (m/ s )=” ,0,”Time ( s )=”) 9 for i=2:2:12 10 v(count)=v(count -1)+(g-c*v(count -1)/m)*(2); 11 disp(v(count),”v (m/ s )=”,i,”Time ( s )=”) 12 count=count +1; 13 end 14 disp(g*m/c,”v (m/ s )=”,” i n f i n i t y ”,”Time ( s )=”) 14
  • 16. Chapter 2 Programming and Software Scilab code Exa 2.1 roots of quadratic 1 clc; 2 clear; 3 response =1; 4 while response ==1 5 a=input(” Input the value of a : ”) 6 b=input(” Input the value of b : ”) 7 c=input(” Input the value of c : ”) 8 if a==0 then 9 if b~=0 then 10 r1=-c/b; 11 disp(r1 ,”The root : ”) 12 else disp(” T r i v i a l S o l u t i o n . ”) 13 end 14 else 15 discr=b^2-4*a*c; 16 if discr >=0 then 17 r1=(-b+sqrt(discr))/(2*a); 18 r2=(-b-sqrt(discr))/(2*a); 19 disp(r2 ,”and”,r1 ,”The r o o t s are : ”) 20 else 21 r1=-b/(2*a); 15
  • 17. 22 r2=r1; 23 i1=sqrt(abs(discr))/(2*a); 24 i2=-i1; 25 disp(r2+i2*sqrt (-1),r1+i1*sqrt (-1),”The r o o t s are : ”) 26 end 27 end 28 response=input(”Do you want to continue ( p r e s s 1 f o r yes and 2 f o r no ) ?”) 29 if response =2 then exit; 30 end 31 end 32 end 16
  • 18. Chapter 3 Approximations and Round off Errors Scilab code Exa 3.1 Calculations of Errors 1 clc; 2 clear; 3 lbm =9999; //cm , measured length of bridge 4 lrm =9; //cm , measured length of r i v e t 5 lbt =10000; //cm , true length of bridge 6 lrt =10; //cm , true length of r i v e t 7 // c a l c u l a t i n g true e r r o r below ; 8 Etb=lbt -lbm;//cm , true e r r o r in bridge 9 Etr=lrt -lrm;//cm , true e r r o r in r i v e t 10 // c a l c u l a t i n g percent r e l a t i v e e r r o r below 11 etb=Etb *100/ lbt;// percent r e l a t i v e e r r o r f o r bridge 12 etr=Etb *100/ lrt;// percent r e l a t i v e e r r o r f o r r i v e t 13 disp(”a . The true e r r o r i s ”) 14 disp(” f o r the bridge ”,”cm”,Etb) 15 disp(” f o r the r i v e t ”,”cm”,Etr) 16 disp(”b . The percent r e l a t i v e e r r o r i s ”) 17 disp(” f o r the bridge ”,”%”,etb) 18 disp(” f o r the r i v e t ”,”%”,etr) 17
  • 19. Scilab code Exa 3.2 Iterative error estimation 1 clc; 2 clear; 3 n=3; //number of s i g n i f i c a n t f i g u r e s 4 es =0.5*(10^(2 -n));// percent , s p e c i f i e d e r r o r c r i t e r i o n 5 x=0.5; 6 f(1) =1; // f i r s t estimate f=e ˆx = 1 7 ft =1.648721; // true value of e ˆ0.5= f 8 et(1)=(ft -f(1))*100/ ft; 9 ea(1) =100; 10 i=2; 11 while ea(i-1) >=es 12 f(i)=f(i-1)+(x^(i-1))/( factorial(i-1)); 13 et(i)=(ft -f(i))*100/ ft; 14 ea(i)=(f(i)-f(i-1))*100/f(i); 15 i=i+1; 16 end 17 for j=1:i-1 18 disp(ea(j),” Approximate estimate of e r r o r (%)=”,et (j),”True % r e l a t i v e e r r o r=”,f(j),” Result=”,j, ” term number=”) 19 disp(” −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 20 end Scilab code Exa 3.3 Range of Integers 1 n=16; // no of b i t s 2 num =0; 18
  • 20. 3 for i=0:(n-2) 4 num=num +(1*(2^i)); 5 end 6 disp ((-1* num),” to ”,num ,”Thus a 16− b i t computer word can s t o r e decimal i n t e g e r s ranging from ”) Scilab code Exa 3.4 Floating Point Numbers 1 clc; 2 clear; 3 n=7; // no . of b i t s 4 // the maximum value of exponents i s given by 5 Max =1*(2^1) +1*(2^0); 6 // mantissa i s found by 7 mantissa =1*(2^ -1) +0*(2^ -3) +0*(2^ -3); 8 num=mantissa *(2^( Max*-1));// s m a l l e s t p o s s i b l e p o s i t i v e number f o r t h i s system 9 disp(num ,”The s m a l l e s t p o s s i b l e p o s i t i v e number f o r t h i s system i s ”) Scilab code Exa 3.5 Machine Epsilon 1 clc; 2 clear; 3 b=2; // base 4 t=3; //number of mantissa b i t s 5 E=2^(1 -t);// e p s i l o n 6 disp(E,” value of e p s i l o n=”) Scilab code Exa 3.6 Interdependent Computations 19
  • 21. 1 clc; 2 clear; 3 num=input(” Input a number : ”) 4 sum =0; 5 for i=1:100000 6 sum=sum+num; 7 end 8 disp(sum ,”The number summed up 100 ,000 times i s=”) Scilab code Exa 3.7 Subtractive Cancellation 1 clc; 2 clear; 3 a=1; 4 b=3000.001; 5 c=3; 6 // the r o o t s of the qu ad ra ti c equation x ˆ2+3000.001∗ x +3=0 are found as 7 D=(b^2) -4*a*c; 8 x1=(-b+(D^0.5))/(2*a); 9 x2=(-b-(D^0.5))/(2*a); 10 disp(x2 ,and ,x1 ,”The r o o t s of the qu ad ra ti c equation ( x ˆ2) +(3000.001∗ x )+3=0 are = ”) Scilab code Exa 3.8 Infinite Series Evaluation 1 clc; 2 clear; 3 function y=f(x) 4 y=exp(x) 5 endfunction 6 sum =1; 7 test =0; 20
  • 22. 8 i=0; 9 term =1; 10 x=input(” Input value of x : ”) 11 while sum~= test 12 disp(sum ,”sum : ”,term ,” term : ”,i,” i : ”) 13 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−”) 14 i=i+1; 15 term=term*x/i; 16 test=sum; 17 sum=sum+term; 18 end 19 disp(f(x),” Exact Value ”) 21
  • 23. Chapter 4 Truncation Errors and the Taylor Series Scilab code Exa 4.1 Polynomial Taylor Series 1 clc; 2 clear; 3 function y=f(x) 4 y= -0.1*x^4 -0.15*x^3 -0.5*x^2 -0.25*x+1.2; 5 endfunction 6 xi=0; 7 xf=1; 8 h=xf -xi; 9 fi=f(xi);// f u n c t i o n value at xi 10 ffa=f(xf);// a c t u a l f u n c t i o n value at xf 11 12 // f o r n=0, i . e , zero order approximation 13 ff=fi; 14 Et(1)=ffa -ff;// t r u n c a t i o n e r r o r at x=1 15 disp(fi ,”The value of f at x=0 : ”) 16 disp(ff ,”The value of f at x=1 due to zero order approximation : ”) 17 disp(Et(1),” Truncation e r r o r : ”) 18 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− 22
  • 24. ”) 19 20 // f o r n=1, i . e , f i r s t order approximation 21 deff( ’ y=f1 ( x ) ’ , ’ y=d e r i v a t i v e ( f , x , order =4) ’ ) 22 f1i=f1(xi);// value of f i r s t d e r i v a t i v e of f u n c t i o n at xi 23 f1f=fi+f1i*h;// value of f i r s t d e r i v a t i v e of f u n c t i o n at xf 24 Et(2)=ffa -f1f;// t r u n c a t i o n e r r o r at x=1 25 disp(f1i ,”The value of f i r s t d e r i v a t i v e of f at x=0 : ”) 26 disp(f1f ,”The value of f at x=1 due to f i r s t order approximation : ”) 27 disp(Et(2),” Truncation e r r o r : ”) 28 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 29 30 31 // f o r n=2, i . e , second order approximation 32 deff( ’ y=f2 ( x ) ’ , ’ y=d e r i v a t i v e ( f1 , x , order =4) ’ ) 33 f2i=f2(xi);// value of second d e r i v a t i v e of f u n c t i o n at xi 34 f2f=f1f+f2i*(h^2)/factorial (2);// value of second d e r i v a t i v e of f u n c t i o n at xf 35 Et(3)=ffa -f2f;// t r u n c a t i o n e r r o r at x=1 36 disp(f2i ,”The value of second d e r i v a t i v e of f at x=0 : ”) 37 disp(f2f ,”The value of f at x=1 due to second order approximation : ”) 38 disp(Et(3),” Truncation e r r o r : ”) 39 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 40 41 // f o r n=3, i . e , t h i r d order approximation 42 deff( ’ y=f3 ( x ) ’ , ’ y=d e r i v a t i v e ( f2 , x , order =4) ’ ) 43 f3i=f3(xi);// value of t h i r d d e r i v a t i v e of f u n c t i o n at xi 44 f3f=f2f+f3i*(h^3)/factorial (3);// value of t h i r d 23
  • 25. d e r i v a t i v e of f u n c t i o n at xf 45 Et(4)=ffa -f3f;// t r u n c a t i o n e r r o r at x=1 46 disp(f3i ,”The value of t h i r d d e r i v a t i v e of f at x=0 : ”) 47 disp(f3f ,”The value of f at x=1 due to t h i r d order approximation : ”) 48 disp(Et(4),” Truncation e r r o r : ”) 49 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 50 51 // f o r n=4, i . e , f o u r t h order approximation 52 deff( ’ y=f4 ( x ) ’ , ’ y=d e r i v a t i v e ( f3 , x , order =4) ’ ) 53 f4i=f4(xi);// value of f o u r t h d e r i v a t i v e of f u n c t i o n at xi 54 f4f=f3f+f4i*(h^4)/factorial (4);// value of f o u r t h d e r i v a t i v e of f u n c t i o n at xf 55 Et(5)=ffa -f4f;// t r u n c a t i o n e r r o r at x=1 56 disp(f4i ,”The value of f o u r t h d e r i v a t i v e of f at x=0 : ”) 57 disp(f4f ,”The value of f at x=1 due to f o u r t h order approximation : ”) 58 disp(Et(5),” Truncation e r r o r : ”) 59 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) Scilab code Exa 4.2 Taylor Series Expansion 1 clc; 2 clear; 3 function y=f(x) 4 y=cos(x) 5 endfunction 6 xi=%pi /4; 7 xf=%pi /3; 8 h=xf -xi; 24
  • 26. 9 fi=f(xi);// f u n c t i o n value at xi 10 ffa=f(xf);// a c t u a l f u n c t i o n value at xf 11 12 // f o r n=0, i . e , zero order approximation 13 ff=fi; 14 et(1)=(ffa -ff)*100/ ffa;// percent r e l a t i v e e r r o r at x =1 15 disp(ff ,”The value of f at x=1 due to zero order approximation : ”) 16 disp(et(1),”% r e l a t i v e e r r o r : ”) 17 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 18 19 // f o r n=1, i . e , f i r s t order approximation 20 deff( ’ y=f1 ( x ) ’ , ’ y=d e r i v a t i v e ( f , x , order =4) ’ ) 21 f1i=f1(xi);// value of f i r s t d e r i v a t i v e of f u n c t i o n at xi 22 f1f=fi+f1i*h;// value of f i r s t d e r i v a t i v e of f u n c t i o n at xf 23 et(2)=(ffa -f1f)*100/ ffa;//% r e l a t i v e e r r o r at x=1 24 disp(f1f ,”The value of f at x=1 due to f i r s t order approximation : ”) 25 disp(et(2),”% r e l a t i v e e r r o r : ”) 26 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 27 28 29 // f o r n=2, i . e , second order approximation 30 deff( ’ y=f2 ( x ) ’ , ’ y=d e r i v a t i v e ( f1 , x , order =4) ’ ) 31 f2i=f2(xi);// value of second d e r i v a t i v e of f u n c t i o n at xi 32 f2f=f1f+f2i*(h^2)/factorial (2);// value of second d e r i v a t i v e of f u n c t i o n at xf 33 et(3)=(ffa -f2f)*100/ ffa;//% r e l a t i v e e r r o r at x=1 34 disp(f2f ,”The value of f at x=1 due to second order approximation : ”) 35 disp(et(3),”% r e l a t i v e e r r o r : ”) 36 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− 25
  • 27. ”) 37 38 39 // f o r n=3, i . e , t h i r d order approximation 40 deff( ’ y=f3 ( x ) ’ , ’ y=d e r i v a t i v e ( f2 , x , order =4) ’ ) 41 f3i=f3(xi);// value of t h i r d d e r i v a t i v e of f u n c t i o n at xi 42 f3f=f2f+f3i*(h^3)/factorial (3);// value of t h i r d d e r i v a t i v e of f u n c t i o n at xf 43 et(4)=(ffa -f3f)*100/ ffa;//% r e l a t i v e e r r o r at x=1 44 disp(f3f ,”The value of f at x=1 due to t h i r d order approximation : ”) 45 disp(et(4),”% r e l a t i v e e r r o r : ”) 46 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 47 48 49 // f o r n=4, i . e , f o u r t h order approximation 50 deff( ’ y=f4 ( x ) ’ , ’ y=d e r i v a t i v e ( f3 , x , order =4) ’ ) 51 f4i=f4(xi);// value of f o u r t h d e r i v a t i v e of f u n c t i o n at xi 52 f4f=f3f+f4i*(h^4)/factorial (4);// value of f o u r t h d e r i v a t i v e of f u n c t i o n at xf 53 et(5)=(ffa -f4f)*100/ ffa;//% r e l a t i v e e r r o r at x=1 54 disp(f4f ,”The value of f at x=1 due to f o u r t h order approximation : ”) 55 disp(et(5),”% r e l a t i v e e r r o r : ”) 56 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 57 58 59 // f o r n=5, i . e , f i f t h order approximation 60 f5i=(f4 (1.1* xi)-f4 (0.9* xi))/(2*0.1);// value of f i f t h d e r i v a t i v e of f u n c t i o n at xi ( c e n t r a l d i f f e r e n c e method ) 61 f5f=f4f+f5i*(h^5)/factorial (5);// value of f i f t h d e r i v a t i v e of f u n c t i o n at xf 62 et(6)=(ffa -f5f)*100/ ffa;//% r e l a t i v e e r r o r at x=1 26
  • 28. 63 disp(f5f ,”The value of f at x=1 due to f i f t h order approximation : ”) 64 disp(et(6),”% r e l a t i v e e r r o r : ”) 65 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 66 67 68 // f o r n=6, i . e , s i x t h order approximation 69 deff( ’ y=f6 ( x ) ’ , ’ y=d e r i v a t i v e ( f5 , x , order =4) ’ ) 70 f6i=(f4 (1.1* xi) -2*f4(xi)+f4 (0.9* xi))/(0.1^2);// value of s i x t h d e r i v a t i v e of f u n c t i o n at xi ( c e n t r a l d i f f e r e n c e method ) 71 f6f=f5f+f6i*(h^6)/factorial (6);// value of s i x t h d e r i v a t i v e of f u n c t i o n at xf 72 et(6)=(ffa -f6f)*100/ ffa;//% r e l a t i v e e r r o r at x=1 73 disp(f6f ,”The value of f at x=1 due to s i x t h order approximation : ”) 74 disp(et(6),”% r e l a t i v e e r r o r : ”) 75 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) Scilab code Exa 4.3 Effect of Nonlinearity and Stepsize on Taylor expansion 1 clc; 2 clear; 3 m=input(” Input value of m: ”) 4 h=input(” Input value of h : ”) 5 function y=f(x) 6 y=x^m 7 endfunction 8 x1=1; 9 x2=x1+h; 10 fx1=f(x1); 11 fx2=fx1+m*(fx1^(m-1))*h; 12 if m==1 then 27
  • 29. 13 R=0; 14 else if m==2 then 15 R=2*(h^2)/factorial (2); 16 end 17 if m==3 then 18 R=(6*( x1)*(h^2)/factorial (2))+(6*(h^3)/ factorial (3)); 19 end 20 if m==4 then 21 R=(12*( x1^2)*(h^2)/factorial (2))+(24*( x1)*(h ^3)/factorial (3))+(24*(h^4)/factorial (4)) ; 22 end 23 end 24 disp(R,” Remainder : ”,fx2 ,”The value by f i r s t order approximation : ”) 25 disp(f(x2),”True Value at x2 : ”) Scilab code Exa 4.4 Finite divided difference approximation of derivatives 1 clc; 2 clear; 3 function y=f(x) 4 y= -0.1*(x^4) -0.15*(x^3) -0.5*(x^2) -0.25*(x)+1.2 5 endfunction 6 x=0.5; 7 h=input(” Input h : ”) 8 x1=x-h; 9 x2=x+h; 10 // forward d i f f e r e n c e method 11 fdx1 =(f(x2)-f(x))/h;// d e r i v a t i v e at x 12 et1=abs((fdx1 -derivative(f,x))/derivative(f,x))*100; 13 // backward d i f f e r e n c e method 14 fdx2 =(f(x)-f(x1))/h;// d e r i v a t i v e at x 15 et2=abs((fdx2 -derivative(f,x))/derivative(f,x))*100; 28
  • 30. 16 // c e n t r a l d i f f e r e n c e method 17 fdx3 =(f(x2)-f(x1))/(2*h);// d e r i v a t i v e at x 18 et3=abs((fdx3 -derivative(f,x))/derivative(f,x))*100; 19 disp(h,”For h=”) 20 disp(et1 ,”and percent e r r o r=”,fdx1 ,” D e r i v a t i v e at x by forward d i f f e r e n c e method=”) 21 disp(et2 ,”and percent e r r o r=”,fdx2 ,” D e r i v a t i v e at x by backward d i f f e r e n c e method=”) 22 disp(et3 ,”and percent e r r o r=”,fdx3 ,” D e r i v a t i v e at x by c e n t r a l d i f f e r e n c e method=”) Scilab code Exa 4.5 Error propagation in function of single variable 1 clc; 2 clear; 3 function y=f(x) 4 y=x^3 5 endfunction 6 x=2.5; 7 delta =0.01; 8 deltafx=abs(derivative(f,x))*delta; 9 fx=f(x); 10 disp(fx+deltafx ,”and”,fx -deltafx ,” true value i s between ”) Scilab code Exa 4.6 Error propagation in multivariable function 1 clc; 2 clear; 3 function y=f(F,L,E,I) 4 y=(F*(L^4))/(8*E*I) 5 endfunction 6 Fbar =50; // lb / f t 29
  • 31. 7 Lbar =30; // f t 8 Ebar =1.5*(10^8);// lb / f t ˆ2 9 Ibar =0.06; // f t ˆ4 10 deltaF =2; // lb / f t 11 deltaL =0.1; // f t 12 deltaE =0.01*(10^8);// lb / f t ˆ2 13 deltaI =0.0006; // f t ˆ4 14 ybar =( Fbar *( Lbar ^4))/(8* Ebar*Ibar); 15 function y=f1(F) 16 y=(F*( Lbar ^4))/(8* Ebar*Ibar) 17 endfunction 18 function y=f2(L) 19 y=( Fbar*(L^4))/(8* Ebar*Ibar) 20 endfunction 21 function y=f3(E) 22 y=( Fbar*( Lbar ^4))/(8*E*Ibar) 23 endfunction 24 function y=f4(I) 25 y=( Fbar*( Lbar ^4))/(8* Ebar*I) 26 endfunction 27 28 deltay=abs(derivative(f1 ,Fbar))*deltaF+abs( derivative(f2 ,Lbar))*deltaL+abs(derivative(f3 , Ebar))*deltaE+abs(derivative(f4 ,Ibar))*deltaI; 29 30 disp(ybar+deltay ,”and”,ybar -deltay ,”The value of y i s between : ”) 31 ymin =((Fbar -deltaF)*((Lbar -deltaL)^4))/(8*( Ebar+ deltaE)*( Ibar+deltaI)); 32 ymax =(( Fbar+deltaF)*(( Lbar+deltaL)^4))/(8*( Ebar - deltaE)*(Ibar -deltaI)); 33 disp(ymin ,”ymin i s c a l c u l a t e d at lower extremes of F , L , E, I v a l u e s as =”) 34 disp(ymax ,”ymax i s c a l c u l a t e d at higher extremes of F , L , E, I v a l u e s as =”) 30
  • 32. Scilab code Exa 4.7 Condition Number 1 clc; 2 clear; 3 function y=f(x) 4 y=tan(x) 5 endfunction 6 x1bar =(%pi /2) +0.1*( %pi /2); 7 x2bar =(%pi /2) +0.01*( %pi /2); 8 // computing c o n d i t i o n number f o r x1bar 9 condnum1=x1bar*derivative(f,x1bar)/f(x1bar); 10 disp(condnum1 ,” i s : ”,x1bar ,”The c o n d i t i o n number of f u n c t i o n f o r x=”) 11 if abs(condnum1) >1 then disp(x1bar ,” Function i s i l l − c o n d i t i o n e d f o r x=”) 12 end 13 // computing c o n d i t i o n number f o r x2bar 14 condnum2=x2bar*derivative(f,x2bar)/f(x2bar); 15 disp(condnum2 ,” i s : ”,x2bar ,”The c o n d i t i o n number of f u n c t i o n f o r x=”) 16 if abs(condnum2) >1 then disp(x2bar ,” Function i s i l l − c o n d i t i o n e d f o r x=”) 17 end 31
  • 33. Chapter 5 Bracketing Methods Scilab code Exa 5.1 Graphical Approach 1 clc; 2 clear; 3 m=68.1; // kg 4 v=40; //m/ s 5 t=10; // s 6 g=9.8; //m/ s ˆ2 7 function y=f(c) 8 y=g*m*(1-exp(-c*t/m))/c - v; 9 endfunction 10 disp(”For v a r i o u s v a l u e s of c and f ( c ) i s found as : ” ) 11 i=0; 12 for c=4:4:20 13 i=i+1; 14 bracket =[c f(c)]; 15 disp(bracket) 16 fc(i)=f(c); 17 end 18 c=[4 8 12 16 20] 32
  • 34. Figure 5.1: Graphical Approach 33
  • 35. Figure 5.2: Computer Graphics to Locate Roots 19 plot2d(c,fc) 20 xtitle( ’ f ( c ) vs c ’ , ’ c ’ , ’ f ( c ) (m/ s ) ’ ) Scilab code Exa 5.2 Computer Graphics to Locate Roots 1 clc; 2 clear; 3 function y=f(x) 4 y=sin (10*x)+cos (3*x); 5 endfunction 6 count =1; 34
  • 36. 7 for i=1:0.05:5 8 val(count)=i; 9 func(count)=f(i); 10 count=count +1; 11 end 12 plot2d(val ,func) 13 xtitle(”x vs f ( x ) ”, ’ x ’ , ’ f ( x ) ’ ) Scilab code Exa 5.3 Bisection 1 clc; 2 clear; 3 m=68.1; // kg 4 v=40; //m/ s 5 t=10; // s 6 g=9.8; //m/ s ˆ2 7 function y=f(c) 8 y=g*m*(1-exp(-c*t/m))/c - v; 9 endfunction 10 x1=12; 11 x2=16; 12 xt =14.7802; // true value 13 e=input(” e n t e r the t o l e r a b l e true percent e r r o r=”) 14 xr=(x1+x2)/2; 15 etemp=abs(xr -xt)/xt *100; // e r r o r 16 while etemp >e 17 if f(x1)*f(xr) >0 then 18 x1=xr; 19 xr=(x1+x2)/2; 20 etemp=abs(xr -xt)/xt *100; 21 end 22 if f(x1)*f(xr) <0 then 23 x2=xr; 24 xr=(x1+x2)/2; 25 etemp=abs(xr -xt)/xt *100; 35
  • 37. 26 end 27 if f(x1)*f(xr)==0 then break; 28 end 29 end 30 31 disp(xr ,”The r e s u l t i s=”) Scilab code Exa 5.4 Error Estimates for Bisection 1 clc; 2 clear; 3 m=68.1; // kg 4 v=40; //m/ s 5 t=10; // s 6 g=9.8; //m/ s ˆ2 7 function y=f(c) 8 y=g*m*(1-exp(-c*t/m))/c - v; 9 endfunction 10 x1=12; 11 x2=16; 12 xt =14.7802; // true value 13 e=input(” e n t e r the t o l e r a b l e approximate e r r o r=”) 14 xr=(x1+x2)/2; 15 i=1; 16 et=abs(xr -xt)/xt *100; // e r r o r 17 disp(i,” I t e r a t i o n : ”) 18 disp(x1 ,” xl : ”) 19 disp(x2 ,”xu : ”) 20 disp(xr ,” xr : ”) 21 disp(et ,” et (%) : ”) 22 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−”) 23 etemp =100; 24 while etemp >e 25 if f(x1)*f(xr) >0 then 26 x1=xr; 36
  • 38. 27 xr=(x1+x2)/2; 28 etemp=abs(xr -x1)/xr *100; 29 et=abs(xr -xt)/xt *100; 30 end 31 if f(x1)*f(xr) <0 then 32 x2=xr; 33 xr=(x1+x2)/2; 34 etemp=abs(xr -x2)/xr *100; 35 et=abs(xr -xt)/xt *100; 36 end 37 if f(x1)*f(xr)==0 then break; 38 end 39 i=i+1; 40 disp(i,” I t e r a t i o n : ”) 41 disp(x1 ,” xl : ”) 42 disp(x2 ,”xu : ”) 43 disp(xr ,” xr : ”) 44 disp(et ,” et (%) : ”) 45 disp(etemp ,” ea (%) ”) 46 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−”) 47 end 48 49 disp(xr ,”The r e s u l t i s=”) Scilab code Exa 5.5 False Position 1 clc; 2 clear; 3 m=68.1; // kg 4 v=40; //m/ s 5 t=10; // s 6 g=9.8; //m/ s ˆ2 7 function y=f(c) 8 y=g*m*(1-exp(-c*t/m))/c - v; 9 endfunction 37
  • 39. 10 x1 =12; 11 x2 =16; 12 xt =14.7802; // true value 13 e=input(” e n t e r the t o l e r a b l e true percent e r r o r=”) 14 xr=x1 -(f(x1)*(x2 -x1))/(f(x2)-f(x1)); 15 etemp=abs(xr -xt)/xt *100; // e r r o r 16 while etemp >e 17 if f(x1)*f(xr) >0 then 18 x1=xr; 19 xr=x1 -(f(x1)*(x2 -x1))/(f(x2)-f(x1)); 20 etemp=abs(xr -xt)/xt *100; 21 end 22 if f(x1)*f(xr) <0 then 23 x2=xr; 24 xr=x1 -(f(x1)*(x2 -x1))/(f(x2)-f(x1)); 25 etemp=abs(xr -xt)/xt *100; 26 end 27 if f(x1)*f(xr)==0 then break; 28 end 29 end 30 disp(xr ,”The r e s u l t i s=”) Scilab code Exa 5.6 Bracketing and False Position Methods 1 clc; 2 clear; 3 function y=f(x) 4 y=x^10 - 1; 5 endfunction 6 x1=0; 7 x2 =1.3; 8 xt=1; 9 10 // using b i s e c t i o n method 11 disp(”BISECTION METHOD: ”) 38
  • 40. 12 xr=(x1+x2)/2; 13 et=abs(xr -xt)/xt *100; // e r r o r 14 disp(1,” I t e r a t i o n : ”) 15 disp(x1 ,” xl : ”) 16 disp(x2 ,”xu : ”) 17 disp(xr ,” xr : ”) 18 disp(et ,” et (%) : ”) 19 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−”) 20 for i=2:5 21 if f(x1)*f(xr) >0 then 22 x1=xr; 23 xr=(x1+x2)/2; 24 ea=abs(xr -x1)/xr *100; 25 et=abs(xr -xt)/xt *100; 26 else if f(x1)*f(xr) <0 then 27 x2=xr; 28 xr=(x1+x2)/2; 29 ea=abs(xr -x2)/xr *100; 30 et=abs(xr -xt)/xt *100; 31 end 32 end 33 if f(x1)*f(xr)==0 then break; 34 end 35 disp(i,” I t e r a t i o n : ”) 36 disp(x1 ,” xl : ”) 37 disp(x2 ,”xu : ”) 38 disp(xr ,” xr : ”) 39 disp(et ,” et (%) : ”) 40 disp(ea ,” ea (%) ”) 41 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−”) 42 end 43 44 // using f a l s e p o s i t i o n method 45 disp(”FALSE POSITION METHOD: ”) 46 x1=0; 47 x2 =1.3; 48 xt=1; 49 xr=x1 -(f(x1)*(x2 -x1))/(f(x2)-f(x1));; 39
  • 41. 50 et=abs(xr -xt)/xt *100; // e r r o r 51 disp(1,” I t e r a t i o n : ”) 52 disp(x1 ,” xl : ”) 53 disp(x2 ,”xu : ”) 54 disp(xr ,” xr : ”) 55 disp(et ,” et (%) : ”) 56 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−”) 57 for i=2:5 58 if f(x1)*f(xr) >0 then 59 x1=xr; 60 xr=x1 -(f(x1)*(x2 -x1))/(f(x2)-f(x1)); 61 ea=abs(xr -x1)/xr *100; 62 et=abs(xr -xt)/xt *100; 63 64 else if f(x1)*f(xr) <0 then 65 x2=xr; 66 xr=x1 -(f(x1)*(x2 -x1))/(f(x2)-f(x1)); 67 ea=abs(xr -x2)/xr *100; 68 et=abs(xr -xt)/xt *100; 69 end 70 end 71 if f(x1)*f(xr)==0 then break; 72 end 73 disp(i,” I t e r a t i o n : ”) 74 disp(x1 ,” xl : ”) 75 disp(x2 ,”xu : ”) 76 disp(xr ,” xr : ”) 77 disp(et ,” et (%) : ”) 78 disp(ea ,” ea (%) ”) 79 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−”) 80 end 40
  • 42. Chapter 6 Open Methods Scilab code Exa 6.1 simple fixed point iteration 1 // c l c ( ) 2 // f ( x ) = exp(−x ) − x ; 3 // using simple f i x e d point i t e r a t i o n , Xi+1 = exp(−Xi ) 4 x = 0; // i n i t i a l guess 5 for i = 1:11 6 if i == 1 then 7 y(i) = x; 8 else 9 y(i) = exp(-y(i-1)); 10 e(i) = (y(i) - y(i-1)) * 100 / y(i); 11 end 12 end 13 disp(y,”x = ”) 14 disp(”%”,e,” e = ”) Scilab code Exa 6.2 The Two curve graphical method 41
  • 43. Figure 6.1: The Two curve graphical method 42
  • 44. 1 // c l c ( ) 2 // y1 = x 3 // y2 = exp(−x ) 4 for i = 1:6 5 if i == 1 then 6 x(i) = 0; 7 else x(i) = x(i-1) + 0.2; 8 end 9 y1(i) = x(i); 10 y2(i) = exp(-x(i)); 11 end 12 disp(x,”x = ”) 13 disp(y1 ,”y1 = ”) 14 disp(y2 ,”y2 = ”) 15 plot(x,y1); 16 plot(x,y2); 17 xtitle(” f ( x ) vs x”,”x”,” f ( x ) ”) 18 // from the graph , we get 19 x7 = 5.7; 20 disp(x7 , ” answer using two curve g r a p h i c a l method = ”); Scilab code Exa 6.3 Newton Raphson Method 1 // c l c ( ) 2 // f ( x ) = exp(−x )−x 3 // f ’ ( x ) = −exp(−x )−1 4 for i = 1:5 5 if i == 1 then 6 x(i) = 0; 7 else 8 x(i) = x(i-1) - (exp(-x(i-1))-x(i-1))/(-exp (-x(i-1)) -1); 9 et(i) = (x(i) - x(i-1)) * 100 / x(i); 10 x(i-1) = x(i); 43
  • 45. 11 end 12 end 13 disp(x) 14 disp(et) Scilab code Exa 6.4 Error analysis of Newton Raphson method 1 // c l c ( ) 2 // f ( x ) = exp(−x ) − x 3 // f ’ ( x ) = −exp(−x ) − 1 4 // f ”( x ) = exp(−x ) 5 xr = 0.56714329; 6 //E( t i +1) = −f ”( x ) ∗ E( t i ) / 2 ∗ f ’ ( x ) 7 Et0 = 0.56714329; 8 Et1 = -exp(-xr)* ((Et0)^2) / (2 * (-exp(-xr) - 1)); 9 disp(” which i s c l o s e to the true e r r o r of 0.06714329 ”,Et1 ,”Et1 = ”) 10 Et1true = 0.06714329; 11 Et2 = -exp(-xr)* (( Et1true)^2) / (2 * (-exp(-xr) - 1)); 12 disp(” which i s c l o s e to the true e r r o r of 0.0008323 ” ,Et2 ,”Et2 = ”) 13 Et2true = 0.0008323; 14 Et3 = -exp(-xr)* (( Et2true)^2) / (2 * (-exp(-xr) - 1)); 15 disp(” which i s c l o s e to the true e r r o r ”,Et3 ,”Et3 = ” ) 16 Et4 = -exp(-xr)* ((Et3)^2) / (2 * (-exp(-xr) - 1)); 17 disp(” which i s c l o s e to the true e r r o r ”,Et4 ,”Et4 = ” ) 18 disp(”Thus i t i l l u s t r a t r e s that the e r r o r of newton raphson method f o r t h i s case i s p r o p o r t i o n a l ( by a f a c t o r of 0.18095) to the square of the e r r o r of the p r e v i o u s i t e r a t i o n ”) 44
  • 46. Scilab code Exa 6.5 slowly converging function with Newton Raphson method 1 // c l c ( ) 2 z = 0.5; 3 // f ( x ) = xˆ10 − 1 4 // f ’ ( x ) = 10∗xˆ9 5 for i = 1:40 6 if i==1 then 7 y(i) = z; 8 else 9 y(i) = y(i-1) - (y(i-1) ^10 - 1) /(10*y(i-1) ^9) 10 end 11 end 12 disp(y) 13 disp(”Thus , a f t e r the f i r s t poor p r e d i c t i o n , the technique i s converging on to the true root of 1 but at a very slow r a t e ”) Scilab code Exa 6.6 The secant method 1 // c l c ( ) 2 funcprot (0) 3 // f ( x ) = exp(−x )−x 4 for i = 1:5 5 if i==1 then 6 x(i) = 0; 7 else 8 if i==2 then 9 x(i) = 1; 10 else 45
  • 47. 11 x(i) =x(i-1) - (exp(-x(i-1))-x(i-1))*(x(i-2) - x(i-1))/(( exp(-x(i-2))-x(i-2)) -(exp(-x (i-1))-x(i-1))) 12 er(i) = (0.56714329 - x(i)) * 100 / 0.56714329; 13 end 14 end 15 end 16 disp(x(1:5) ,”x =”) 17 disp(er (3:5) ,” et =”) Scilab code Exa 6.7 secant and false position techniques 1 // c l c ( ) 2 funcprot (0) 3 // f ( x ) = l og ( x ) 4 disp(” secant method”) 5 for i = 1:4 6 if i==1 then 7 x(i) = 0.5; 8 else 9 if i==2 then 10 x(i) = 5; 11 else 12 x(i) =x(i-1) - log(x(i-1)) * (x(i-2) - x(i -1))/(log(x(i-2)) - log(x(i-1))); 13 end 14 end 15 end 16 disp(x(1:4) ,”x =”) 17 disp(” thus , secant method i s d i v e r g e n t ”) 18 disp(”Now, False p o s i t i o n method”) 19 xl = 0.5; 20 xu = 5; 21 for i = 1:3 46
  • 48. 22 m = log(xl); 23 n = log(xu); 24 xr = xu - n*(xl - xu)/(m - n); 25 disp(xr ,” xr = ”) 26 w = log(xr); 27 if m*w < 0 then 28 xu = xr; 29 else 30 xl = xr; 31 end 32 end 33 34 disp(” thus , f a l s e p o s i t i o n method i s convergent ”) Scilab code Exa 6.8 Modified secant method 1 // c l c ( ) 2 del = 0.01; 3 z = 0.56714329 4 x1 = 1; 5 // f ( x ) = exp(−x ) − x 6 disp(x1 ,”x1 = ”) 7 for i = 1:4 8 if i == 1 then 9 x(i) = 1 10 else 11 w = x(i-1); 12 m = exp(-x(i-1)) - x(i-1); 13 x(i-1) = x(i-1) *(1+ del); 14 n = exp(-x(i-1)) - x(i-1); 15 x(i) = w - (x(i-1)- w) * m/(n-m); 16 em = (z - x(i))*100/z; 17 disp(x(i),”x = ”) 18 disp(”%”,em ,” e r r o r = ”) 19 end 47
  • 49. 20 end Scilab code Exa 6.9 modified newton raphson method 1 // c l c ( ) 2 // f ( x ) = xˆ3 − 5∗xˆ2 + 7∗x −3 3 // f ’ ( x ) = 3∗xˆ2 − 10∗x + 7 4 disp(” standard Newton Raphson method”) 5 for i = 1:7 6 if i == 1 then 7 x(i) = 0; 8 else 9 x(i) = x(i-1) - ((x(i-1))^3 - 5*(x(i-1))^2 + 7*x(i-1) -3)/(3*(x(i-1))^2 - 10*(x(i -1)) + 7); 10 et(i) = (1 - x(i)) * 100 / 1; 11 disp(x(i),”x = ”) 12 disp(”%”,et(i),” e r r o r = ”) 13 x(i-1) = x(i); 14 end 15 end 16 disp(” Modified Newton Raphson method”) 17 // f ”( x ) = 6∗x − 10 18 for i = 1:4 19 if i == 1 then 20 x(i) = 0; 21 else 22 x(i) = x(i-1) - ((x(i-1))^3 - 5*(x(i-1))^2 + 7*x(i-1) -3)*((3*(x(i-1))^2 - 10*(x(i -1)) + 7))/((3*(x(i-1))^2 - 10*(x(i-1)) + 7)^2 - ((x(i-1))^3 - 5*(x(i-1))^2 + 7* x(i-1) -3) * (6*x(i-1) - 10)); 23 et(i) = (1 - x(i)) * 100 / 1; 24 disp(x(i),”x = ”) 25 disp(”%”,et(i),” e r r o r = ”) 48
  • 50. 26 x(i-1) = x(i); 27 end 28 end Scilab code Exa 6.10 fixed point iteration for nonlinear system 1 // c l c ( ) 2 //u ( x , y ) = xˆ2 + x∗y − 10 3 //v ( x , y ) = y + 3∗x∗yˆ2 −57 4 for i = 1:4 5 if i == 1 then 6 x(i) = 1.5; 7 y(i) = 3.5; 8 else 9 x(i) = sqrt (10 - (x(i-1))*(y(i-1))); 10 y(i) = sqrt ((57 - y(i-1))/(3*x(i))); 11 disp(x(i),”x =”) 12 disp(y(i),”y =”) 13 end 14 end 15 disp(”Thus the approaching to the true value 0 f x = 2 and y = 3”) Scilab code Exa 6.11 Newton Raphson for a nonlinear Problem 1 clc; 2 clear; 3 function z=u(x,y) 4 z=x^2+x*y-10 5 endfunction 6 function z=v(x,y) 7 z=y+3*x*y^2-57 8 endfunction 49
  • 51. 9 x=1.5; 10 y=3.5; 11 e=[100 100]; 12 while e(1) >0.0001 & e(2) >0.0001 13 J=[2*x+y x; 3*y^2 1+6*x*y]; 14 deter=determ(J); 15 u1=u(x,y); 16 v1=v(x,y); 17 x=x-((u1*J(2,2)-v1*J(1,2))/deter); 18 y=y-((v1*J(1,1)-u1*J(2,1))/deter); 19 e(1)=abs(2-x); 20 e(2)=abs(3-y); 21 end 22 bracket =[x y]; 23 disp(bracket) 50
  • 52. Chapter 7 Roots of Polynomials Scilab code Exa 7.1 Polynomial Deflation 1 clc; 2 clear; 3 function y=f(x) 4 y=(x-4)*(x+6) 5 endfunction 6 n=2; 7 a(1) =-24; 8 a(2) =2; 9 a(3) =1; 10 t=4; 11 r=a(3); 12 a(3) =0; 13 for i=(n):-1:1 14 s=a(i); 15 a(i)=r; 16 r=s+r*t; 17 end 18 disp(”The quptient i s a (1 )+a (2 ) ∗x where : ”) 19 disp(a(1),”a (1 )=”) 20 disp(a(2),”a (2 )=”) 21 disp(r,” remainder=”) 51
  • 53. Scilab code Exa 7.2 Mullers Method 1 clc; 2 clear; 3 function y=f(x) 4 y=x^3 - 13*x - 12 5 endfunction 6 7 x1t=-3; 8 x2t=-1; 9 x3t =4; 10 x0 =4.5; 11 x1 =5.5; 12 x2=5; 13 disp(0,” i t e r a t i o n : ”) 14 disp(x2 ,” xr : ”) 15 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−” ) 16 for i=1:4 17 18 h0=x1 -x0; 19 h1=x2 -x1; 20 d0=(f(x1)-f(x0))/(x1 -x0); 21 d1=(f(x2)-f(x1))/(x2 -x1); 22 a=(d1 -d0)/(h1+h0); 23 b=a*h1+d1; 24 c=f(x2); 25 d=(b^2 - 4*a*c)^0.5; 26 if abs(b+d)>abs(b-d) then x3=x2+(( -2*c)/(b+d)); 27 else x3=x2+(( -2*c)/(b-d)); end 28 ea=abs(x3 -x2)*100/ x3; 29 x0=x1; 30 x1=x2; 31 x2=x3; 52
  • 54. 32 disp(i,” i t e r a t i o n : ”) 33 disp(x2 ,” xr : ”) 34 disp(ea ,” ea (%) : ”) 35 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−” ) 36 end Scilab code Exa 7.3 Bairstows Method 1 clc; 2 clear; 3 function y=f(x) 4 y=x^5 -3.5*x^4+2.75*x^3+2.125*x^2 -3.875*x+1.25 5 endfunction 6 r=-1; 7 s=-1; 8 es=1; //% 9 n=6; 10 count =1; 11 ear =100; 12 eas =100; 13 a=[1.25 -3.875 2.125 2.75 -3.5 1]; 14 while (ear >es) and (eas >es) 15 16 b(n)=a(n); 17 b(n-1)=a(n-1)+r*b(n); 18 for i=n-2: -1:1 19 b(i)=a(i)+r*b(i+1)+s*b(i+2); 20 end 21 c(n)=b(n); 22 c(n-1)=b(n-1)+r*c(n); 23 for i=(n-2) :-1:2 24 c(i)=b(i)+r*c(i+1)+s*c(i+2); 25 end 26 // c ( 3) ∗ dr+c ( 4) ∗ ds=−b (2 ) 53
  • 55. 27 // c ( 2) ∗ dr+c ( 3) ∗ ds=−b (1 ) 28 ds=((-b(1))+(b(2)*c(2)/c(3)))/(c(3) -(c(4)*c(2 )/c(3))); 29 dr=(-b(2)-c(4)*ds)/c(3); 30 r=r+dr; 31 s=s+ds; 32 ear=abs(dr/r)*100; 33 eas=abs(ds/s)*100; 34 disp(count ,” I t e r a t i o n : ”) 35 disp(dr ,” d e l a t a r : ”) 36 disp(ds ,” d e l a t a s : ”) 37 disp(r,” r : ”) 38 disp(s,” s : ”) 39 disp(ear ,” Error in r : ”) 40 disp(eas ,” Error in s : ”) 41 disp(” −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 42 count=count +1; 43 end 44 x1=(r+(r^2 + 4*s)^0.5) /2; 45 x2=(r-(r^2 + 4*s)^0.5) /2; 46 bracket =[x1 x2]; 47 disp(bracket ,”The r o o t s are : ”) 48 disp(”xˆ3 − 4∗xˆ2 + 5.25∗ x − 2. 5 ”,”The q u o t i e n t i s : ” ) 49 disp(” −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) Scilab code Exa 7.4 Locate single root 1 clc; 2 clear; 3 function y=f(x) 54
  • 56. 4 y=x-cos(x) 5 endfunction 6 x1=0; 7 if f(x1) <0 then 8 x2=x1 +0.001; 9 while f(x2) <0 10 x2=x2 +0.001; 11 end 12 elseif x2=x1 +0.001; 13 while f(x2) >0 14 x2=x2 +0.001; 15 end 16 else disp(x1 ,”The root i s=”) 17 end 18 x=x2 -(x2 -x1)*f(x2)/(f(x2)-f(x1)); 19 disp(x,”The root i s=”) Scilab code Exa 7.5 Solving nonlinear system 1 clc; 2 clear; 3 function z=u(x,y) 4 z=x^2+x*y-10 5 endfunction 6 function z=v(x,y) 7 z=y+3*x*y^2-57 8 endfunction 9 x=1; 10 y=3.5; 11 while u(x,y)~=v(x,y) 12 x=x+1; 13 y=y -0.5; 14 end 15 disp(x,”x=”) 16 disp(y,”y=”) 55
  • 57. Scilab code Exa 7.6 Root Location 1 clc; 2 clear; 3 x=poly(0, ’ s ’ ); 4 p=x^10 -1; 5 disp(”The r o o t s of the polynomial are : ”) 6 disp(roots(p)) Scilab code Exa 7.7 Roots of Polynomials 1 clc; 2 clear; 3 x=poly(0, ’ s ’ ); 4 p=x^5 - 3.5*x^4 +2.75*x^3 +2.125*x^2 - 3.875*x + 1.25; 5 disp(”The r o o t s of the polynomial are : ”) 6 disp(roots(p)) Scilab code Exa 7.8 Root Location 1 clc; 2 clear; 3 function y=f(x) 4 y=x-cos(x) 5 endfunction 6 x1=0; 7 if f(x1) <0 then 8 x2=x1 +0.00001; 56
  • 58. 9 while f(x2) <0 10 x2=x2 +0.00001; 11 end 12 elseif x2=x1 +0.00001; 13 while f(x2) >0 14 x2=x2 +0.00001; 15 end 16 else disp(x1 ,”The root i s=”) 17 end 18 x=x2 -(x2 -x1)*f(x2)/(f(x2)-f(x1)); 19 disp(x,”The root i s=”) 57
  • 59. Chapter 9 Gauss Elimination Scilab code Exa 9.1 Graphical Method for two Equations 1 clc; 2 clear; 3 // the equations are : 4 // 3∗ x1 + 2∗ x2=18 and −x1 + 2∗ x2=2 5 6 // equation 1 becomes , 7 // x2=−(3/2)∗x1 + 9 8 // equation 2 becomes , 9 // x2=−(1/2)∗x1 + 1 10 11 // p l o t t i n g equation 1 12 for x1 =1:6 13 x2(x1)= -(3/2)*x1 + 9; 14 end 15 x1=[1 2 3 4 5 6]; 16 // p l o t t i n g equation 2 17 for x3 =1:6 18 x4(x3)=(1/2)*x3 + 1; 19 end 58
  • 60. Figure 9.1: Graphical Method for two Equations 59
  • 61. 20 x3=[1 2 3 4 5 6]; 21 plot(x1 ,x2) 22 plot(x3 ,x4) 23 xtitle(”x2 vs x1”,”x1”,”x2”) 24 // the l i n e s meet at x1=4 amd x2=3 25 disp(3,”x2=”,”and” ,4,”x1=”,”The l i n e s meet at=”) Scilab code Exa 9.2 Deterinants 1 clc; 2 clear; 3 // For f i g 9 . 1 4 a=[3 2;-1 2]; 5 disp(determ(a),”The value of determinant f o r system re pe sen te d in f i g 9.1 =”) 6 // For f i g 9 . 2 ( a ) 7 a=[ -0.5 1; -0.5 1]; 8 disp(determ(a),”The value of determinant f o r system re pe sen te d in f i g 9.2 ( a ) =”) 9 // For f i g 9 . 2 ( b ) 10 a=[ -0.5 1;-1 2]; 11 disp(determ(a),”The value of determinant f o r system re pe sen te d in f i g 9.2 ( b ) =”) 12 // For f i g 9 . 2 ( c ) 13 a=[ -0.5 1; -2.3/5 1]; 14 disp(determ(a),”The value of determinant f o r system re pe sen te d in f i g 9.2 ( c ) =”) Scilab code Exa 9.3 Cramers Rule 1 clc; 2 clear; 3 // the matrix or the system 60
  • 62. 4 b1= -0.01; 5 b2 =0.67; 6 b3= -0.44; 7 a=[0.3 0.52 1;0.5 1 1.9;0.1 0.3 0.5]; 8 a1=[a(2,2) a(2,3);a(3,2) a(3,3)]; 9 A1=determ(a1); 10 a2=[a(2,1) a(2,3);a(3,1) a(3,3)]; 11 A2=determ(a2); 12 a3=[a(2,1) a(2,2);a(3,1) a(3,2)]; 13 A3=determ(a3); 14 D=a(1,1)*A1 -a(1,2)*A2+a(1,3)*A3; 15 p=[b1 0.52 1;b2 1 1.9; b3 0.3 0.5]; 16 q=[0.3 b1 1;0.5 b2 1.9;0.1 b3 0.5]; 17 r=[0.3 0.52 b1 ;0.5 1 b2 ;0.1 0.3 b3]; 18 x1=det(p)/D; 19 x2=det(q)/D; 20 x3=det(r)/D; 21 disp(”The v a l u e s are : ”) 22 disp(x1 ,”x1=”) 23 disp(x2 ,”x2=”) 24 disp(x3 ,”x3=”) Scilab code Exa 9.4 Elimination of Unknowns 1 clc; 2 clear; 3 // the equations are : 4 // 3∗ x1+2∗x2=18 5 //−x1+2∗x2=2 6 a11 =3; 7 a12 =2; 8 b1=18; 9 a21=-1; 10 a22 =2; 11 b2=2; 61
  • 63. 12 x1=(b1*a22 -a12*b2)/(a11*a22 -a12*a21); 13 x2=(b2*a11 -a21*b1)/(a11*a22 -a12*a21); 14 disp(x1 ,”x1=”) 15 disp(x2 ,”x2=”) Scilab code Exa 9.5 Naive Gauss Elimination 1 clc; 2 clear; 3 n=3; 4 b(1) =7.85; 5 b(2) = -19.3; 6 b(3) =71.4; 7 a=[3 -0.1 -0.2;0.1 7 -0.3;0.3 -0.2 10]; 8 for k=1:n-1 9 for i=k+1:n 10 fact=a(i,k)/a(k,k); 11 for j=k+1:n 12 a(i,j)=a(i,j)-fact*(a(k,j)); 13 end 14 b(i)=b(i)-fact*b(k); 15 end 16 end 17 x(n)=b(n)/a(n,n); 18 for i=n-1: -1:1 19 s=b(i); 20 for j=i+1:n 21 s=s-a(i,j)*x(j) 22 end 23 x(i)=b(i)/a(i,i); 24 end 25 disp(x(1),”x1=”) 26 disp(x(2),”x2=”) 27 disp(x(3),”x3=”) 62
  • 64. Scilab code Exa 9.6 ill conditioned systems 1 clc; 2 clear; 3 a11 =1; 4 a12 =2; 5 b1=10; 6 a21 =1.1; 7 a22 =2; 8 b2 =10.4; 9 x1=(b1*a22 -a12*b2)/(a11*a22 -a12*a21); 10 x2=(b2*a11 -a21*b1)/(a11*a22 -a12*a21); 11 disp(”For the o r i g i n a l system : ”) 12 disp(x1 ,”x1=”) 13 disp(x2 ,”x2=”) 14 a21 =1.05; 15 x1=(b1*a22 -a12*b2)/(a11*a22 -a12*a21); 16 x2=(b2*a11 -a21*b1)/(a11*a22 -a12*a21); 17 disp(”For the new system : ”) 18 disp(x1 ,”x1=”) 19 disp(x2 ,”x2=”) Scilab code Exa 9.7 Effect of Scale on Determinant 1 clc; 2 clear; 3 // part a 4 a=[3 2;-1 2]; 5 b1=18; 6 b2=2; 7 disp(determ(a),”The determinant f o r part ( a )=”) 8 // part b 63
  • 65. 9 a=[1 2;1.1 2]; 10 b1 =10; 11 b2 =10.4; 12 disp(determ(a),”The determinant f o r part ( b )=”) 13 // part c 14 a1=a*10; 15 b1 =100; 16 b2 =104; 17 disp(determ(a1),”The determinant f o r part ( c )=”) Scilab code Exa 9.8 Scaling 1 clc; 2 clear; 3 // part a 4 a=[1 0.667; -0.5 1]; 5 b1=6; 6 b2=1; 7 disp(determ(a),”The determinant f o r part ( a )=”) 8 // part b 9 a=[0.5 1;0.55 1]; 10 b1=5; 11 b2 =5.2; 12 disp(determ(a),”The determinant f o r part ( b )=”) 13 // part c 14 b1=5; 15 b2 =5.2; 16 disp(determ(a),”The determinant f o r part ( c )=”) Scilab code Exa 9.9 Partial Pivoting 1 // c l c ( ) 2 // 0.0003∗ x1 + 3∗ x2 = 2.0001 64
  • 66. 3 // 1∗ x1 + 1∗ x2 = 1 4 a11 = 0.000; 5 // m u l t i p l y i n g f i r s t equation by 1/0.0003 , we get , x1 + 10000∗ x2 = 6667 6 x2 = (6667 -1) /(10000 -1); 7 x1 = 6667 - 10000* x2; 8 disp(x2 ,”x2 = ”) 9 disp(x1 ,”x1 = ”) 10 disp(”The e r r o r v a r i e s depending on the no . of s i g n i f i c a n t f i g u r e s used ”) Scilab code Exa 9.10 Effect of scaling on Pivoting and round off 1 // c l c ( ) 2 // 2∗ x1 + 10000∗ x2 = 10000 3 // x1 + x2 = 2 4 x1 = 1; 5 x2 = 1; 6 disp(” without s c a l i n g , applying forward e l i m i n a t i o n ” ) 7 // x1 i s too small and can be n e g l e c t e d 8 x21 = 10000/10000; 9 x11 = 0; 10 e1 = (x1 - x11)*100/ x1; 11 disp(x21 ,”x2 = ”) 12 disp(x11 ,”x1 = ”) 13 disp(e1 ,” e r r o r f o r x1 = ”) 14 disp(” with s c a l i n g ”) 15 // 0.00002∗ x1 + x2 = 1 16 //now x1 i s n e g l e c t e d because of the co e f f i c i e n t 17 x22 = 1; 18 x12 = 2 - x1; 19 disp(x12 ,”x1 = ”) 20 disp(x22 ,”x2 = ”) 21 // using o r i g i n a l co e f f i c i e n t 65
  • 67. 22 // x1 can be n e g l e c t e d 23 disp(” pivot and r e t a i n i n g o r i g i n a l c o e f f i c i e n t s ”) 24 x22 = 10000/10000; 25 x12 = 2 - x1; 26 disp(x12 ,”x1 = ”) 27 disp(x22 ,”x2 = ”) Scilab code Exa 9.11 Solution of Linear Algebraic Equations 1 clc; 2 clear; 3 a=[70 1 0;60 -1 1;40 0 -1]; 4 b=[636;518;307]; 5 x=abs(linsolve(a,b)); 6 disp(”m/ s ˆ2”,x(1,1),”a=”) 7 disp(”N”,x(2,1),”T=”) 8 disp(”N”,x(3,1),”R=”) Scilab code Exa 9.12 Gauss Jordan method 1 // c l c ( ) 2 // 3∗ x1 − 0.1∗ x2 − 0.2∗ x3 = 7.85 3 // 0.1∗ x1 + 7∗ x2 − 0.3∗ x3 = −19.3 4 // 0.3∗ x1 − 0.2∗ x2 + 10∗ x3 = 71.4 5 // t h i s can be w r i t t e n in matrix form as 6 A = [3 , -0.1 , -0.2 ,7.85;0.1 ,7 , -0.3 , -19.3;0.3 , -0.2 ,10 ,71.4]; 7 disp(A,” Equation in matrix form can be w r i t t e n as ”) 8 X = A(1,:) / det(A(1,1)); 9 Y = A(2,:) - 0.1*X; 10 Z = A(3,:) - 0.3*X; 11 Y = Y/det(Y(1,2)); 66
  • 68. 12 X = X - Y * det(X(1,2)); 13 Z = Z - Y * det(Z(1,2)); 14 Z = Z/det(Z(1,3)); 15 X = X - Z*det(X(1,3)); 16 Y = Y - Z*det(Y(1,3)); 17 A = [X;Y;Z]; 18 disp(A,” f i n a l matrix = ”) 19 disp(det(A(1,4)),”x1 = ”) 20 disp(det(A(2,4)),”x2 = ”) 21 disp(det(A(3,4)),”x3 = ”) 67
  • 69. Chapter 10 LU Decomposition and matrix inverse Scilab code Exa 10.1 LU decomposition with gauss elimination 1 // c l c ( ) 2 A = [3 , -0.1 , -0.2;0.1 ,7 , -0.3;0.3 , -0.2 ,10]; 3 U = A; 4 disp(A,”A =”) 5 m = det(U(1,1)); 6 n = det(U(2,1)); 7 a = n/m; 8 U(2,:) = U(2,:) - U(1,:) / (m/n); 9 n = det(U(3,1)); 10 b = n/m; 11 U(3,:) = U(3,:) - U(1,:) / (m/n); 12 m = det(U(2,2)); 13 n = det(U(3,2)); 14 c = n/m; 15 U(3,:) = U(3,:) - U(2,:) / (m/n); 16 disp(U,”U = ”) 17 L = [1,0,0;a,1,0;b,c ,1]; 18 disp(L,”L c a l c u l a t e d based on gauss e l i m i n a t i o n method = ”) 68
  • 70. Scilab code Exa 10.2 The substitution steps 1 // c l c ( ) 2 A = [3 , -0.1 , -0.2;0.1 ,7 , -0.3;0.3 , -0.2 ,10]; 3 B = [7.85; -19.3;71.4]; 4 X = inv(A) * B; 5 disp(X,”X = ”) Scilab code Exa 10.3 Matrix inversion 1 // c l c ( ) 2 A = [3 , -0.1 , -0.2;0.1 ,7 , -0.3;0.3 , -0.2 ,10]; 3 //B = inv (A) 4 L = [1 ,0 ,0;0.033333 ,1 ,0;0.1 , -0.02713 ,1]; 5 U = [3 , -0.1 , -0.2;0 ,7.0033 , -0.293333;0 ,0 ,10.012]; 6 for i =1:3 7 if i==1 then 8 m = [1;0;0]; 9 else 10 if i==2 then 11 m = [0;1;0]; 12 else 13 m = [0;0;1]; 14 end 15 end 16 d = inv(L) * m; 17 x = inv(U) * d; 18 B(:,i) = x 19 end 20 disp(B) 69
  • 71. Scilab code Exa 10.4 Matrix condition evaluation 1 // c l c ( ) 2 A = [1 ,1/2 ,1/3;1/2 ,1/3 ,1/4;1/3 ,1/4 ,1/5]; 3 n = det(A(2,1)); 4 A(2,:) = A(2,:)/n; 5 n = det(A(3,1)); 6 A(3,:) = A(3,:)/n; 7 B = inv(A); 8 disp(A,”A = ”) 9 for j = 1:3 10 a = 0; 11 for i = 1:3 12 m(i) = det(A(j,i)); 13 su(j) = a + m(i); 14 a = su(j); 15 end 16 end 17 if su(1) < su(2) then 18 if su(2) < su(3) then 19 z = su(3); 20 else 21 z = su(2); 22 end 23 else 24 if su(1) < su(3) then 25 z = su(3); 26 else 27 z = su(1); 28 end 29 end 30 for j = 1:3 31 a = 0; 32 for i = 1:3 70
  • 72. 33 m(i) = det(B(j,i)); 34 sm(j) = a + abs(m(i)); 35 a = sm(j); 36 end 37 end 38 if sm(1) < sm(2) then 39 if sm(2) < sm(3) then 40 y = sm(3); 41 else 42 y = sm(2); 43 end 44 else 45 if sm(1) < sm(3) then 46 y = sm(3); 47 else 48 y = sm(1); 49 end 50 end 51 C = z*y; 52 disp(C,” Condition number f o r the matrix =”) 71
  • 73. Chapter 11 Special Matrices and gauss seidel Scilab code Exa 11.1 Tridiagonal solution with Thomas algorithm 1 // c l c ( ) 2 A = [2.04 , -1 ,0 ,0; -1 ,2.04 , -1 ,0;0 ,-1,2.04 ,-1;0 ,0, -1 ,2.04]; 3 B = [40.8;0.8;0.8;200.8]; 4 g = det(A(1,2)); 5 f1 =det(A(1,1)); 6 e2 = det(A(2,1))/f1; 7 f2 = det(A(1,1)) - e2 * det(A(2,1)); 8 e3 = det(A(2,1))/f2; 9 f3 = det(A(1,1)) - e3 * det(A(2,1)); 10 e4 = det(A(2,1))/f3; 11 f4 = det(A(1,1)) - e4 * det(A(2,1)); 12 M = [f1 ,g,0,0;e2 ,f2 ,g,0;0,e3 ,f3 ,g;0,0,e4 ,f4]; 13 L = [1,0,0,0;det(M(2,1)) ,1,0,0;0,det(M(3,2)) ,1,0;0,0,det(M(4,3)) ,1]; 14 U = [det(M(1,1)),g,0,0;0,det(M(2,2)),g,0;0,0,det(M (3,3)),g;0,0,0,det(M(4,4))]; 15 r1 = det(B(1,1)); 72
  • 74. 16 r2 = det(B(2,1)) - e2*det(B(1,1)); 17 r3 = det(B(3,1)) - e3*r2; 18 r4= det(B(4,1)) - e4*r3; 19 N = [r1;r2;r3;r4]; 20 T4 = r4/det(U(4,4)); 21 T3 = (r3 - g*T4)/det(U(3,3)); 22 T2 = (r2 - g*T3)/det(U(2,2)); 23 T1 = (r1 - g*T2)/det(U(1,1)); 24 disp(T1 ,”T1 = ”) 25 disp(T2 ,”T2 = ”) 26 disp(T3 ,”T3 = ”) 27 disp(T4 ,”T4 = ”) Scilab code Exa 11.2 Cholesky Decomposition 1 // c l c ( ) 2 A = [6 ,15 ,55;15 ,55 ,225;55 ,225 ,979]; 3 sl = 0 4 l11 = (det(A(1,1)))^(1/2); 5 // f o r second row 6 l21 = (det(A(2,1)))/l11; 7 l22 = (det(A(2,2)) - l21 ^2) ^(0.5); 8 // f o r t h i r d row 9 l31 = (det(A(3,1)))/l11; 10 l32 = (det(A(3,2)) - l21*l31)/l22; 11 l33 = (det(A(3,3)) - l31^2 - l32 ^2) ^(0.5); 12 L = [l11 ,0,0;l21 ,l22 ,0;l31 ,l32 ,l33]; 13 disp(L,”L = ”) Scilab code Exa 11.3 Gauss Seidel method 1 // c l c ( ) 2 //3x − 0. 1 y − 0. 2 z = 7.85 73
  • 75. 3 // 0. 1 x + 7y − 0. 3 z = −19.3 4 // 0. 3 x − 0. 2 y + 10 z = 71.4 5 Y = 0; 6 Z = 0; 7 for i = 1:2 8 x(i) = (7.85 +0.1*Y+0.2*Z)/3; 9 X = x(i); 10 y(i) = ( -19.3 - 0.1*X +0.3*Z)/7; 11 Y = y(i); 12 z(i) = (71.4 - 0.3*X+0.2*Y)/10; 13 Z = z(i); 14 if i==2 then 15 ex = (x(i) - x(i-1))*100/x(i); 16 ey = (y(i) - y(i-1))*100/y(i); 17 ez = (z(i) - z(i-1))*100/z(i); 18 end 19 end 20 disp(x(1:2) ,”x through two i t e r a t i o n s =”) 21 disp(y(1:2) ,”y through two i t e r a t i o n s =”) 22 disp(z(1:2) ,” z through two i t e r a t i o n s =”) 23 disp(”%”,ex ,” e r r o r of x = ”) 24 disp(”%”,ey ,” e r r o r of y = ”) 25 disp(”%”,ez ,” e r r o r of z = ”) Scilab code Exa 11.4 Linear systems 1 // c l c ( ) 2 A = [1 ,0.5 ,1/3;1 ,2/3 ,1/2;1 ,3/4 ,3/5]; 3 B = [1.833333;2.166667;2.35]; 4 U = inv(A); 5 X = U*B; 6 x = det(X(1,1)); 7 y = det(X(2,1)); 8 z = det(X(3,1)); 9 disp(x,”x = ”) 74
  • 76. 10 disp(y,”y = ”) 11 disp(z,” z = ”) Scilab code Exa 11.5 Manipulate linear algebraic equations 1 // c l c ( ) 2 A = [1 ,0.5 ,1/3;1 ,2/3 ,1/2;1 ,3/4 ,3/5]; 3 B = [1.833333;2.166667;2.35]; 4 U = inv(A); 5 X = U*B; 6 x = det(X(1,1)); 7 y = det(X(2,1)); 8 z = det(X(3,1)); 9 disp(x,”x = ”) 10 disp(y,”y = ”) 11 disp(z,” z = ”) Scilab code Exa 11.6 Analyze and solve Hilbert matrix 1 // c l c ( ) 2 A = [1 ,0.5 ,1/3;1/2 ,1/3 ,1/4;1/3 ,1/4 ,1/5]; 3 B = [1.833333;1.083333;0.783333]; 4 U = inv(A); 5 X = U*B; 6 x = det(X(1,1)); 7 y = det(X(2,1)); 8 z = det(X(3,1)); 9 disp(x,”x = ”) 10 disp(y,”y = ”) 11 disp(z,” z = ”) 75
  • 77. Chapter 13 One dimensional unconstrained optimization Scilab code Exa 13.1 Golden section method 1 // c l c ( ) 2 // f ( x ) = 2 sin x − x ˆ2/10 3 xl(1) = 0; 4 xu(1) = 4; 5 for i = 1:10 6 d(i) = ((5) ^(0.5) - 1)*(xu(i) - xl(i))/2; 7 x1(i) = xl(i) + d(i); 8 x2(i) = xu(i) - d(i); 9 m(i) = 2*sin(x1(i)) - (x1(i)^2) /10; 10 n(i) = 2*sin(x2(i)) - (x2(i)^2) /10; 11 if n(i) > m(i) then 12 xu(i+1) = x1(i); 13 xl(i+1) = xl(i); 14 else 15 xl(i+1) = x2(i); 16 xu(i+1) = xu(i); 17 end 18 end 19 disp(xl ,” xl = ”) 76
  • 78. 20 disp(x2 ,”x2 = ”) 21 disp(x1 ,”x1 = ”) 22 disp(xu ,”xu = ”) Scilab code Exa 13.2 Quadratic interpolation 1 // c l c ( ) 2 // f ( x ) = 2 sin x − x ˆ2/10 3 x0(1) = 0; 4 x1(1) = 1 5 x2(1) = 4; 6 for i = 1:6 7 m(i) = 2*sin(x0(i)) - (x0(i)^2) /10; 8 n(i) = 2*sin(x1(i)) - (x1(i)^2) /10; 9 r(i) = 2*sin(x2(i)) - (x2(i)^2) /10; 10 x3(i) = ((m(i)*(x1(i) ^ 2 -x2(i) ^ 2)) + (n(i)*( x2(i) ^ 2 -x0(i) ^ 2)) + (r(i)*(x0(i) ^ 2 -x1 (i) ^ 2)))/((2*m(i)*(x1(i) -x2(i)))+(2*n(i)*( x2(i) -x0(i)))+(2*r(i)*(x0(i) -x1(i)))); 11 s(i) = 2*sin(x3(i)) - (x3(i)^2) /10; 12 if x1(i) > x3(i) then 13 if n(i)<s(i) then 14 x0(i+1) = x0(i); 15 x1(i+1) = x3(i); 16 x2(i+1) = x1(i); 17 else 18 x0(i+1) = x1(i); 19 x1(i+1) = x3(i); 20 x2(i+1) = x2(i); 21 end 22 else 23 if n(i)>s(i) then 24 x0(i+1) = x0(i); 25 x1(i+1) = x3(i); 26 x2(i+1) = x1(i); 77
  • 79. 27 else 28 x0(i+1) = x1(i); 29 x1(i+1) = x3(i); 30 x2(i+1) = x2(i); 31 end 32 end 33 end 34 disp(x0 (1:6) ,”x0 = ”) 35 disp(x1 (1:6) ,”x1 = ”) 36 disp(x3 (1:6) ,”x3 = ”) 37 disp(x2 (1:6) ,”x2 = ”) Scilab code Exa 13.3 Newtons method 1 // c l c ( ) 2 // f ( x ) = 2 sin x − x ˆ2/10 3 x(1) = 2.5; 4 // f ’ ( x ) = 2 cosx − x/5 5 // f ”( x ) = −2sin x − 1/5 6 for i = 2:10 7 x(i) = x(i-1) - (2* cos(x(i-1)) - x(i-1) /5) /(-2* sin (x(i-1)) - 1/5); 8 end 9 disp(x,”x = ”) 78
  • 80. Chapter 14 Multidimensional Unconstrainted Optimization Scilab code Exa 14.1 Random Search Method 1 clc; 2 clear; 3 function z=f(x,y) 4 z=y-x -(2*(x^2)) -(2*x*y) -(y^2); 5 endfunction 6 x1=-2; 7 x2=2; 8 y1=1; 9 y2=3; 10 fmax = -1*10^( -15); 11 n=10000; 12 for j=1:n 13 r=rand (1,2); 14 x=x1+(x2 -x1)*r(1,1); 15 y=y1+(y2 -y1)*r(1,2); 16 fn=f(x,y); 17 if fn >fmax then 18 fmax=fn; 19 xmax=x; 79
  • 81. 20 ymax=y; 21 end 22 if modulo(j ,1000) ==0 then 23 24 disp(j,” I t e r a t i o n : ”) 25 disp(x,”x : ”) 26 disp(y,”y : ”) 27 disp(fn ,” f u n c t i o n value : ”) 28 disp(”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 29 end 30 end Scilab code Exa 14.2 Path of Steepest Descent 1 clc; 2 clear; 3 function z=f(x,y) 4 z=x*y*y 5 endfunction 6 p1=[2 2]; 7 elevation=f(p1(1),p1(2)); 8 dfx=p1(1)*p1(1); 9 dfy =2*p1(1)*p1(2); 10 theta=atan(dfy/dfx); 11 slope =(dfx^2 + dfy ^2) ^0.5; 12 disp(elevation ,” Elevation : ”) 13 disp(theta ,” Theta : ”) 14 disp(slope ,” s l o p e : ”) Scilab code Exa 14.3 1 D function along Gradient 1 clc; 80
  • 82. 2 clear; 3 function z=f(x,y) 4 z=2*x*y + 2*x - x^2 - 2*y^2 5 endfunction 6 x=-1; 7 y=1; 8 dfx =2*y+2-2*x; 9 dfy =2*x-4*y; 10 // the f u n c t i o n can thus be expressed along h a x i s as 11 // f ( ( x+dfx ∗h ) ,( y+dfy ∗h ) ) 12 disp(” 180∗hˆ2 + 72∗h − 7”,”The f i n a l equation i s=”) Scilab code Exa 14.4 Optimal Steepest Descent 1 clc; 2 clear; 3 function z=f(x,y) 4 z=2*x*y + 2*x - x^2 - 2*y^2 5 endfunction 6 x=-1; 7 y=1; 8 d2fx =-2; 9 d2fy =-4; 10 d2fxy =2; 11 12 modH=d2fx*d2fy -( d2fxy)^2; 13 14 for i=1:25 15 dfx =2*y+2-2*x; 16 dfy =2*x - 4*y; 17 // the f u n c t i o n can thus be expressed along h a x i s as 18 // f ( ( x+dfx ∗h ) ,( y+dfy ∗h ) ) 19 function d=g(h) 20 d=2*(x+dfx*h)*(y+dfy*h) + 2*(x+dfx*h) - (x+dfx*h )^2 - 2*(y+dfy*h)^2 81
  • 83. 21 endfunction 22 // 2∗ dfx ∗( y+dfy ∗h )+2∗dfy ∗( x+dfx ∗h )+2∗dfx −2∗(x+dfx ∗h ) ∗ dfx −4∗(y+dfy ∗h ) ∗ dfy=g ’ ( h )=0 23 // 2∗ dfx ∗y + 2∗ dfx ∗ dfy ∗h + 2∗ dfy ∗x + 2∗ dfy ∗ dfx ∗h + 2∗ dfx − 2∗x∗ dfx − 2∗ dfx ∗ dfx ∗h − 4∗y∗ dfy − 4∗ dfy ∗ dfy ∗h=0 24 //h (2∗ dfx ∗ dfy+2∗dfy ∗ dfx −2∗dfx ∗ dfx −4∗dfy ∗ dfy )=−(2∗dfx ∗y+2∗dfy ∗x−2∗x∗ dfx −4∗y∗ dfy ) 25 h=(2* dfx*y+2* dfy*x-2*x*dfx -4*y*dfy +2* dfx)/( -1*(2* dfx *dfy +2* dfy*dfx -2* dfx*dfx -4* dfy*dfy)); 26 x=x+dfx*h; 27 y=y+dfy*h; 28 end 29 disp ([x y],”The f i n a l v a l u e s are : ”) 82
  • 84. Chapter 15 Constrained Optimization Scilab code Exa 15.1 Setting up LP problem 1 clc; 2 clear; 3 regular =[7 10 9 150]; 4 premium =[11 8 6 175]; 5 res_avail =[77 80]; 6 // t o t a l p r o f i t ( to be maximized )=z =150∗x1+175∗x2 7 // t o t a l gas used=7∗x1+11∗x2 ( has to be l e s s than 77 mˆ3/ week ) 8 // s i m i l a r l y other c o n s t r a i n t s are developed 9 disp(” Maximize z=150∗x1+175∗x2”) 10 disp(” s u b j e c t to ”) 11 disp(”7∗ x1+11∗x2<=77 ( Material c o n s t r a i n t ) ”) 12 disp(” 10∗ x1+8∗x2<=80 ( Time c o n s t r a i n t ) ”) 13 disp(”x1<=9 ( Regular s t o r a g e c o n s t r a i n t ) ”) 14 disp(”x2<=6 ( Premium s t o r a g e c o n s t r a i n t ) ”) 15 disp(”x1 , x2>=0 ( P o s i t i v i t y c o n s t r a i n t ) ”) 83
  • 85. Figure 15.1: Graphical Solution 84
  • 86. Scilab code Exa 15.2 Graphical Solution 1 clc; 2 clear; 3 for x1 =0:8 4 x21(x1+1) = -(7/11)*x1+7; 5 x22(x1+1) =(80 -10* x1)/8; 6 x23(x1+1) =6; 7 x24(x1+1) = -150*x1 /175; 8 x25(x1+1) =(600 -150* x1)/175; 9 x26(x1+1) =(1400 -150* x1)/175; 10 end 11 x1 =0:8; 12 13 plot(x1 ,x24 , ’ o− ’ ) 14 plot(x1 ,x25 , ’ .− ’ ) 15 plot(x1 ,x26 , ’x− ’ ) 16 h1=legend ([ ’Z=0 ’ ; ’Z=600 ’ ; ’Z=1400 ’ ]) 17 plot(x1 ,x21); 18 plot(x1 ,x22); 19 plot(x1 ,x23); 20 xtitle( ’ x2 vs x1 ’ , ’ x1 ( tonnes ) ’ , ’ x2 ( tonnes ) ’ ) Scilab code Exa 15.3 Linear Programming Problem 1 clc; 2 clear; 3 x1 =[4.888889 3.888889]; 4 x2=[7 11]; 5 x3=[10 8]; 6 x4 =[150 175]; 7 x5=[77 80 9 6]; 8 profit =[x1(1)*x4(1) x1(2)*x4(2)]; 9 total =[x1(1)*x3(1)+x1(2)*x3(2) x1(1)*x3(1)+x1(2)*x3 (2) x1(1) x1(2) profit (1)+profit (2)]; 85
  • 87. 10 e=1000; 11 12 while e>total (5) 13 if total (1) <=x5(1) then 14 if total (2) <=x5(2) then 15 if total (3) <=x5(3) then 16 if total (4) <=x5(4) then 17 l=1; 18 end 19 end 20 end 21 end 22 if l==1 then 23 x1(1)=x1(1) +4.888889; 24 x1(2)=x1(2) +3.888889; 25 profit =[x1(1)*x4(1) x1(2)*x4(2)]; 26 total (5)=profit (1)+profit (2); 27 end 28 end 29 disp(total (5),”The maximized p r o f i t i s=”) Scilab code Exa 15.4 Nonlinear constrained optimization 1 clc; 2 clear; 3 Mt =2000; // kg 4 g=9.8; //m/ s ˆ2 5 c0 =200; //$ 6 c1=56; //$/m 7 c2 =0.1; //$/mˆ2 8 vc=20; //m/ s 9 kc=3; // kg /( s ∗mˆ2) 10 z0 =500; //m 11 t=27; 12 r=2.943652; 86
  • 88. 13 n=6; 14 A=2* %pi*r*r; 15 l=(2^0.5)*r; 16 c=3*A; 17 m=Mt/n; 18 function y=f(t) 19 y=(z0+g*m*m/(c*c)*(1-exp(-c*t/m)))*c/(g*m); 20 endfunction 21 22 while abs(f(t)-t) >0.00001 23 t=t+0.00001; 24 end 25 v=g*m*(1-exp(-c*t/m))/c; 26 disp(v,”The f i n a l value of v e l o c i t y=”) 27 disp(n,”The f i n a l no . of load p a r c e l s=”) 28 disp(”m”,r,”The chute r a d i u s=”) 29 disp ((c0+c1*l+c2*A*A)*n,”The minimum c o s t ( $ )=”) Scilab code Exa 15.5 One dimensional Optimization 1 clc; 2 clear; 3 function y=fx(x) 4 y=-(2* sin(x))+x^2/10 5 endfunction 6 x=fminsearch(fx ,0) 7 disp(” After maximization : ”) 8 disp(x,”x=”) 9 disp(fx(x),” f ( x )=”) Scilab code Exa 15.6 Multidimensional Optimization 1 clc; 87
  • 89. 2 clear; 3 function f=fx(x) 4 f=-(2*x(1)*x(2) +2*x(1)-x(1)^2-2*x(2) ^2) 5 endfunction 6 x=fminsearch(fx ,[-1 1]) 7 disp(” After maximization : ”) 8 disp(x,”x=”) 9 disp(fx(x),” f ( x )=”) Scilab code Exa 15.7 Locate Single Optimum 1 clc; 2 clear; 3 function y=fx(x) 4 y=-(2* sin(x)-x^2/10) 5 endfunction 6 x=fminsearch(fx ,0) 7 disp(” After maximization : ”) 8 disp(x,”x=”) 9 disp(fx(x),” f ( x )=”) 88
  • 90. Chapter 17 Least squares regression Scilab code Exa 17.1 Linear regression 1 // c l c ( ) 2 x = [1,2,3,4,5,6,7]; 3 y = [0.5 ,2.5 ,2 ,4 ,3.5 ,6 ,5.5]; 4 n = 7; 5 s = 0; 6 xsq = 0; 7 xsum = 0; 8 ysum = 0; 9 for i = 1:7 10 s = s + (det(x(1,i)))*(det(y(1,i))); 11 xsq = xsq + (det(x(1,i))^2); 12 xsum = xsum + det(x(1,i)); 13 ysum = ysum + det(y(1,i)); 14 end 15 disp(s,”sum of product of x and y =”) 16 disp(xsq ,”sum of square of x = ”) 17 disp(xsum ,”sum of a l l the x = ”) 18 disp(ysum ,”sum of a l l the y = ”) 19 a = xsum/n; 20 b = ysum/n; 21 a1 = (n*s - xsum*ysum)/(n*xsq -xsum ^2); 89
  • 91. 22 a0 = b - a*a1; 23 disp(a1 ,”a1 = ”) 24 disp(a0 ,”a0 = ”) 25 disp(”The equation of the l i n e obtained i s y = a0 + a1∗x”) Scilab code Exa 17.2 Estimation of errors for the linear least square fit 1 // c l c ( ) 2 x = [1,2,3,4,5,6,7]; 3 y = [0.5 ,2.5 ,2 ,4 ,3.5 ,6 ,5.5]; 4 n = 7; 5 s = 0; 6 ssum = 0; 7 xsq = 0; 8 xsum = 0; 9 ysum = 0; 10 msum = 0; 11 for i = 1:7 12 s = s + (det(x(1,i)))*(det(y(1,i))); 13 xsq = xsq + (det(x(1,i))^2); 14 xsum = xsum + det(x(1,i)); 15 ysum = ysum + det(y(1,i)); 16 end 17 a = xsum/n; 18 b = ysum/n; 19 a1 = (n*s - xsum*ysum)/(n*xsq -xsum ^2); 20 a0 = b - a*a1; 21 for i = 1:7 22 m(i) = (det(y(1,i)) - ysum /7) ^2; 23 msum = msum +m(i); 24 si(i) = (det(y(1,i)) - a0 - a1*det(x(1,i)))^2; 25 ssum = ssum + si(i); 26 end 27 disp(ysum ,”sum of a l l y =”) 90
  • 92. 28 disp(m,” ( yi − yavg ) ˆ2 = ”) 29 disp(msum ,” t o t a l ( yi − yavg ) ˆ2 = ”) 30 disp(si ,” ( yi − a0 − a1∗x ) ˆ2 = ”) 31 disp(ssum ,” t o t a l ( yi − a0 − a1∗x ) ˆ2 = ”) 32 sy = (msum / (n-1))^(0.5); 33 sxy = (ssum /(n-2))^(0.5); 34 disp(sy ,” sy = ”) 35 disp(sxy ,” sxy = ”) 36 r2 = (msum - ssum)/( msum); 37 r = r2 ^0.5; 38 disp(r,” r = ”) 39 disp(”The r e s u l t i n d i c a t e that 86.8 percent of the o r i g i n a l u n c e r t a i n t y has been e xp la in ed by l i n e a r model ”) Scilab code Exa 17.3.a linear regression using computer 1 // c l c ( ) 2 s = [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]; 3 v = [10 ,16.3 ,23 ,27.5 ,31 ,35.6 ,39 ,41.5 ,42.9 ,45 ,46 ,45.5 ,46 ,49 ,50]; 4 g = 9.8 //m/ s ˆ2 5 m = 68.1; // kg 6 c = 12.5 // kg/ s 7 for i = 1:15 8 v1(i) = g*m*(1 - exp(-c*s(i)/m))/c; 9 v2(i) = g*m*s(i)/(c*(3.75+s(i))); 10 end 11 disp(s,” time = ”) 12 disp(v,” measured v =”) 13 disp(v1 ,” using equation ( 1 . 1 0 ) v1 = ”) 14 disp(v2 ,” using equation ( ( 1 7 . 3 ) ) v2 = ”) 15 plot2d(v,v1); 16 xtitle( ’ v vs v1 ’ , ’ v ’ , ’ v1 ’ ); 91
  • 93. Figure 17.1: linear regression using computer Scilab code Exa 17.3.b linear regression using computer 1 // c l c ( ) 2 s = [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]; 3 v = [10 ,16.3 ,23 ,27.5 ,31 ,35.6 ,39 ,41.5 ,42.9 ,45 ,46 ,45.5 ,46 ,49 ,50]; 4 g = 9.8 //m/ s ˆ2 92
  • 94. Figure 17.2: linear regression using computer 93
  • 95. 5 m = 68.1; // kg 6 c = 12.5 // kg/ s 7 for i = 1:15 8 v1(i) = g*m*(1 - exp(-c*s(i)/m))/c; 9 v2(i) = g*m*s(i)/(c*(3.75+s(i))); 10 end 11 disp(s,” time = ”) 12 disp(v,” measured v =”) 13 disp(v1 ,” using equation ( 1 . 1 0 ) v1 = ”) 14 disp(v2 ,” using equation ( ( 1 7 . 3 ) ) v2 = ”) 15 plot2d(v,v2); 16 xtitle( ’ v vs v2 ’ , ’ v ’ , ’ v2 ’ ); Scilab code Exa 17.4 Linearization of a power function 1 // c l c ( ) 2 //y = a∗xˆb 3 a1 = -0.3000; 4 a = 10^( a1); 5 b = 1.75; 6 disp(a) 7 for i=1:5 8 x(i) = i; 9 y(i) = a*x(i)^b; 10 m(i) = log10(x(i)); 11 n(i) = log10(y(i)); 12 end 13 disp(x(1:5) ,”x = ”) 14 disp(y(1:5) ,”y = ”) 15 disp(m(1:5) ,”m = ”) 16 disp(n(1:5) ,”n = ”) 17 plot2d(x(1:5) ,y(1:5)) 18 zoom_rect ([0 ,0 ,7 ,7]) 19 xtitle( ’ y vs x ’ , ’ x ’ , ’ y ’ ) 20 plot2d(m(1:5) ,n(1:5)) 94
  • 96. Figure 17.3: Linearization of a power function 21 zoom_rect ([0,-1,1,1]) 22 xtitle( ’ l og y vs lo g x ’ , ’ l og x ’ , ’ log y ’ ) Scilab code Exa 17.5 polynomial regression 1 // c l c ( ) 2 x = [0,1,2,3,4,5]; 95
  • 97. Figure 17.4: Linearization of a power function 96
  • 98. Figure 17.5: polynomial regression 97
  • 99. 3 y = [2.1 ,7.7 ,13.6 ,27.2 ,40.9 ,61.1]; 4 sumy = 0; 5 sumx = 0; 6 m = 2; 7 n = 6; 8 s = 1.12; 9 xsqsum = 0; 10 xcsum = 0; 11 x4sum = 0; 12 xysum = 0; 13 x2ysum = 0; 14 rsum = 0; 15 usum = 0; 16 for i=1:6 17 sumy = sumy+y(i); 18 sumx = sumx+x(i); 19 r(i) = (y(i) - s/n)^2; 20 xsqsum = xsqsum + x(i)^2; 21 xcsum = xcsum +x(i)^3; 22 x4sum = x4sum + x(i)^4; 23 xysum = xysum + x(i)*y(i); 24 x2ysum = x2ysum + y(i)*x(i)^2; 25 rsum = r(i) + rsum; 26 end 27 disp(sumy ,”sum y =”) 28 disp(sumx ,”sum x =”) 29 xavg = sumx/n; 30 yavg = sumy/n; 31 disp(xavg ,” xavg = ”) 32 disp(yavg ,” yavg = ”) 33 disp(xsqsum ,”sum xˆ2 =”) 34 disp(xcsum ,”sum xˆ3 =”) 35 disp(x4sum ,”sum xˆ4 =”) 36 disp(xysum ,”sum x∗y =”) 37 disp(x2ysum ,”sum xˆ2 ∗ y =”) 38 J = [n,sumx ,xsqsum;sumx ,xsqsum ,xcsum;xsqsum ,xcsum , x4sum ]; 39 I = [sumy;xysum;x2ysum ]; 98
  • 100. 40 X = inv(J) * I; 41 a0 = det(X(1,1)); 42 a1 = det(X(2,1)); 43 a2 = det(X(3,1)); 44 for i=1:6 45 u(i) = (y(i) - a0 - a1*x(i) - a2*x(i)^2) ^2; 46 usum = usum + u(i); 47 end 48 disp(r,” ( yi − yavg ) ˆ2 = ”) 49 disp(u,” ( yi − a0 − a1∗x − a2∗x ˆ2) ˆ2 = ”) 50 plot(x,y); 51 xtitle( ’ x vs y ’ , ’ x ’ , ’ y ’ ); 52 syx = (usum /(n-3))^0,5; 53 disp(syx ,”The standard e r r o r of the estimate based on r e g r e s s i o n polynomial =”) 54 R2 = (rsum - usum)/( rsum); 55 disp(”%”,R2*100,” Percentage of o r i g i n a l u n c e r t a i n t y that has been e xp la in ed by the model = ”) Scilab code Exa 17.6 Multiple linear regression 1 // c l c ( ) 2 x1 = [0,2,2.5,1,4,7]; 3 x2 = [0,1,2,3,6,2]; 4 x1sum = 0; 5 x2sum = 0; 6 ysum = 0; 7 x12sum = 0; 8 x22sum = 0; 9 x1ysum = 0; 10 x2ysum = 0; 11 x1x2sum = 0; 12 n = 6; 13 for i=1:6 14 y(i) = 5 + 4*x1(i) - 3*x2(i); 99
  • 101. 15 x12(i) = x1(i)^2; 16 x22(i) = x2(i)^2; 17 x1x2(i) = x1(i) * x2(i); 18 x1y(i) = x1(i) * y(i); 19 x2y(i) = x2(i) * y(i); 20 x1sum = x1sum + x1(i); 21 x2sum = x2sum + x2(i); 22 ysum = ysum + y(i); 23 x1ysum = x1ysum + x1y(i); 24 x2ysum = x2ysum + x2y(i); 25 x1x2sum = x1x2sum + x1x2(i); 26 x12sum = x12sum + x12(i); 27 x22sum = x22sum + x22(i); 28 end 29 X = [n,x1sum ,x2sum;x1sum ,x12sum ,x1x2sum;x2sum , x1x2sum ,x22sum ]; 30 Y = [ysum;x1ysum;x2ysum ]; 31 Z = inv(X)*Y; 32 a0 = det(Z(1,1)); 33 a1 = det(Z(2,1)); 34 a2 = det(Z(3,1)); 35 disp(a0 ,”a0 = ”) 36 disp(a1 ,”a1 = ”) 37 disp(a2 ,”a2 = ”) 38 disp(”Thus , y = a0 + a1∗x1 + a2∗x2”) Scilab code Exa 17.7 Confidence interval for linear regression 1 // c l c ( ) 2 //y = −0.859 + 1.032∗ x 3 Z = [1 ,10;1 ,16.3;1 ,23;1 ,27.5;1 ,31;1 ,35.6;1 ,39;1 ,41.5;1 ,42.9;1 ,45;1 ,46; 4 for i = 1:15 5 Y(i) = 9.8*68.1*(1 - exp ( -12.5*i/68.1))/12.5; 100
  • 102. 6 end 7 M = Z’; 8 R = M*Z; 9 S = M*Y; 10 P = inv(R); 11 X = inv(R)*S; 12 a0 = det(X(1,1)); 13 a1 = det(X(2,1)); 14 disp(a0 ,”a0 = ”) 15 disp(a1 ,”a1 = ”) 16 sxy = 0.863403; 17 sa0 = (det(P(1,1)) * sxy ^2) ^0.5; 18 sa1 = (det(P(2,2)) * sxy ^2) ^0.5; 19 disp(sa0 ,” standard e r r o r of co e f f i c i e n t a0 = ”) 20 disp(sa1 ,” standard e r r o r of co e f f i c i e n t a1 = ”) 21 TINV = 2.160368; 22 a0 = [a0 - TINV *(sa0),a0 + TINV *(sa0)]; 23 a1 = [a1 - TINV *(sa1),a1 + TINV *(sa1)]; 24 disp(a0 ,” i n t e r v a l of a0 = ”) 25 disp(a1 ,” i n t e r v a l of a1 = ”) Scilab code Exa 17.8 Gauss Newton method 1 // c l c ( ) 2 x = [0.25 ,0.75 ,1.25 ,1.75 ,2.25]; 3 y = [0.28 ,0.57 ,0.68 ,0.74 ,0.79]; 4 a0 = 1; 5 a1 = 1; 6 sr = 0.0248; 7 for i = 1:5 8 pda0(i) = 1 - exp(-a1 * x(i)); 9 pda1(i) = a0 * x(i)*exp(-a1*x(i)); 10 end 11 Z0 = [pda0 (1),pda1 (1);pda0 (2),pda1 (2);pda0 (3),pda1 (3);pda0 (4),pda1 (4);pda0 (5),pda1 (5)] 101
  • 103. 12 disp(Z0 ,”Z0 = ”) 13 R = Z0 ’*Z0; 14 S = inv(R); 15 for i = 1:5 16 y1(i) = a0 * (1-exp(-a1*x(i))); 17 D(i) = y(i) - y1(i); 18 end 19 disp(D,”D = ”) 20 M = Z0 ’*D; 21 X = S *M; 22 disp(X,”X = ”) 23 a0 = a0 + det(X(1,1)); 24 a1 = a1 + det(X(2,1)); 25 disp(a0 ,”The value of a0 a f t e r 1 s t i t e r a t i o n = ”) 26 disp(a1 ,”The value of a1 a f t e r 1 s t i t e r a t i o n = ”) 102
  • 104. Chapter 18 Interpolation Scilab code Exa 18.1 Linear interpolation 1 // c l c ( ) 2 // f1 ( x ) = f0 ( x ) +( f ( x1 ) − f ( x0 ) ∗( x − x0 ) / ( x1 − x0 ) 3 x = 2; 4 x0 = 1; 5 x1 = 6; 6 m = 1.791759; 7 n = 0; 8 r = log (2); 9 f = 0 + (m - n) * (x - x0) / (x1 - x0); 10 disp(f,” value of ln2 f o r i n t e r p o l a t i o n r e g i o n 1 to 6 =”) 11 e = (r - f) * 100/r; 12 disp(”%”,e,” e r r o r by i n t e r p o l a t i o n f o r i n t e r v a l [ 1 , 6 ] =”) 13 x2 = 4; 14 p = 1.386294; 15 f = 0 + (p - n) * (x - x0) / (x2 - x0); 16 disp(f,” value of ln2 f o r i n t e r p o l a t i o n r e g i o n 1 to 6 =”) 17 e = (r - f) * 100/r; 18 disp(”%”,e,” e r r o r by i n t e r p o l a t i o n f o r i n t e r v a l [ 1 , 6 ] 103
  • 105. =”) Scilab code Exa 18.2 Quadratic interpolation 1 // c l c ( ) 2 x = 2; 3 x0 = 1; 4 m = 0; 5 x1 = 4; 6 n = 1.386294; 7 x2 = 6; 8 p = 1.791759; 9 b0 = m; 10 b1 = (n - m)/(x1 - x0); 11 b2 = ((p - n)/(x2 - x1) - (n - m)/(x1 - x0))/(x2 - x0); 12 disp(b0 ,”b0 = ”) 13 disp(b1 ,”b1 = ”) 14 disp(b2 ,”b2 = ”) 15 f = b0 + b1*(x - x0) + b2*(x - x0)*(x - x1); 16 disp(f,” f (2) = ”) 17 r = log (2); 18 e = (r -f)*100/r; 19 disp(”%”,e,” e r r o r = ”) Scilab code Exa 18.3 Newtons divided difference Interpolating polynomials 1 // c l c ( ) 2 x = 2; 3 x0 = 1; 4 m = 0; 5 x1 = 4; 6 n = 1.386294; 104
  • 106. 7 x3 = 5; 8 p = 1.609438; 9 x2 = 6; 10 o = 1.791759; 11 f01 = (m - n)/(x0 - x1); 12 f12 = (n - o)/(x1 - x2); 13 f23 = (p - o)/(x3 - x2); 14 f210 = (f12 - f01)/(x2 - x0); 15 f321 = (f23 - f12)/(x3 - x1); 16 f0123 = (f321 - f210) / (x3 - x0); 17 b0 = m; 18 b1 = f01; 19 b2 = f210; 20 b3 = f0123; 21 disp(b0 ,”b0 = ”) 22 disp(b1 ,”b1 = ”) 23 disp(b2 ,”b2 = ”) 24 disp(b3 ,”b3 = ”) 25 f = b0 + b1*(x - x0) + b2*(x - x0)*(x - x1) + b3*(x - x0)*(x - x1)*(x - x2); 26 disp(f,” f (2) = ”) 27 r = log (2); 28 e = (r -f)*100/r; 29 disp(”%”,e,” e r r o r = ”) Scilab code Exa 18.4 Error estimation for Newtons polynomial 1 // c l c ( ) 2 x = 2; 3 x0 = 1; 4 m = 0; 5 x1 = 4; 6 n = 1.386294; 7 x3 = 5; 8 p = 1.609438; 105
  • 107. 9 x2 = 6; 10 o = 1.791759; 11 f01 = (m - n)/(x0 - x1); 12 f12 = (n - o)/(x1 - x2); 13 f23 = (p - o)/(x3 - x2); 14 f210 = (f12 - f01)/(x2 - x0); 15 f321 = (f23 - f12)/(x3 - x1); 16 f0123 = (f321 - f210) / (x3 - x0); 17 b0 = m; 18 b1 = f01; 19 b2 = f210; 20 b3 = f0123; 21 R2 = b3 * (x - x0) * (x - x1)*(x-x2); 22 disp(R2 ,” e r r o r R2 = ”) Scilab code Exa 18.5 Error Estimates for Order of Interpolation 1 clc; 2 clear; 3 x=[1 4 6 5 3 1.5 2.5 3.5]; 4 y=[0 1.3862944 1.7917595 1.6094379 1.0986123 0.4054641 0.9162907 1.2527630]; 5 n=8; 6 for i=1:n 7 fdd(i,1)=y(i); 8 end 9 for j=2:n 10 for i=1:n-j+1 11 fdd(i,j)=(fdd(i+1,j-1)-fdd(i,j-1))/(x(i+j-1) -x(i)); 12 end 13 end 14 xterm =1; 15 yint (1)=fdd (1,1); 16 106
  • 108. 17 for order =2:n 18 xterm=xterm *(2-x(order -1)); 19 yint2=yint(order -1)+fdd(1,order)*xterm; 20 Ea(order -1)=yint2 -yint(order -1); 21 yint(order)=yint2; 22 23 end 24 disp(yint ,”F( x )=”) 25 disp(Ea ,”Ea=”) Scilab code Exa 18.6 Lagrange interpolating polynomials 1 // c l c ( ) 2 x = 2; 3 x0 = 1; 4 m = 0; 5 x1 = 4; 6 n = 1.386294; 7 x2 = 6; 8 p = 1.791759; 9 f1 = (x - x1)*m/((x0 - x)) + (x- x0) * n/(x1 - x0); 10 disp(f1 ,” f i r s t order polynomial f1 (2 ) = ”) 11 f2 = (x - x1)*(x - x2)*m/((x0 - x1)*(x0 - x2)) + (x - x0)*(x - x2)*n/((x1 -x0)*(x1 -x2)) + (x - x0)*(x - x1)*p/((x2 - x0)*(x2 - x1)); 12 disp(f2 ,” second order polynomial f2 ( 2) = ”) Scilab code Exa 18.7 Lagrange interpolation using computer 1 // c l c ( ) 2 z = 10; 3 x =[1 ,3 ,5 ,7 ,13]; 4 v = [800 ,2310 ,3090 ,3940 ,4755]; 107
  • 109. 5 f1 = (z - x(5)) * v(4) / (x(4) - x(5)) + (z - x(4)) * v(5) / (x(5) - x(4)); 6 f2 = (z - x(4))*(z - x(5)) *v(3) /((x(3) - x(4))*(x (3) - x(5)))+(z - x(3))*(z - x(5)) *v(4) /((x(4) - x(3))*(x(4) - x(5)))+(z - x(4))*(z - x(3)) *v(5) /((x(5) - x(4))*(x(5) - x(3))); 7 f3 = (z - x(3))*(z - x(4))*(z - x(5)) *v(2) /((x(2) - x(4))*(x(2) - x(5))*(x(2) - x(3)))+(z - x(4))*(z - x(2))*(z - x(5)) *v(3) /((x(3) - x(2))*(x(3) - x(5))*(x(3) - x(4)))+(z - x(2))*(z - x(3))*(z - x (5)) *v(4) /((x(4) - x(3))*(x(4) - x(2))*(x(4) - x (5)))+ (z - x(3))*(z - x(4))*(z - x(2)) *v(5) /((x (5) - x(4))*(x(5) - x(2))*(x(5) - x(3))); 8 f4 = (z - x(2))*(z - x(3))*(z - x(4))*(z - x(5)) *v (1) /((x(1) - x(2))*(x(1) - x(4))*(x(1) - x(5))*(x (1) - x(3)))+ (z - x(1))*(z - x(3))*(z - x(4))*(z - x(5)) *v(2) /((x(2) - x(1))*(x(2) - x(4))*(x(2) - x(5))*(x(2) - x(3)))+(z - x(1))*(z - x(4))*(z - x(2))*(z - x(5)) *v(3) /((x(3) - x(1))*(x(3) - x (2))*(x(3) - x(5))*(x(3) - x(4)))+(z - x(1))*(z - x(2))*(z - x(3))*(z - x(5)) *v(4) /((x(4) - x(1)) *(x(4) - x(3))*(x(4) - x(2))*(x(4) - x(5)))+ (z - x(1))*(z - x(3))*(z - x(4))*(z - x(2)) *v(5) /((x (5) - x(1))*(x(5) - x(4))*(x(5) - x(2))*(x(5) - x (3))); 9 disp(f1 ,” V e l o c i t y at 10 sec by f i r s t order i n t e r p o l a t i o n = ”) 10 disp(f2 ,” V e l o c i t y at 10 sec by second order i n t e r p o l a t i o n = ”) 11 disp(f3 ,” V e l o c i t y at 10 sec by t h i r d order i n t e r p o l a t i o n = ”) 12 disp(f4 ,” V e l o c i t y at 10 sec by f o u r t h order i n t e r p o l a t i o n = ”) 108
  • 110. Figure 18.1: First order splines 109
  • 111. Scilab code Exa 18.8 First order splines 1 // c l c ( ) 2 x = [3 ,4.5 ,7 ,9]; 3 fx = [2.5 ,1 ,2.5 ,0.5]; 4 m1 = (fx(2) - fx(1))/(x(2) - x(1)); 5 m2 = (fx(3) - fx(2))/(x(3) - x(2)); 6 m3 = (fx(4) - fx(3))/(x(4) - x(3)); 7 x1 = [3 ,4.5]; 8 x2 = [4.5 ,7]; 9 x3 = [7 ,9]; 10 plot2d(x1 ,m1*x1 +5.5); 11 plot2d(x2 ,m2*x2 -1.7); 12 plot2d(x3 ,m3*x3 +9.5); 13 xtitle(” f ( x ) vs x”,”x”,” f ( x ) ”) 14 r = 5 15 z = m2*r -1.7; 16 disp(z,”The value at x = 5 i s ”) Scilab code Exa 18.9 Quadratic splines 1 // c l c ( ) 2 x = [3 ,4.5 ,7 ,9]; 3 fx = [2.5 ,1 ,2.5 ,0.5]; 4 p = 4; 5 n = 3; 6 uk = n*(p-1); 7 c = 2*n - 2; 8 // 20.25∗ a1 + 4.5∗ b1 + c1 = 1 9 // 20.25∗ a2 + 4.5∗ b2 + c2 = 1 10 // 49∗ a2 + 7∗b2 + c2 = 2.5 11 // 49∗ a3 + 7∗b3 + c3 = 2.5 12 //9a1 + 3b1 + c1 = 2.5 110
  • 112. Figure 18.2: Quadratic splines 111
  • 113. 13 // 81∗ a3 + 9∗b3 + c3 = 0.5 14 // 9∗ a1 + b1 = 9∗ a2 + b2 15 // 14 a2 + b2 = 14 ∗a3 + b3 16 a1 = 0; 17 // thus we have above 9 equations and 9 unknowns a1 , a2 , a3 , b1 , b2 , b3 , c1 , c2 , c3 18 // thus we get 19 A = [20.25,4.5,1,0,0,0,0,0,0;0,0,0,20.25,4.5,1,0,0,0;0,0,0,49,7,1,0,0, 20 disp(A,”A = ”) 21 B = [1;1;2.5;2.5;2.5;0.5;0;0;0]; 22 disp(B,”B =”) 23 X = inv(A)*B; 24 a1 = det(X(1,1)); 25 b1 = det(X(2,1)); 26 c1 = det(X(3,1)); 27 a2 = det(X(4,1)); 28 b2 = det(X(5,1)); 29 c2 = det(X(6,1)); 30 a3 = det(X(7,1)); 31 b3 = det(X(8,1)); 32 c3 = det(X(9,1)); 33 disp(a1 ,”a1 = ”) 34 disp(b1 ,”b1 = ”) 35 disp(c1 ,” c1 = ”) 36 disp(a2 ,”a2 = ”) 37 disp(b2 ,”b2 = ”) 38 disp(c2 ,” c2 = ”) 39 disp(a3 ,”a3 = ”) 40 disp(b3 ,”b3 = ”) 41 disp(c3 ,” c3 = ”) 42 // thus , f1 ( x ) = −x + 5.5 3 < x < 4. 5 43 // f2 ( x ) = 0.64∗ xˆ2 −6.76∗x + 18.46 4.5 < x < 7 44 // f3 ( x ) = −1.6∗xˆ2 + 24.6∗ x − 91.3 7 < x < 9 112
  • 114. Figure 18.3: Cubic splines 45 x1 = 3:0.1:4.5; 46 x2 = 4.5:0.1:7; 47 x3 = 7:0.1:9; 48 plot2d(x1 ,-x1 + 5.5); 49 plot2d(x2 ,0.64* x2^2 -6.76*x2+ 18.46); 50 plot2d(x3 , -1.6*x3^2 + 24.6* x3 - 91.3); 51 xtitle(” f ( x ) vs x”,”x”,” f ( x ) ”) 52 x = 5; 53 fx = 0.64*x^2 -6.76*x + 18.46; 54 disp(fx ,”The value at x = 5 i s ”) 113
  • 115. Scilab code Exa 18.10 Cubic splines 1 // c l c ( ) 2 x = [3 ,4.5 ,7 ,9]; 3 fx = [2.5 ,1 ,2.5 ,0.5]; 4 //we get the f o l l o w i n g equations f o r cubic s p l i n e s 5 // 8∗ f ” ( 4 . 5 ) + 2.5∗ f ”(7) = 9.6 6 // 2.5∗ f ” ( 4 . 5 ) + 9∗ f ”(7) = −9.6 7 // above two equations g i v e 8 m = 1.67909; // (m = f ” ( 4 . 5 ) ) 9 n = -1.53308; // ( n = f ”(7) ) 10 // t h i s v a l u e s can be used to y i e l d the f i n a l equation 11 // f1 ( x ) = 0.186566 ∗ ( x − 3) ˆ3 + 1 . 6 6 6 6 7 ∗ ( 4 . 5 − x ) + 0.246894∗( x − 3) 12 // in s i m i l a r manner other equations can be found too 13 // f2 ( x ) = 0.111939(7 − x ) ˆ3 − 0.102205∗( x − 4 . 5 ) ˆ3 − 0.299621 ∗ (7 − x ) + 1.638783 ∗ ( x − 4 . 5 ) 14 // f3 ( x ) = −0.127757∗(9 − x ) ˆ3 + 1.761027 ∗(9 − x ) + 0 . 2 5 ∗ ( x − 7) 15 x1 = 3:0.1:4.5; 16 x2 = 4.5:0.1:7; 17 x3 = 7:0.1:9; 18 plot2d(x1 ,0.186566 * (x1 - 3)^3 + 1.66667*(4.5 - x1) + 0.246894*( x1 - 3)); 19 plot2d(x2 ,0.111939*(7 - x2)^3 - 0.102205*( x2 - 4.5) ^3 - 0.299621 * (7 - x2) + 1.638783 * (x2 - 4.5)) ; 20 plot2d(x3 , -0.127757*(9 - x3)^3 + 1.761027 *(9 - x3) + 0.25*( x3 - 7)); 21 xtitle(” f ( x ) vs x”,”x”,” f ( x ) ”) 22 x = 5; 23 fx = 0.111939*(7 - x)^3 - 0.102205*(x - 4.5) ^3 - 0.299621 * (7 - x) + 1.638783 * (x - 4.5); 24 disp(fx ,”The value at x = 5 i s ”) 114
  • 116. Chapter 19 Fourier Approximation Scilab code Exa 19.1 Least Square Fit 1 clc; 2 clear; 3 function y=f(t) 4 y=1.7+ cos (4.189*t+1.0472) 5 endfunction 6 deltat =0.15; 7 t1=0; 8 t2 =1.35; 9 omega =4.189; 10 del=(t2 -t1)/9; 11 for i=1:10 12 t(i)=t1+del*(i-1); 13 end 14 sumy =0; 15 suma =0; 16 sumb =0; 17 for i=1:10 18 y(i)=f(t(i)); 19 a(i)=y(i)*cos(omega*t(i)); 20 b(i)=y(i)*sin(omega*t(i)); 21 sumy=sumy+y(i); 115
  • 117. 22 suma=suma+a(i); 23 sumb=sumb+b(i); 24 end 25 A0=sumy /10; 26 A1=2* suma /10; 27 B1=2* sumb /10; 28 disp(”The l e a s t square f i t i s y=A0+A1∗ cos (w0∗ t )+A2∗ s i n (w0∗ t ) , where ”) 29 disp(A0 ,”A0=”) 30 disp(A1 ,”A1=”) 31 disp(B1 ,”B1=”) 32 theta=atan(-B1/A1); 33 C1=(A1^2 + B1^2) ^0.5; 34 disp(” A l t e r n a t i v e l y , the l e a s t square f i t can be expressed as ”) 35 disp(”y=A0+C1∗ cos (w0∗ t + theta ) , where ”) 36 disp(A0 ,”A0=”) 37 disp(theta ,” Theta=”) 38 disp(C1 ,”C1=”) 39 disp(”Or”) 40 disp(”y=A0+C1∗ s i n (w0∗ t + theta + pi /2) , where ”) 41 disp(A0 ,”A0=”) 42 disp(theta ,” Theta=”) 43 disp(C1 ,”C1=”) Scilab code Exa 19.2 Continuous Fourier Series Approximation 1 clc; 2 clear; 3 a0=0; 4 // f ( t )=−1 f o r −T/2 to −T/4 5 // f ( t )=1 f o r −T/4 to T/4 6 // f ( t )=−1 f o r T/4 to T/2 7 // ak=2/T∗ ( i n t e g r a t i o n of f ( t ) ∗ cos (w0∗ t ) from −T/2 to T/2) 116
  • 118. 8 // ak=2/T∗(( i n t e g r a t i o n of f ( t ) ∗ cos (w0∗ t ) from −T/2 to −T/4) + ( i n t e g r a t i o n of f ( t ) ∗ cos (w0∗ t ) from −T /4 to T/4) + ( i n t e g r a t i o n of f ( t ) ∗ cos (w0∗ t ) from T/4 to T/2) ) 9 // Therefore , 10 // ak=4/(k∗%pi ) f o r k = 1 , 5 , 9 , . . . . . 11 // ak=−4/(k∗%pi ) f o r k = 3 , 7 , 1 1 , . . . . . 12 // ak=0 f o r k=even i n t e g e r s 13 // s i m i l a r l y we f i n d the b ’ s . 14 // a l l the b ’ s=0 15 disp(”The f o u r i e r approximtion i s : ”) 16 disp(” 4/( %pi ) ∗ cos (w) ∗ t ) − 4/(3∗ %pi ) ∗ cos (3∗(w) ∗ t ) + 4/(5∗ %pi ) ∗ cos (5∗(w) ∗ t ) − 4/(7∗ %pi ) ∗ cos (7∗(w) ∗ t ) + . . . . . ”) Scilab code Exa 19.3 Trendline 1 clc; 2 clear; 3 x=0.5:0.5:5.5; 4 for i=1:11 5 y(i)=0.9846* log(x(i))+1.0004; 6 end 7 plot(x,y) 8 xtitle(”y vs x”,”x”,”y”) Scilab code Exa 19.4 Data Analysis 1 clc; 2 clear; 3 s=[0.0002 0.0002 0.0005 0.0005 0.001 0.001]; 117
  • 120. 4 r=[0.2 0.5 0.2 0.5 0.2 0.5]; 5 u=[0.25; 0.5; 0.4; 0.75; 0.5; 1]; 6 logs=log10(s); 7 logr=log10(r); 8 logu=log10(u); 9 for i=1:6 10 m(i,1) =1; 11 m(i,2)=logs(i); 12 m(i,3)=logr(i); 13 end 14 15 a=mlogu; 16 disp (10^a(1),” alpha=”) 17 disp(a(2),” sigma=”) 18 disp(a(3),” rho=”) Scilab code Exa 19.5 Curve Fitting 1 clc; 2 clear; 3 x=0:10; 4 y=sin(x); 5 xi =0:.25:10; 6 // part a 7 yi=interp1(x,y,xi); 8 plot2d(xi ,yi) 9 xtitle(”y vs x ( part a ) ”,”x”,”y”) 10 // part b 11 // f i t t i n g x and y in a f i f t h order polynomial 12 clf(); 13 p=[0.0008 -0.0290 0.3542 -1.6854 2.586 -0.0915]; 14 15 for i=1:41 16 yi(i)=p(1)*(xi(i)^5)+p(2)*(xi(i)^4)+p(3)*(xi(i) ^3)+p(4)*(xi(i)^2)+p(5)*(xi(i))+p(6); 119
  • 121. Figure 19.2: Curve Fitting 17 end 18 plot2d(xi ,yi); 19 xtitle(”y vs x ( part b ) ”,”x”,”y”) 20 // part c 21 clf(); 22 d=splin(x,y) 23 yi -interp(xi ,x,y,d) 24 plot2d(xi ,yi) 25 xtitle(”y vs x ( part c ) ”,”x”,”y”) 120
  • 122. Figure 19.3: Curve Fitting 121
  • 123. Scilab code Exa 19.6 Polynomial Regression 1 clc; 2 clear; 3 x=[0.05 0.12 0.15 0.3 0.45 0.7 0.84 1.05]; 4 y=[0.957 0.851 0.832 0.72 0.583 0.378 0.295 0.156]; 5 sx=sum(x); 6 sxx=sum(x.*x); 7 sx3=sum(x.*x.*x); 8 sx4=sum(x.*x.*x.*x); 9 sx5=sum(x.*x.*x.*x.*x); 10 sx6=sum(x.*x.*x.*x.*x.*x); 11 n=8; 12 sy=sum(y); 13 sxy=sum(x.*y); 14 sx2y=sum(x.*x.*y); 15 sx3y=sum(x.*x.*x.*y); 16 m=[n sx sxx sx3;sx sxx sx3 sx4;sxx sx3 sx4 sx5;sx3 sx4 sx5 sx6]; 17 p=[sy;sxy;sx2y;sx3y ]; 18 a=mp; 19 disp(”The cubic polynomial i s y=a0 + a1∗x + a2∗xˆ2 + a3∗x ˆ3 , where a0 , a1 , a2 and a3 are ”) 20 disp(a) 122
  • 124. Chapter 21 Newton Cotes Integration Formulas Scilab code Exa 21.1 Single trapezoidal rule 1 clc; 2 clear; 3 function y=f(x) 4 y=(0.2+25*x -200*x^2+675*x^3 -900*x^4+400*x^5) 5 endfunction 6 tval =1.640533; 7 a=0; 8 b=0.8; 9 fa=f(a); 10 fb=f(b); 11 l=(b-a)*((fa+fb)/2); 12 Et=tval -l;// e r r o r 13 et=Et *100/ tval;// percent r e l a t i v e e r r o r 14 15 //by using approximate e r r o r estimate 16 17 // the second d e r i v a t i v e of f 18 function y=g(x) 19 y= -400+4050*x -10800*x^2+8000*x^3 123
  • 125. 20 endfunction 21 f2x=intg (0,0.8,g)/(b-a);// average value of second d e r i v a t i v e 22 Ea = -(1/12) *(f2x)*(b-a)^3; 23 disp(Et ,”The Error Et=”) 24 disp(”%”,et ,”The percent r e l a t i v e e r r o r et=”) 25 disp(Ea ,”The approximate e r r o r estimate without using the true value=”) Scilab code Exa 21.2 Multiple trapezoidal rule 1 clc; 2 clear; 3 function y=f(x) 4 y=(0.2+25*x -200*x^2+675*x^3 -900*x^4+400*x^5) 5 endfunction 6 a=0; 7 b=0.8; 8 tval =1.640533; 9 n=2; 10 h=(b-a)/n; 11 fa=f(a); 12 fb=f(b); 13 fh=f(h); 14 l=(b-a)*(fa+2*fh+fb)/(2*n); 15 Et=tval -l;// e r r o r 16 et=Et *100/ tval;// percent r e l a t i v e e r r o r 17 18 //by using approximate e r r o r estimate 19 20 // the second d e r i v a t i v e of f 21 function y=g(x) 22 y= -400+4050*x -10800*x^2+8000*x^3 23 endfunction 24 f2x=intg (0,0.8,g)/(b-a);// average value of second 124
  • 126. d e r i v a t i v e 25 Ea = -(1/12) *(f2x)*(b-a)^3/(n^2); 26 disp(Et ,”The Error Et=”) 27 disp(”%”,et ,”The percent r e l a t i v e e r r o r et=”) 28 disp(Ea ,”The approximate e r r o r estimate without using the true value=”) Scilab code Exa 21.3 Evaluating Integrals 1 clc; 2 clear; 3 g=9.8; //m/ s ˆ2; a c c e l e r a t i o n due to g r a v i t y 4 m=68.1; // kg 5 c=12.5; // kg/ sec ; drag c o e f f i c i e n t 6 function v=f(t) 7 v=g*m*(1-exp(-c*t/m))/c 8 endfunction 9 tval =289.43515; //m 10 a=0; 11 b=10; 12 fa=f(a); 13 fb=f(b); 14 for i=10:10:20 15 n=i; 16 h=(b-a)/n; 17 disp(i,”No . of segments=”) 18 disp(h,”Segment s i z e=”) 19 j=a+h; 20 s=0; 21 while j<b 22 s=s+f(j); 23 j=j+h; 24 end 25 l=(b-a)*(fa+2*s+fb)/(2*n); 26 Et=tval -l;// e r r o r 125
  • 127. 27 et=Et *100/ tval;// percent r e l a t i v e e r r o r 28 disp(”m”,l,” Estimated d=”) 29 disp(et ,” et (%) ”) 30 disp(” −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 31 end 32 for i=50:50:100 33 n=i; 34 h=(b-a)/n; 35 disp(i,”No . of segments=”) 36 disp(h,”Segment s i z e=”) 37 j=a+h; 38 s=0; 39 while j<b 40 s=s+f(j); 41 j=j+h; 42 end 43 l=(b-a)*(fa+2*s+fb)/(2*n); 44 Et=tval -l;// e r r o r 45 et=Et *100/ tval;// percent r e l a t i v e e r r o r 46 disp(”m”,l,” Estimated d=”) 47 disp(et ,” et (%) ”) 48 disp(” −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 49 end 50 for i=100:100:200 51 n=i; 52 h=(b-a)/n; 53 disp(i,”No . of segments=”) 54 disp(h,”Segment s i z e=”) 55 j=a+h; 56 s=0; 57 while j<b 58 s=s+f(j); 59 j=j+h; 60 end 126
  • 128. 61 l=(b-a)*(fa+2*s+fb)/(2*n); 62 Et=tval -l;// e r r o r 63 et=Et *100/ tval;// percent r e l a t i v e e r r o r 64 disp(”m”,l,” Estimated d=”) 65 disp(et ,” et (%) ”) 66 disp(” −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 67 end 68 for i=200:300:500 69 n=i; 70 h=(b-a)/n; 71 disp(i,”No . of segments=”) 72 disp(h,”Segment s i z e=”) 73 j=a+h; 74 s=0; 75 while j<b 76 s=s+f(j); 77 j=j+h; 78 end 79 l=(b-a)*(fa+2*s+fb)/(2*n); 80 Et=tval -l;// e r r o r 81 et=Et *100/ tval;// percent r e l a t i v e e r r o r 82 disp(”m”,l,” Estimated d=”) 83 disp(et ,” et (%) ”) 84 disp(” −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 85 end 86 for i=1000:1000:2000 87 n=i; 88 h=(b-a)/n; 89 disp(i,”No . of segments=”) 90 disp(h,”Segment s i z e=”) 91 j=a+h; 92 s=0; 93 while j<b 94 s=s+f(j); 127
  • 129. 95 j=j+h; 96 end 97 l=(b-a)*(fa+2*s+fb)/(2*n); 98 Et=tval -l;// e r r o r 99 et=Et *100/ tval;// percent r e l a t i v e e r r o r 100 disp(”m”,l,” Estimated d=”) 101 disp(et ,” et (%) ”) 102 disp(” −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 103 end 104 for i=2000:3000:5000 105 n=i; 106 h=(b-a)/n; 107 disp(i,”No . of segments=”) 108 disp(h,”Segment s i z e=”) 109 j=a+h; 110 s=0; 111 while j<b 112 s=s+f(j); 113 j=j+h; 114 end 115 l=(b-a)*(fa+2*s+fb)/(2*n); 116 Et=tval -l;// e r r o r 117 et=Et *100/ tval;// percent r e l a t i v e e r r o r 118 disp(”m”,l,” Estimated d=”) 119 disp(et ,” et (%) ”) 120 disp(” −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 121 end 122 for i=5000:5000:10000 123 n=i; 124 h=(b-a)/n; 125 disp(i,”No . of segments=”) 126 disp(h,”Segment s i z e=”) 127 j=a+h; 128 s=0; 128
  • 130. 129 while j<b 130 s=s+f(j); 131 j=j+h; 132 end 133 l=(b-a)*(fa+2*s+fb)/(2*n); 134 Et=tval -l;// e r r o r 135 et=Et *100/ tval;// percent r e l a t i v e e r r o r 136 disp(”m”,l,” Estimated d=”) 137 disp(et ,” et (%) ”) 138 disp(” −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 139 end Scilab code Exa 21.4 Single Simpsons 1 by 3 rule 1 clc; 2 clear; 3 function y=f(x) 4 y=(0.2+25*x -200*x^2+675*x^3 -900*x^4+400*x^5) 5 endfunction 6 a=0; 7 b=0.8; 8 tval =1.640533; 9 n=2; 10 h=(b-a)/n; 11 fa=f(a); 12 fb=f(b); 13 fh=f(h); 14 l=(b-a)*(fa+4*fh+fb)/(3*n); 15 disp(l,” l=”) 16 Et=tval -l;// e r r o r 17 et=Et *100/ tval;// percent r e l a t i v e e r r o r 18 19 //by using approximate e r r o r estimate 129
  • 131. 20 21 // the f o u r t h d e r i v a t i v e of f 22 function y=g(x) 23 y= -21600+48000*x 24 endfunction 25 f4x=intg (0,0.8,g)/(b-a);// average value of f o u r t h d e r i v a t i v e 26 Ea = -(1/2880) *(f4x)*(b-a)^5; 27 disp(Et ,”The Error Et=”) 28 disp(”%”,et ,”The percent r e l a t i v e e r r o r et=”) 29 disp(Ea ,”The approximate e r r o r estimate without using the true value=”) Scilab code Exa 21.5 Multiple Simpsons 1 by 3 rule 1 clc; 2 clear; 3 function y=f(x) 4 y=(0.2+25*x -200*x^2+675*x^3 -900*x^4+400*x^5) 5 endfunction 6 a=0; 7 b=0.8; 8 tval =1.640533; 9 n=4; 10 h=(b-a)/n; 11 fa=f(a); 12 fb=f(b); 13 j=a+h; 14 s=0; 15 count =1; 16 while j<b 17 if (-1)^count ==-1 then 18 s=s+4*f(j); 19 else 20 s=s+2*f(j); 130
  • 132. 21 end 22 count=count +1; 23 j=j+h; 24 end 25 l=(b-a)*(fa+s+fb)/(3*n); 26 disp(l,” l=”) 27 Et=tval -l;// e r r o r 28 et=Et *100/ tval;// percent r e l a t i v e e r r o r 29 30 //by using approximate e r r o r estimate 31 32 // the f o u r t h d e r i v a t i v e of f 33 function y=g(x) 34 y= -21600+48000*x 35 endfunction 36 f4x=intg (0,0.8,g)/(b-a);// average value of f o u r t h d e r i v a t i v e 37 Ea = -(1/(180*4^4))*(f4x)*(b-a)^5; 38 disp(Et ,”The Error Et=”) 39 disp(”%”,et ,”The percent r e l a t i v e e r r o r et=”) 40 disp(Ea ,”The approximate e r r o r estimate without using the true value=”) Scilab code Exa 21.6 Simpsons 3 by 8 rule 1 clc; 2 clear; 3 function y=f(x) 4 y=(0.2+25*x -200*x^2+675*x^3 -900*x^4+400*x^5) 5 endfunction 6 a=0; 7 b=0.8; 8 tval =1.640533; 9 // part a 10 n=3; 131
  • 133. 11 h=(b-a)/n; 12 fa=f(a); 13 fb=f(b); 14 j=a+h; 15 s=0; 16 count =1; 17 while j<b 18 s=s+3*f(j); 19 count=count +1; 20 j=j+h; 21 end 22 l=(b-a)*(fa+s+fb)/(8); 23 disp(” Part A: ”) 24 disp(l,” l=”) 25 Et=tval -l;// e r r o r 26 et=Et *100/ tval;// percent r e l a t i v e e r r o r 27 28 //by using approximate e r r o r estimate 29 30 // the f o u r t h d e r i v a t i v e of f 31 function y=g(x) 32 y= -21600+48000*x 33 endfunction 34 f4x=intg (0,0.8,g)/(b-a);// average value of f o u r t h d e r i v a t i v e 35 Ea = -(1/6480) *(f4x)*(b-a)^5; 36 disp(Et ,”The Error Et=”) 37 disp(”%”,et ,”The percent r e l a t i v e e r r o r et=”) 38 disp(Ea ,”The approximate e r r o r estimate without using the true value=”) 39 40 // part b 41 n=5; 42 h=(b-a)/n; 43 l1=(a+2*h-a)*(fa+4*f(a+h)+f(a+2*h))/6; 44 l2=(a+5*h-a-2*h)*(f(a+2*h)+3*(f(a+3*h)+f(a+4*h))+fb) /8; 45 l=l1+l2; 132
  • 134. 46 disp(” −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 47 disp(” Part B: ”) 48 disp(l,” l=”) 49 Et=tval -l;// e r r o r 50 et=Et *100/ tval;// percent r e l a t i v e e r r o r 51 disp(Et ,”The Error Et=”) 52 disp(”%”,et ,”The percent r e l a t i v e e r r o r et=”) Scilab code Exa 21.7 Unequal Trapezoidal segments 1 clc; 2 clear; 3 function y=f(x) 4 y=(0.2+25*x -200*x^2+675*x^3 -900*x^4+400*x^5) 5 endfunction 6 tval =1.640533; 7 x=[0 0.12 0.22 0.32 0.36 0.4 0.44 0.54 0.64 0.7 0.8]; 8 for i=1:11 9 func(i)=f(x(i)); 10 end 11 l=0; 12 for i=1:10 13 l=l+(x(i+1)-x(i))*( func(i)+func(i+1))/2; 14 end 15 disp(l,” l=”) 16 Et=tval -l;// e r r o r 17 et=Et *100/ tval;// percent r e l a t i v e e r r o r 18 disp(Et ,”The Error Et=”) 19 disp(”%”,et ,”The percent r e l a t i v e e r r o r et=”) 133
  • 135. Scilab code Exa 21.8 Simpsons Uneven data 1 clc; 2 clear; 3 function y=f(x) 4 y=(0.2+25*x -200*x^2+675*x^3 -900*x^4+400*x^5) 5 endfunction 6 tval =1.640533; 7 x=[0 0.12 0.22 0.32 0.36 0.4 0.44 0.54 0.64 0.7 0.8]; 8 for i=1:11 9 func(i)=f(x(i)); 10 end 11 l1=(x(2)-x(1))*((f(x(1))+f(x(2)))/2); 12 l2=(x(4)-x(2))*(f(x(4))+4*f(x(3))+f(x(2)))/6; 13 l3=(x(7)-x(4))*(f(x(4))+3*(f(x(5))+f(x(6)))+f(x(7))) /8; 14 l4=(x(9)-x(7))*(f(x(7))+4*f(x(8))+f(x(9)))/6; 15 l5=(x(10) -x(9))*((f(x(10))+f(x(9)))/2); 16 l6=(x(11) -x(10))*((f(x(11))+f(x(10)))/2); 17 l=l1+l2+l3+l4+l5+l6; 18 disp(l,” l=”) 19 Et=tval -l;// e r r o r 20 et=Et *100/ tval;// percent r e l a t i v e e r r o r 21 disp(Et ,”The Error Et=”) 22 disp(”%”,et ,”The percent r e l a t i v e e r r o r et=”) Scilab code Exa 21.9 Average Temperature Determination 1 clc; 2 clear; 3 function t=f(x,y) 4 t=2*x*y+2*x-x^2-2*y^2+72 5 endfunction 6 len =8; //m, length 134
  • 136. 7 wid =6; //m, width 8 a=0; 9 b=len; 10 n=2; 11 h=(b-a)/n; 12 a1=0; 13 b1=wid; 14 h1=(b1 -a1)/n; 15 16 fa=f(a,0); 17 fb=f(b,0); 18 fh=f(h,0); 19 lx1=(b-a)*(fa+2*fh+fb)/(2*n); 20 21 fa=f(a,h1); 22 fb=f(b,h1); 23 fh=f(h,h1); 24 lx2=(b-a)*(fa+2*fh+fb)/(2*n); 25 26 fa=f(a,b1); 27 fb=f(b,b1); 28 fh=f(h,b1); 29 lx3=(b-a)*(fa+2*fh+fb)/(2*n); 30 31 l=(b1 -a1)*(lx1 +2* lx2+lx3)/(2*n); 32 33 avg_temp=l/(len*wid); 34 disp(avg_temp ,”The average termperature i s=”) 135
  • 137. Chapter 22 Integration of equations Scilab code Exa 22.1 Error corrections of the trapezoidal rule 1 // c l c ( ) 2 h = [0.8 ,0.4 ,0.2]; 3 I = [0.1728 ,1.0688 ,1.4848]; 4 E = [89.5 ,34.9 ,9.5]; 5 I1 = 4 * I(2) / 3 - I(1) / 3; 6 t = 1.640533; 7 et1 = t - I1 ; 8 Et1 = et1 * 100/t; 9 disp(”%”,Et1 ,” Error of the improved i n t e g r a l f o r segment 1 and 2 = ”) 10 I2 = 4 * I(3) / 3 - I(2) / 3; 11 et2 = t - I2 ; 12 Et2 = et2 * 100/t; 13 disp(”%”,Et2 ,” Error of the improved i n t e g r a l f o r segment 4 and 2 = ”) Scilab code Exa 22.2 Higher order error correction of integral estimates 136
  • 138. 1 // c l c ( ) 2 I1 = 1.367467; 3 I2 = 1.623467; 4 I = 16 * I2 /15 - I1 / 15; 5 disp(I,” Obtained i n t e g r a l which i s the c o r r e c t answer t i l l the seventh decimal ”) Scilab code Exa 22.3 Two point gauss legendre formulae 1 // c l c ( ) 2 // f ( x ) = 0 .2 + 25∗x − 200∗ xˆ2 + 675∗ xˆ3 − 900∗ xˆ4 + 400∗ xˆ5 3 // f o r using two point gauss l e g e n d r e formulae , the i n t e r v a l s have to be changed to −1 and 1 4 // t h e r e f o r e , x = 0 .4 + 0 .4 ∗ xd 5 // thus the i n t e g r a l i s t r a n s f e r r e d to 6 // ( 0 . 2 + 25∗(0.4+0.4∗ x ) − 200∗(0.4 + 0.4∗ x ) ˆ2 + 675∗(0.4 + 0.4∗ x ) ˆ3 − 900∗(0.4 + 0.4∗ x ) ˆ4 + 400∗(0.4 + 0.4∗ x ) ˆ5) ∗0.4 7 // f o r t h r e e point gauss l e g e n d r e formulae 8 x1 = -(1/3) ^ 0.5; 9 x2 = (1/3) ^ 0.5; 10 I1 = (0.2 + 25*(0.4+0.4* x1) - 200*(0.4 + 0.4*x1)^2 + 675*(0.4 + 0.4* x1)^3 - 900*(0.4 + 0.4* x1)^4 + 400*(0.4 + 0.4* x1)^5) *0.4; 11 I2 = (0.2 + 25*(0.4+0.4* x2) - 200*(0.4 + 0.4*x2)^2 + 675*(0.4 + 0.4* x2)^3 - 900*(0.4 + 0.4* x2)^4 + 400*(0.4 + 0.4* x2)^5) *0.4; 12 I = I1 + I2; 13 disp(I,” I n t e g r a l obtained using gauss l e g e n d r e formulae =”) 14 t = 1.640533; 15 e = (t - I)*100/t; 16 disp(”%”,e,” e r r o r = ”) 137
  • 139. Scilab code Exa 22.4 Three point gauss legendre method 1 // c l c ( ) 2 // f ( x ) = 0 .2 + 25∗x − 200∗ xˆ2 + 675∗ xˆ3 − 900∗ xˆ4 + 400∗ xˆ5 3 // f o r using t h r e e point gauss l e g e n d r e formulae , the i n t e r v a l s have to be changed to −1 and 1 4 // t h e r e f o r e , x = 0 .4 + 0 .4 ∗ xd 5 // thus the i n t e g r a l i s t r a n s f e r r e d to 6 // ( 0 . 2 + 25∗(0.4+0.4∗ x ) − 200∗(0.4 + 0.4∗ x ) ˆ2 + 675∗(0.4 + 0.4∗ x ) ˆ3 − 900∗(0.4 + 0.4∗ x ) ˆ4 + 400∗(0.4 + 0.4∗ x ) ˆ5) ∗0.4 7 // f o r t h r e e point gauss l e g e n d r e formulae 8 x1 = -0.7745967; 9 x2 = 0; 10 x3 = 0.7745967; 11 c0 = 0.5555556; 12 c1 = 0.8888889; 13 c2 = 0.5555556; 14 I1 = (0.2 + 25*(0.4+0.4* x1) - 200*(0.4 + 0.4*x1)^2 + 675*(0.4 + 0.4* x1)^3 - 900*(0.4 + 0.4* x1)^4 + 400*(0.4 + 0.4* x1)^5) *0.4; 15 I2 = (0.2 + 25*(0.4+0.4* x2) - 200*(0.4 + 0.4*x2)^2 + 675*(0.4 + 0.4* x2)^3 - 900*(0.4 + 0.4* x2)^4 + 400*(0.4 + 0.4* x2)^5) *0.4; 16 I3 = (0.2 + 25*(0.4+0.4* x3) - 200*(0.4 + 0.4*x3)^2 + 675*(0.4 + 0.4* x3)^3 - 900*(0.4 + 0.4* x3)^4 + 400*(0.4 + 0.4* x3)^5) *0.4; 17 I = c0 * I1 + c1 * I2 + c2 * I3; 18 disp(I,” i n t e g r a l obtained using t h r e e point gauss l e g e n d r e formulae = ”) 138
  • 140. Scilab code Exa 22.5 Applying Gauss Quadrature to the falling Parachutist problem 1 // c l c ( ) 2 // f ( t ) = g∗m∗( i n t (0 ,10 ,(1 − exp(−c∗ t /m) ) ) ) / c 3 // f o r using gauss quadrature method , l i m i t s are changed to −1 to 1 by r e p l c i n g x = 5 + 5∗xd 4 // the new i n t e g r a l obtained i s 5 // (1 − exp(−c ∗(5 + 5∗x ) /m ) ) ∗5 6 g = 9.8; 7 c = 12.5; 8 m = 68.1; 9 // f o r two point method 10 x1 = -(1/3) ^0.5; 11 x2 = (1/3) ^0.5; 12 I1 = g*m*(1 - exp(-c*(5 + 5*x1)/m ))*5 / c; 13 I2 = g*m*(1 - exp(-c*(5 + 5*x2)/m ))*5 / c; 14 I = I1 + I2; 15 disp(I,” i n t e g r a l by two point method = ”) 16 x1 = -0.7745967; 17 x2 = 0; 18 x3 = 0.7745967; 19 c0 = 0.5555556; 20 c1 = 0.8888889; 21 c2 = 0.5555556; 22 I1 = g*m*(1 - exp(-c*(5 + 5*x1)/m ))*5 / c; 23 I2 = g*m*(1 - exp(-c*(5 + 5*x2)/m ))*5 / c; 24 I3 = g*m*(1 - exp(-c*(5 + 5*x3)/m ))*5 / c; 25 I = c0*I1 + c1 * I2 + c2 * I3; 26 disp(I,” i n t e g r a l by t h r e e point method =”) 27 x1 = -0.861136312; 28 x2 = -0.339981044; 29 x3 = 0.339981044; 30 x4 = 0.861136312; 31 c1 = 0.3478548; 32 c2 = 0.6521452; 33 c3 = 0.6521452; 34 c4 = 0.3478548; 35 I1 = g*m*(1 - exp(-c*(5 + 5*x1)/m ))*5 / c; 139
  • 141. 36 I2 = g*m*(1 - exp(-c*(5 + 5*x2)/m ))*5 / c; 37 I3 = g*m*(1 - exp(-c*(5 + 5*x3)/m ))*5 / c; 38 I4 = g*m*(1 - exp(-c*(5 + 5*x4)/m ))*5 / c; 39 I = c1*I1 + c2 * I2 + c3 * I3 + c4 * I4; 40 disp(I,” i n t e g r a l by f o u r point method =”) 41 x1 = -0.906179846; 42 x2 = -0.538469310; 43 x3 = 0; 44 x4 = 0.538469310; 45 x5 = 0.906179846 46 c1 = 0.2369269; 47 c2 = 0.4786287; 48 c3 = 0.5688889; 49 c4 = 0.4786287; 50 c5 = 0.2369269; 51 I1 = g*m*(1 - exp(-c*(5 + 5*x1)/m ))*5 / c; 52 I2 = g*m*(1 - exp(-c*(5 + 5*x2)/m ))*5 / c; 53 I3 = g*m*(1 - exp(-c*(5 + 5*x3)/m ))*5 / c; 54 I4 = g*m*(1 - exp(-c*(5 + 5*x4)/m ))*5 / c; 55 I5 = g*m*(1 - exp(-c*(5 + 5*x5)/m ))*5 / c; 56 I = c1*I1 + c2 * I2 + c3 * I3 + c4 * I4 + c5 * I5; 57 disp(I,” i n t e g r a l by f i v e point method =”) Scilab code Exa 22.6 Evaluation of improper integral 1 // c l c ( ) 2 //N( x ) = ( i n t (− i n f i n i t y , −2 , exp (−(x ˆ2) /2) ) + i n t ( −2 ,1 , exp (−(x ˆ2) /2) ) ) /(2∗ pi ) ˆ0.5 3 // f i r s t i n t e g r a l can be s o l v e d as 4 // i n t (− i n f i n i t y , −2 , exp (−(x ˆ2) /2) ) = i n t ( −0.5 ,0 , exp ( −1/(2∗ t ˆ2) ) / t ˆ2) 5 h = 1/8; 6 // i n t ( −0.5 ,0 , exp ( −1/(2∗ t ˆ2) ) / t ˆ2) = h ∗( f ( x −7/16) + f ( x −5/16) + f ( x −3/16) + f ( x −1/16) ) 7 t1 = -7/16; 140
  • 142. 8 t2 = -5/16; 9 t3 = -3/16; 10 t4 = -1/16; 11 m1 = exp ( -1/(2*t1^2))/t1^2; 12 m2 = exp ( -1/(2*t2^2))/t2^2; 13 m3 = exp ( -1/(2*t3^2))/t3^2; 14 m4 = exp ( -1/(2*t4^2))/t4^2; 15 I1 = h*(m1 + m2 + m3 + m4); 16 disp(I1 ,” the value of f i r s t i n t e g r a l = ”) 17 // simpsons 1/3 rd s u l e i s a p p l i e d f o r the second i n t e g r a l 18 h1 = 0.5; 19 x1 = -2; 20 x2 = -1.5; 21 x3 = -1; 22 x4 = -0.5; 23 x5 = 0; 24 x6 = 0.5; 25 x7 = 1; 26 n1 = exp(-(x1^2) /2); 27 n2 = exp(-(x2^2) /2); 28 n3 = exp(-(x3^2) /2); 29 n4 = exp(-(x4^2) /2); 30 n5 = exp(-(x5^2) /2); 31 n6 = exp(-(x6^2) /2); 32 n7 = exp(-(x7^2) /2); 33 I2 =(1-(-2)) * (n1 + 4 *(n2 + n4 + n6) + 2*(n3 + n5) + n7)/(18); 34 disp(I2 ,”The value of second i n t e g r a l = ”) 35 f = (I1 + I2)/(2 * %pi)^0.5; 36 disp(f,” Therefore the f i n a l r e s u l t can be computed as ”) 37 N = 0.8413; 38 e = (N - f) * 100 / N; 39 disp(”%”,e,” e r r o r = ”) 141
  • 143. Chapter 23 Numerical differentiation Scilab code Exa 23.1 High accuracy numerical differentiation formulas 1 // c l c ( ) 2 // f ( x ) = −0.1∗xˆ4 − 0.15∗ xˆ3 − 0. 5 ∗ xˆ2 − 0.25 ∗x + 1. 2 3 h = 0.25; 4 t = -0.9125; 5 x = 0:h:1; 6 disp(x,”x = ”) 7 fx = -0.1*x^4 - 0.15*x^3 - 0.5 * x^2 - 0.25 *x + 1.2; 8 disp(fx ,” f ( x ) = ”) 9 fd = (- fx(5) + 4*fx(4) - 3 * fx(3))/(2 * h); 10 efd = (t - fd) * 100 / t; 11 disp(fd ,”by forward d i f f e r e n c e ”) 12 disp(”%”,efd ,” e r r o r in forward d i f f e r e n c e method = ” ) 13 bd = (3 * fx(3) - 4 * fx(2) + fx(1))/ (2*h); 14 ebd = (t - bd) * 100 / t; 15 disp(bd ,”by backward d i f f e r e n c e ”) 16 disp(”%”,ebd ,” e r r o r in backward d i f f e r e n c e method = ”) 17 cdm = (-fx(5) + 8*(fx(4)) -8*fx(2) + fx(1) ) / (12*h 142
  • 144. ); 18 ecdm = (t - cdm) * 100 / t; 19 disp(cdm ,”by c e n t r a l d i f f e r e n c e ”) 20 disp(”%”,ecdm ,” e r r o r in c e n t r a l d i f f e r e n c e method = ”) Scilab code Exa 23.2 Richardson extrapolation 1 // c l c ( ) 2 // f ( x ) = −0.1∗xˆ4 − 0.15∗ xˆ3 − 0. 5 ∗ xˆ2 − 0.25 ∗x + 1. 2 3 h = 0.5; 4 t = -0.9125; 5 x = 0:h:1; 6 disp(x,”x with h = 0 .5 i s ”) 7 fx = -0.1*x^4 - 0.15*x^3 - 0.5 * x^2 - 0.25 *x + 1.2; 8 disp(fx ,” f ( x ) with h = 0. 5 i s ”) 9 cdm = (fx(3) - fx(1))/ 1; 10 ecdm = (t - cdm) * 100 / t; 11 disp(cdm ,”by c e n t r a l d i f f e r e n c e ( h = 0.5 ) ”) 12 disp(”%”,ecdm ,” e r r o r in c e n t r a l d i f f e r e n c e method ( h = 0. 5 ) = ”) 13 h1 = 0.25; 14 x1 = 0:h1:1; 15 disp(x1 ,”x with h = 0.25 i s ”) 16 fx1 = -0.1*x1^4 - 0.15* x1^3 - 0.5 * x1^2 - 0.25 *x1 + 1.2; 17 disp(fx1 ,” fx with h = 0.25 i s ”) 18 cdm1 = (fx1 (4) - fx1 (2))/ (2*h1); 19 ecdm1 = (t - cdm1) * 100 / t; 20 disp(cdm1 ,”by c e n t r a l d i f f e r e n c e ( h = 0.25 ) ”) 21 disp(”%”,ecdm1 ,” e r r o r in c e n t r a l d i f f e r e n c e method ( h = 0.25 ) = ”) 22 D = 4 * cdm1 /3 - cdm / 3; 143
  • 145. 23 disp(D,” improved estimate =”) Scilab code Exa 23.3 Differentiating unequally spaced data 1 // c l c ( ) 2 //q ( z = 0) = −k∗p∗C∗(dT/dz ) /( z = 0) 3 k = 3.5 * 10^ - 7; //mˆ2/ s 4 p = 1800; // kg/mˆ3 5 C = 840; // ( J /( kg .C) ) 6 x = 0; 7 fx0 = 13.5; 8 fx1 = 12; 9 fx2 = 10; 10 x0 = 0; 11 x1 = 1.25; 12 x2 = 3.75; 13 dfx = fx0 *(2*x - x1 - x2)/((x0 - x1)*(x0 - x2)) + fx1 *(2*x - x0 - x2)/((x1 - x0)*(x1 - x2)) + fx2 *(2*x - x1 - x0)/((x2 - x1)*(x2 - x0)); 14 disp(”C/cm”,dfx ,” (dT/dz ) = ”) 15 q = - k * p *C * dfx *100; 16 disp(”W/mˆ2”,q,”q ( z = 0) =”) Scilab code Exa 23.4 Integration and Differentiation 1 clc; 2 clear; 3 function y=f(x) 4 y=0.2+25*x -200*x^2+675*x^3 -900*x^4+400*x^5 5 endfunction 6 a=0; 7 b=0.8; 8 Q=intg (0,0.8,f); 144
  • 146. 9 disp(Q,”Q=”) 10 x=[0 0.12 0.22 0.32 0.36 0.4 0.44 0.54 0.64 0.7 0.8]; 11 y=f(x); 12 integral=inttrap(x,y) 13 disp(integral ,” i n t e r g r a l=”) 14 disp(diff(x),” d i f f ( x )=”) 15 d=diff(y)./ diff(x); 16 disp(d,”d=”) Scilab code Exa 23.5 Integrate a function 1 clc; 2 clear; 3 function y=f(x) 4 y=0.2+25*x -200*x^2+675*x^3 -900*x^4+400*x^5 5 endfunction 6 a=0; 7 b=0.8; 8 Qt =1.640533; 9 Q=intg (0,0.8,f); 10 disp(Q,”Computed=”) 11 disp(abs(Q-Qt)*100/Qt ,” Error estimate=”) 145
  • 147. Chapter 25 Runga Kutta methods Scilab code Exa 25.1 Eulers method 1 // c l c ( ) 2 //dy/dx = −2∗xˆ3 + 12∗xˆ2 − 20∗x + 8.5 3 // t h e r e f o r e , y = −0.5∗xˆ4 + 4∗xˆ3 − 10∗xˆ2 + 8 .5 + c 4 x1 = 0; 5 y1 = 1; 6 h = 0.5; 7 c =-(-0.5*x1^4 + 4*x1^3 - 10*x1^2 + 8.5* x1 - y1); 8 x = 0:0.5:4; 9 disp(x,”x = ”) 10 y = -0.5*x^4 + 4*x^3 - 10*x^2 + 8.5*x + c; 11 disp(y,” true v a l u e s of y = ”) 12 fxy = -2*x^3 + 12*x^2 - 20*x + 8.5; 13 y2(1) = y(1); 14 e(1) = (y(1) - y2(1)) * 100 / y(1); 15 for i = 2:9 16 y2(i) = y2(i-1) + fxy(i-1)*h; 17 e(i) = (y(i) - y2(i))*100/y(i); 18 end 19 disp(y2 ,”y by e u l e r method =”) 20 disp(e,” e r r o r =”) 146
  • 148. Scilab code Exa 25.2 Taylor series estimate for error by eulers method 1 // c l c ( ) 2 // f ( x , y ) = dy/dx = −2∗xˆ3 + 12∗xˆ2 − 20∗x + 8.5 3 // f ’ ( x , y ) = −6∗xˆ2 + 24∗x − 20 4 // f ”( x , y ) = −12∗x + 24 5 // f ” ’( x , y ) = −12 6 x = 0; 7 Et2 = (-6*x^2 + 24*x - 20) * 0.5^2 / 2; 8 Et3 = (-12*x + 24) * (0.5) ^3 / 6; 9 Et4 = (-12) *(0.5 ^ 4) / 24; 10 Et = Et2 + Et3 + Et4; 11 disp(Et ,” Total t r u n c a t i o n e r r o r =”) Scilab code Exa 25.3 Effect of reduced step size on Eulers method 1 // c l c ( ) 2 //dy/dx = −2∗xˆ3 + 12∗xˆ2 − 20∗x + 8.5 3 // t h e r e f o r e , y = −0.5∗xˆ4 + 4∗xˆ3 − 10∗xˆ2 + 8 .5 + c 4 x1 = 0; 5 y1 = 1; 6 h = 0.25; 7 c =-(-0.5*x1^4 + 4*x1^3 - 10*x1^2 + 8.5* x1 - y1); 8 x = 0:h:4; 9 disp(x,”x = ”) 10 y = -0.5*x^4 + 4*x^3 - 10*x^2 + 8.5*x + c; 11 disp(y,” true v a l u e s of y = ”) 12 fxy = -2*x^3 + 12*x^2 - 20*x + 8.5; 13 y2(1) = y(1); 14 e(1) = (y(1) - y2(1)) * 100 / y(1); 15 for i = 2:17 16 y2(i) = y2(i-1) + fxy(i-1)*h; 147
  • 149. Figure 25.1: Solving ODEs 17 e(i) = (y(i) - y2(i))*100/y(i); 18 end 19 disp(y2 ,”y by e u l e r method =”) 20 disp(e,” e r r o r =”) Scilab code Exa 25.4 Solving ODEs 1 clc; 2 clear; 3 m=68.1; 4 g=9.8; 148
  • 150. 5 c=12.5; 6 a=8.3; 7 b=2.2; 8 vmax =46; 9 function yp=f(t,v) 10 yp=g-c*v/m; 11 endfunction 12 v0=0; 13 t=0:15; 14 x=ode(v0 ,0,t,f); 15 disp(x) 16 plot(t,x, ’ .− ’ ) 17 18 function yp=f1(t,v) 19 yp=g-(c/m)*(v+a*(v/vmax)^b) 20 endfunction 21 x1=ode(v0 ,0,t,f1); 22 plot(t,x1) 23 xtitle(” v e l o c i t y vs time ”,” t ( s ) ”,”v (m/ s ) ”) 24 h1=legend ([” Linear ”;” Nonlinear ”]) Scilab code Exa 25.5 Heuns method 1 // c l c ( ) 2 //y ’ = 4∗ exp ( 0 . 8 ∗ x ) − 0.5∗ y 3 //y = 4∗( exp ( 0 . 8 ∗ x ) − exp ( −0.5∗x ) ) /1.3 + 2∗ exp ( −0.5∗ x ) 4 x = 0:1:4; 5 disp(x) 6 x1 = 0; 7 y1 = 2; 8 y2(1) = y1; 9 for i = 1:5 10 y(i) = 4*( exp (0.8*x(i)) - exp ( -0.5*x(i)))/1.3 + 2*exp ( -0.5*x(i)); 149
  • 151. 11 dy(i) = 4*exp (0.8*x(i)) - 0.5* y2(i); 12 y2(i + 1) = y2(i) + dy(i); 13 if i>1 then 14 m(i) = (dy(i) + dy(i-1))/2; 15 y2(i) = y2(i-1) + m(i); 16 dy(i) = 4*exp (0.8*x(i)) - 0.5* y2(i); 17 end 18 e(i) = (y(i) - y2(i)) * 100 / y(i); 19 end 20 disp(y2 (1:5) ,”By heuns method (1 i t e r a t i o n ) ”) 21 disp(”%”,e(1:5) ,” e r r o r = ”) Scilab code Exa 25.6 Comparison of various second order RK 4 method 1 // c l c ( ) 2 // f ’ ( x , y ) = −2∗xˆ3 + 12∗xˆ2 −20∗x + 8.5 3 // f ( x , y ) = −xˆ4 / 2 + 4∗xˆ3 − 10∗xˆ2 + 8.5∗ x + 1 4 h = 0.5; 5 x = 0:h:4; 6 y1 = -x^4 / 2 + 4*x^3 - 10*x^2 + 8.5*x + 1; 7 y(1) = 1; 8 disp(x,”x =”) 9 disp(y1 ,” true value of y =”) 10 for i = 1:8 11 k1(i) = -2*x(i)^3 + 12*x(i)^2 -20*x(i) + 8.5; 12 x1(i) = x(i) + h/2; 13 k2(i) = -2*x1(i)^3 + 12*x1(i)^2 -20*x1(i) + 8.5; 14 y(i+1) = y(i) + k2(i)*h; 15 e(i) = (y1(i) - y(i))*100/ y1(i); 16 end 17 disp(y(1:9) ,”y by midpoint method”) 18 disp(e,” e r r o r = ”) 19 for i = 1:8 20 k1(i) = -2*x(i)^3 + 12*x(i)^2 -20*x(i) + 8.5; 21 x(i) = x(i) + 3*h/4; 150
  • 152. 22 k2(i) = -2*x(i)^3 + 12*x(i)^2 -20*x(i) + 8.5; 23 y(i+1) = y(i) + (k1(i)/3 + 2*k2(i)/3)*h; 24 e(i) = (y1(i) - y(i))*100/ y1(i); 25 end 26 disp(y(1:9) ,”y by second order Ralston RK”) 27 disp(e,” e r r o r = ”) Scilab code Exa 25.7 Classical fourth order RK method 1 // c l c ( ) 2 // f ’ ( x , y ) = −2∗xˆ3 + 12∗xˆ2 −20∗x + 8.5 3 // f ( x , y ) = −xˆ4 / 2 + 4∗xˆ3 − 10∗xˆ2 + 8.5∗ x + 1 4 h = 0.5; 5 x = 0:h:4; 6 y1 = -x^4 / 2 + 4*x^3 - 10*x^2 + 8.5*x + 1; 7 y(1) = 1; 8 for i=1:8 9 k1(i) = -2*x(i)^3 + 12*x(i)^2 -20*x(i) + 8.5; 10 x1(i) = x(i) + h/2; 11 k2(i) = -2*x1(i)^3 + 12*x1(i)^2 -20*x1(i) + 8.5; 12 k3(i) = -2*x1(i)^3 + 12*x1(i)^2 -20*x1(i) + 8.5; 13 x2(i) = x(i) + h; 14 k4(i) = -2*x2(i)^3 + 12*x2(i)^2 -20*x2(i) + 8.5; 15 y(i+1) = y(i) + (k1(i)+2*k2(i)+2*k3(i)+k4(i))*h /6; 16 e(i) = (y1(i) - y(i))*100/ y1(i); 17 end 18 disp(” f ( x , y ) = −2∗xˆ3 + 12∗xˆ2 −20∗x + 8.5 ”) 19 disp(y(1:9) ,”y by f o u r t h order Ralston RK”) 20 disp(” f ( x , y ) = 4∗ exp ( 0 . 8 ∗ x ) − 0.5∗ y”) 21 x = 0:h:0.5; 22 y(1) = 2; 23 k1 = 4*( exp (0.8*x(1))) -0.5*y(1); 24 x1 = x(1) + 0.5*h; 25 y1 = y(1) + 0.5* k1*h; 151
  • 153. 26 k2 = 4*exp (0.8* x1) - 0.5* y1; 27 y2 = y(1) + 0.5* k2*h; 28 k3 = 4*exp (0.8* x1) - 0.5* y2; 29 x1 = x(1) + h; 30 y1 = y(1) + k3*h; 31 k4 = 4*exp (0.8* x1) - 0.5* y1; 32 y(2) = y(1) + (k1+2*k2+2*k3+k4)*h/6; 33 disp(y(1:2) ,”y = ”) Scilab code Exa 25.8 Comparison of Runga Kutta methods 1 // c l c ( ) 2 disp(” f ( x , y ) = 4∗ exp ( 0 . 8 ∗ x ) − 0.5∗ y”) 3 h = 1; 4 x = 0:h:4; 5 y(1) = 2; 6 for i = 1:5 7 k1(i) = 4*( exp (0.8*x(i))) -0.5*y(i); 8 x1 = x(i) + h; 9 y1 = y(i) + k1(i)*h; 10 k2(i) = 4*( exp (0.8* x1)) -0.5*y1; 11 y(i+1) = y(i) + (k1(i)/2 + k2(i)/2)*h; 12 end 13 disp(y(1:5) ,”y ( second order RK method ) = ”) 14 for i = 1:5 15 k1(i) = 4*( exp (0.8*x(i))) -0.5*y(i); 16 x1 = x(i) + 0.5*h; 17 y1 = y(i) + 0.5*h*k1(i); 18 k2(i) = 4*( exp (0.8* x1)) -0.5*y1; 19 x1 = x(i) + h; 20 y1 = y(i) -k1(i)*h + 2*k2(i)*h; 21 k3(i) = 4*( exp (0.8* x1)) -0.5*y1; 22 y(i+1) = y(i) + (k1(i) + 4*k2(i) + k3(i))*h/6; 23 end 24 disp(y(1:5) ,”y ( t h i r d order RK method ) = ”) 152
  • 154. 25 for i = 1:5 26 k1(i) = 4*( exp (0.8*x(i))) -0.5*y(i); 27 x1 = x(i) + 0.5*h; 28 y1 = y(i) + 0.5* k1(i)*h; 29 k2(i) = 4*exp (0.8* x1) - 0.5* y1; 30 y2 = y(i) + 0.5* k2(i)*h; 31 k3(i) = 4*exp (0.8* x1) - 0.5* y2; 32 x1 = x(i) + h; 33 y1 = y(i) + k3(i)*h; 34 k4(i) = 4*exp (0.8* x1) - 0.5* y1; 35 y(i+1) = y(i) + (k1(i)+2*k2(i)+2*k3(i)+k4(i))*h/6; 36 end 37 disp(y(1:5) ,”y ( f o u r t h order RK method ) = ”) 38 for i = 1:5 39 40 k1(i) = 4*( exp (0.8*x(i))) -0.5*y(i); 41 x1 = x(i) + 0.25*h; 42 y1 = y(i) + 0.25* k1(i)*h; 43 k2(i) = 4*exp (0.8* x1) - 0.5* y1; 44 y2 = y(i) + 0.125* k2(i)*h + 0.125* k1(i)*h; 45 k3(i) = 4*exp (0.8* x1) - 0.5* y2; 46 x1 = x(i) + 0.5*h; 47 y1 = y(i) -0.5*k2(i)*h + k3(i)*h; 48 k4(i) = 4*exp (0.8* x1) - 0.5* y1; 49 x1 = x(i) + 0.75*h; 50 y1 = y(i) + 3*k1(i)*h/16 + 9*k4(i)*h/16; 51 k5(i) = 4*exp (0.8* x1) - 0.5* y1; 52 x1 = x(i) + h; 53 y1 = y(i) - 3*k1(i)*h/7 + 2*k2(i)*h/7 + 12*k3(i)*h/7 - 12*k4(i)*h/7 + 8*k5(i)*h/7; 54 k6(i) = 4*exp (0.8* x1) - 0.5* y1; 55 y(i+1) = y(i) + (7*k1(i)+32* k3(i)+12* k4(i) + 32*k5(i ) + 7*k6(i))*h/90; 56 end 57 disp(y(1:5) ,”y ( f i f t h order RK method ) ”) 153
  • 155. Scilab code Exa 25.9 Solving systems of ODE using Eulers method 1 // c l c ( ) 2 // dy1/dx = −0.5∗ y1 3 // dy2/dx = 4 − 0.3∗ y2 − 0.1∗ y1 4 x1 = 0; 5 h =0.5; 6 y1(1) = 4; 7 y2(1) = 6; 8 x = 0:h:2; 9 for i = 2:5 10 y1(i) = y1(i-1) -0.5*y1(i-1)*h; 11 y2(i) = y2(i-1) + (4 - 0.3* y2(i-1) - 0.1* y1(i-1) )*h; 12 end 13 disp(x,”x = ”) 14 disp(y1 ,”y1 = ”) 15 disp(y2 ,”y2 = ”) Scilab code Exa 25.10 Solving systems of ODE using RK 4 method 1 // c l c ( ) 2 // dy1/dx = −0.5∗ y1 3 // dy2/dx = 4 − 0.3∗ y2 − 0.1∗ y1 4 x1 = 0; 5 h =0.5; 6 y1(1) = 4; 7 y2(1) = 6; 8 x = 0:h:2; 9 for i = 1:5 10 k11(i) = -0.5*y1(i); 11 k12(i) = 4 - 0.3* y2(i) - 0.1* y1(i); 154
  • 156. 12 y11 = y1(i) + k11(i) * h/2; 13 y21 = y2(i) + k12(i) * h/2; 14 k21(i) = -0.5* y11; 15 k22(i) = 4 - 0.3* y21 - 0.1* y11; 16 y11 = y1(i) + k21(i) * h/2; 17 y21 = y2(i) + k22(i) * h/2; 18 k31(i) = -0.5* y11; 19 k32(i) = 4 - 0.3* y21 - 0.1* y11; 20 y11 = y1(i) + k31(i) * h; 21 y21 = y2(i) + k32(i) * h; 22 k41(i) = -0.5* y11; 23 k42(i) = 4 - 0.3* y21 - 0.1* y11; 24 y1(i+1) = y1(i) + ( k11(i) + 2*k21(i) + 2*k31(i) + k41(i) )*h / 6; 25 y2(i+1) = y2(i) + ( k12(i) + 2*k22(i) + 2*k32(i) + k42(i) ) *h/ 6; 26 end 27 disp(” using f o u r t h order RK method”) 28 disp(x,”x =”) 29 disp(y1 (1:5) ,”y1 = ”) 30 disp(y2 (1:5) ,”y2 = ”) Scilab code Exa 25.11 Solving systems of ODEs 1 clc; 2 clear; 3 function yp=f(x,y) 4 yp=[y(2) ; -16.1*y(1)]; 5 endfunction 6 x=0:0.1:4 7 y0 =[0.1 0]; 8 sol=ode(y0 ,0,x,f); 9 count =1; 10 disp(sol) 11 for i=1:2:81 155
  • 157. 12 a(count)=sol(i); 13 b(count)=sol(i+1); 14 count=count +1; 15 end 16 plot(x,a) 17 plot(x,b,”.−”) 18 h1=legend ([”y1 , y3”,”y2 , y4”]) 19 xtitle(”y vs x”,”x”,”y”) 20 function yp=g(x,y) 21 yp=[y(2) ; -16.1* sin(y(1))]; 22 endfunction 23 sol=ode(y0 ,0,x,g); 24 count =1; 25 disp(sol) 26 for i=1:2:81 27 a(count)=sol(i); 28 b(count)=sol(i+1); 29 count=count +1; 30 end 31 plot(x,a) 32 plot(x,b,”.−”) 33 clf(); 34 y0=[%pi/4 0]; 35 sol=ode(y0 ,0,x,f); 36 count =1; 37 disp(sol) 38 for i=1:2:81 39 a(count)=sol(i); 40 b(count)=sol(i+1); 41 count=count +1; 42 end 43 plot(x,a) 44 plot(x,b,”.−”) 45 46 xtitle(”y vs x”,”x”,”y”) 47 sol=ode(y0 ,0,x,g); 48 count =1; 49 disp(sol) 156
  • 158. Figure 25.2: Solving systems of ODEs 50 for i=1:2:81 51 a(count)=sol(i); 52 b(count)=sol(i+1); 53 count=count +1; 54 end 55 plot(x,a, ’ o− ’ ) 56 plot(x,b,”x−”) 57 h1=legend ([”y1”,”y2”,”y3”,”y4”]) 157
  • 159. Figure 25.3: Solving systems of ODEs 158
  • 160. Scilab code Exa 25.12 Adaptive fourth order RK method 1 // c l c ( ) 2 disp(” f ( x , y ) = 4∗ exp ( 0 . 8 ∗ x ) − 0.5∗ y”) 3 // f ’ ( x , y ) = 4∗ exp ( 0 . 8 ∗ x ) − 0.5∗ y 4 x = 0:2:2; 5 y(1) = 2; 6 h =2; 7 t = 14.84392; 8 k1 = 4*exp (0.8*x(1)) - 0.5*y(1); 9 x1 = x(1) + h/2; 10 y1 = y(1) + k1*h/2; 11 k2 = 4*exp (0.8* x1) - 0.5* y1; 12 x1 = x(1) + h/2; 13 y1 = y(1) + k2*h/2; 14 k3 = 4*exp (0.8* x1) - 0.5* y1; 15 x1 = x(1) + h; 16 y1 = y(1) + k3*h; 17 k4 = 4*exp (0.8* x1) - 0.5* y1; 18 y(2) = y(1) + (k1 + 2*k2 + 2*k3 + k4)*h/6; 19 e = (t - y(2))/(t); 20 disp(y(1:2) ,”y by h = 2 i s ”) 21 disp(e,” e r r o r = ”) 22 h = 1; 23 x = 0:h:2; 24 for i=1:3 25 k1(i) = 4*exp (0.8*x(i)) - 0.5*y(i); 26 x1 = x(i) + h/2; 27 y1 = y(i) + k1(i)*h/2; 28 k2(i) = 4*exp (0.8* x1) - 0.5* y1; 29 x1 = x(i) + h/2; 30 y1 = y(i) + k2(i)*h/2; 31 k3(i) = 4*exp (0.8* x1) - 0.5* y1; 32 x1 = x(i) + h; 33 y1 = y(i) + k3(i)*h; 34 k4(i) = 4*exp (0.8* x1) - 0.5* y1; 35 y(i+1) = y(i) + (k1(i) + 2*k2(i) + 2*k3(i) + k4( i))*h/6; 159
  • 161. 36 end 37 e = (t - (y(3)))/t; 38 disp(y(1:3) ,”y by h = 1 i s ”) 39 disp(e,” e r r o r = ”) Scilab code Exa 25.13 Runga kutta fehlberg method 1 // c l c ( ) 2 disp(” f ( x , y ) = 4∗ exp ( 0 . 8 ∗ x ) − 0.5∗ y”) 3 // f ’ ( x , y ) = 4∗ exp ( 0 . 8 ∗ x ) − 0.5∗ y 4 h =2; 5 x = 0:h:2; 6 y(1) = 2; 7 t = 14.84392; 8 k1 = 4*exp (0.8*x(1)) - 0.5*y(1); 9 x1 = x(1) + h/5; 10 y1 = y(1) + k1*h/5; 11 k2 = 4*exp (0.8* x1) - 0.5* y1; 12 x1 = x(1) + 3*h/10; 13 y1 = y(1) + 3*k1*h/40 + 9*k2*h/40; 14 k3 = 4*exp (0.8* x1) - 0.5* y1; 15 x1 = x(1) + 3*h/5; 16 y1 = y(1) + 3*k1*h/10 - 9*k2*h/10 + 6*k3*h/5; 17 k4 = 4*exp (0.8* x1) - 0.5* y1; 18 x1 = x(1) + h; 19 y1 = y(1) -11*k1*h/54 + 5*k2*h/2 - 70*k3*h/27 + 35* k4*h/27; 20 k5 = 4*exp (0.8* x1) - 0.5* y1; 21 x1 = x(1) + 7*h/8; 22 y1 = y(1) + 1631* k1*h/55296 + 175* k2*h/512 + 575* k3* h/13824 + 44275* k4*h/110592 +253* k5*h/4096; 23 k6 = 4*exp (0.8* x1) - 0.5* y1; 24 y1 = y(1) + (37* k1/378 + 250* k3 /621 + 125* k4 /594 + 512* k6 /1771)*h; 25 y2 = y(1) + (2825* k1 /27648 + 18575* k3 /48384 + 13525* 160
  • 162. Figure 25.4: Adaptive Fourth order RK scheme k4 /55296 + 277* k5 /14336 + k6/4)*h; 26 disp(y1 ,”y ( f o u r t h order p r e d i c t i o n ) = ”) 27 disp(y2 ,”y ( f i f t h order p r e d i c t i o n ) = ”) Scilab code Exa 25.14 Adaptive Fourth order RK scheme 1 clc; 2 clear; 3 function yp=f(x,y) 4 yp =10* exp(-(x-2) ^2/(2*(0.075^2))) -0.6*y 5 endfunction 161
  • 163. 6 x=0:0.1:4 7 y0 =0.5; 8 sol=ode(y0 ,0,x,f); 9 plot(x,sol) 10 xtitle(”y vs x”,”x”,”y”) 162
  • 164. Chapter 26 Stiffness and multistep methods Scilab code Exa 26.1 Explicit and Implicit Euler 1 clc; 2 clear; 3 function yp=f(t,y) 4 yp= -1000*y+3000 -2000* exp(-t) 5 endfunction 6 y0=0; 7 // e x p l i c i t e u l e r 8 h1 =0.0005; 9 y1(1)=y0; 10 count =2; 11 t=0:0.0001:0.006 12 for i=0:0.0001:0.0059 13 y1(count)=y1(count -1)+f(i,y1(count -1))*h1 14 count=count +1; 15 end 16 plot(t,y1) 17 h2 =0.0015; 18 y2(1)=y0; 19 count =2; 20 t=0:0.0001:0.006 21 for i=0:0.0001:0.0059 163
  • 165. 22 y2(count)=y2(count -1)+f(i,y2(count -1))*h2 23 count=count +1; 24 end 25 plot(t,y2 , ’ .− ’ ) 26 xtitle(”y vs t ”,” t ”,”y”) 27 h=legend ([”h −0.0005 ”,”h=0.0015 ”]) 28 clf(); 29 // i m p l i c i t order 30 h3 =0.05; 31 y3(1)=y0; 32 count =2; 33 t=0:0.01:0.4 34 for j=0:0.01:0.39 35 y3(count)=(y3(count -1) +3000*h3 -2000* h3*exp(-(j +0.01)))/(1+1000* h3) 36 count=count +1; 37 end 38 39 plot(t,y3) 40 xtitle(”y vs t ”,” t ”,”y”) Scilab code Exa 26.2 Non self starting Heun method 1 // c l c ( ) 2 disp(” f ( x , y ) = 4∗ exp ( 0 . 8 ∗ x ) − 0.5∗ y”) 3 // f ’ ( x , y ) = 4∗ exp ( 0 . 8 ∗ x ) − 0.5∗ y 4 h = 1; 5 x=0:h:4; 6 y(1) = 2; 7 x1 = -1; 8 y1 = -0.3929953; 164
  • 166. Figure 26.1: Explicit and Implicit Euler 165
  • 167. Figure 26.2: Explicit and Implicit Euler 166
  • 168. 9 y10 = y1 + (4* exp (0.8*x(1)) - 0.5*y(1))*2; 10 y11 = y(1) + (4* exp (0.8*x(1)) - 0.5*y(1) + 4*exp (0.8*x(2)) - 0.5* y10)*h/2; 11 y12 = y(1) + (3 + 4*exp (0.8*x(2)) - 0.5* y11)*h/2; 12 t = 6.360865; 13 y20 = y(1) + (4* exp (0.8*x(2)) - 0.5*t) *2; 14 y21 = t + (4* exp (0.8*x(2)) - 0.5*t + 4*exp (0.8*x(3)) - 0.5* y20)*h/2; 15 disp(y21 ,” the f i r s t c o r r e c t o r y i e l d s y = ”) 16 t = 14.84392 17 e = (t - y21)*100/t; 18 disp(”%”,e,” e r r o r = ”) Scilab code Exa 26.3 Estimate of per step truncation error 1 // c l c ( ) 2 x1 = 1; 3 x2 = 2; 4 y1 = 6.194631; 5 y2 = 14.84392; 6 y10 = 5.607005; 7 y11 = 6.360865; 8 y20 = 13.44346; 9 y21 = 15.30224; 10 Ec1 = -(y11 - y10)/5; 11 disp(Ec1 ,”Ec ( x = 1) = ”) 12 e1 = y1 - y11; 13 disp(e1 ,” true e r r o r ( x = 1) = ”) 14 Ec2 = -(y21 - y20)/5; 15 disp(Ec2 ,”Ec ( x = 2) = ”) 16 e2 = y2 - y21; 17 disp(e2 ,” true e r r o r ( x = 2) = ”) 167
  • 169. Scilab code Exa 26.4 Effect of modifier on Predictor Corrector results 1 // c l c ( ) 2 // c l c ( ) 3 x0 = 0; 4 x1 = 1; 5 x2 = 2; 6 y1 = 6.194631; 7 y2 = 14.84392; 8 y10 = 5.607005; 9 y11 = 6.360865; 10 y1m = y11 - (y11 - y10)/5; 11 e = (y1 - y1m)*100/ y1; 12 disp(y1m ,”ym”) 13 disp(”%”,e,” e r r o r = ”) 14 y20 =2+(4* exp (0.8* x1) - 0.5* y1m)*2; 15 e2 = (y2 - y20)*100/ y2; 16 disp(y20 ,” y20 = ”) 17 disp(”%”,e2 ,” e r r o r = ”) 18 y2o = y20 + 4* (y11 - y10)/5; 19 e2 = (y2 - y2o)*100/ y2; 20 disp(y2o ,” y20 = ”) 21 disp(”%”,e2 ,” e r r o r = ”) 22 y21 = 15.21178; 23 y23 = y21 - (y21 - y20)/5; 24 disp(y23 ,”y2 = ”) 25 e3 = (y2 - y23)*100/ y2; 26 disp(”%”,e3 ,” e r r o r = ”) Scilab code Exa 26.5 Milnes Method 1 // c l c ( ) 2 disp(” f ( x , y ) = 4∗ exp ( 0 . 8 ∗ x ) − 0.5∗ y”) 3 // f ’ ( x , y ) = 4∗ exp ( 0 . 8 ∗ x ) − 0.5∗ y 4 h = 1; 168
  • 170. 5 x = -3:h:4; 6 y(1) = -4.547302; 7 y(2) = -2.306160; 8 y(3) = -0.3929953; 9 y(4) = 2; 10 y1(4) = 2; 11 for i = 4:7; 12 y(i+1) = y(i-3) + 4*h*(2*(4* exp (0.8*x(i)) - 0.5* y(i)) - 4*exp (0.8*x(i-1)) + 0.5*y(i-1) + 2*(4* exp (0.8*x(i-2)) - 0.5*y(i-2)))/3; 13 y1(i+1) = y(i-1) + h*(4* exp (0.8*x(i-1)) - 0.5*y( i-1) +4 * (4* exp (0.8*x(i)) - 0.5*y(i)) + 4* exp (0.8*x(i+1)) - 0.5*y(i+1))/3; 14 end 15 disp(x(4:8) ,”x = ”) 16 disp(y(4:8) ,”y0 = ”) 17 disp(y1 (4:8) ,” c o r r e c t e d y1 = ”) Scilab code Exa 26.6 Fourth order Adams method 1 // c l c ( ) 2 disp(” f ( x , y ) = 4∗ exp ( 0 . 8 ∗ x ) − 0.5∗ y”) 3 // f ’ ( x , y ) = 4∗ exp ( 0 . 8 ∗ x ) − 0.5∗ y 4 h = 1; 5 x = -3:h:4; 6 y(1) = -4.547302; 7 y(2) = -2.306160; 8 y(3) = -0.3929953; 9 y(4) = 2; 10 m(4) = y(4); 11 for i =4:7 12 y(i+1) = y(i) + h *(55* (4* exp (0.8*x(i)) - 0.5*y (i)) / 24 - 59 * (4* exp (0.8*x(i-1)) - 0.5*y(i -1)) / 24 + 37*(4* exp (0.8*x(i-2)) - 0.5*y(i -2))/24 - 9*(4* exp (0.8*x(i-3)) - 0.5*y(i-3)) 169
  • 171. /24); 13 m(i+1) = y(i+1) 14 y(i+1) = y(i) + h*(9*(4* exp (0.8*x(i+1)) - 0.5*y( i+1))/24 +19*(4* exp (0.8*x(i)) - 0.5*y(i))/24 - 5*(4* exp (0.8*x(i-1)) - 0.5*y(i-1))/24 + (4* exp (0.8*x(i-2)) - 0.5*y(i-2))/24); 15 end 16 disp(x(4:8) ,”x = ”) 17 disp(m(4:8) ,”y0 = ”) 18 disp(y(4:8) ,”y1 = ”) Scilab code Exa 26.7 stability of Milnes and Fourth order Adams method 1 // c l c ( ) 2 //dy/dx = −y 3 //y = exp(−x ) 4 h = 0.5; 5 x = -1.5:h:10; 6 y(1) = exp(-x(1)); 7 y(2) = exp(-x(2)); 8 y(3) = exp(-x(3)); 9 y(4) = 1; 10 m(4) = y(4); 11 for i = 4:23; 12 y(i+1) = y(i-3) + 4*h*(2*( -y(i)) + y(i-1) + 2*(- y(i-2)))/3; 13 m(i+1) = y(i+1); 14 y(i+1) = y(i-1) + h*(-y(i-1) +4 * (-y(i)) + (-y( i+1)))/3; 15 end 16 disp(x(4:24) ,”x = ”) 17 disp(m(4:24) ,”y0 ( milnes method ) = ”) 18 disp(y(4:24) ,” c o r r e c t e d y1 ( milnes method ) = ”) 19 for i =4:23 20 y(i+1) = y(i) + h *(55* (-y(i)) / 24 - 59 * (-y( 170
  • 172. i-1)) / 24 + 37*(-y(i-2))/24 - 9*(-y(i-3)) /24); 21 m(i+1) = y(i+1) 22 y(i+1) = y(i) + h*(9*( -y(i+1))/24 +19*( -y(i))/24 - 5*(-y(i-1))/24 + (-y(i-2))/24); 23 end 24 disp(m(4:23) ,”y0 ( f o u r t h order adams method ) = ”) 25 disp(y(4:23) ,”y1 ( f o u r t h order adams method ) = ”) 171
  • 173. Chapter 27 Boundary Value and Eigen Value problems Scilab code Exa 27.1 The shooting method 1 // c l c ( ) 2 l = 10; //m 3 h = 0.01; //mˆ( −2) 4 Ta = 20; 5 T0 = 40; 6 T10 = 200; 7 //dT/dx = z 8 // dz/dx = h ’ ( T−Ta) 9 // we use z = 10 as i n i t i a l guess and i n t e g r a t i n g above equations simultaneously , s o l v i n g using RK4 method , we get T10 = 168.3797 10 // s i m i l a r l y , with z = 20 , T10 = 285.898 11 z01 = 10; 12 z02 = 20; 13 T10a = 168.3797; 14 T10b = 285.898; 15 z0 = z01 + (z02 - z01)*(T10 - T10a)/( T10b - T10a); 16 disp(z0 ,” z0 = ”) 17 disp(” t h i s value of z can be used to get the c o r r e c t 172
  • 174. Figure 27.1: The shooting method for non linear problems value ”) Scilab code Exa 27.2 The shooting method for non linear problems 1 // c l c ( ) 2 z(1) = 10.0035; 3 T(1) = 40; 4 Ta = 20; 5 h = 0.5; 6 for i = 1:20 7 k11(i) = z(i); 173
  • 175. 8 k12(i) = 5*10^ -8*(T(i) - Ta)^4; 9 z1 = z(i) + h/2; 10 T1 = T(i) + h/2; 11 k21(i) = z1; 12 k22(i) = 5*10^ -8*( T1 - Ta)^4; 13 z1 = z(i) + h/2; 14 T1 = T(i) + h/2; 15 k31(i) = z1; 16 k32(i) = 5*10^ -8*( T1 - Ta)^4; 17 z1 = z(i) + h; 18 T1 = T(i) + h; 19 k41(i) = z1; 20 k42(i) = 5*10^ -8*( T1 - Ta)^4; 21 T(i+1) = T(i) + ( k11(i) + 2* k21(i) + 2*k31(i) + k41(i))*h/6; 22 z(i+1) = z(i) + ( k12(i) + 2* k22(i) + 2*k32(i) + k42(i))*h/6; 23 end 24 x=0:0.5:10; 25 plot(x,T(1:21)) 26 xtitle(”T vs x”,”x”,”T”) Scilab code Exa 27.3 Finite Difference Approximation 1 clc; 2 clear; 3 h=0.01; 4 delx =2; 5 x=2+h*delx ^2; 6 a=[x -1 0 0;-1 x -1 0;0 -1 x -1;0 0 -1 x]; 7 b=[40.8; 0.8; 0.8; 200.8]; 8 T=linsolve(a,b); 9 disp(”The temperature at the i n t e r i o r nodes : ”) 10 disp(abs(T)) 174
  • 176. Scilab code Exa 27.4 Mass Spring System 1 clc; 2 clear; 3 m1=40; // kg 4 m2=40; // kg 5 k=200; //N/m 6 sqw=poly(0,” s ”); 7 p=sqw ^2 -20* sqw +75; 8 r=roots(p); 9 f1=(r(1))^0.5; 10 f2=(r(2))^0.5; 11 Tp1 =(2* %pi)/f1; 12 Tp2 =(2* %pi)/f2; 13 // f o r f i r s t mode 14 disp(”For f i r s t mode : ”) 15 disp(Tp1 ,” Period of o s c i l l a t i o n : ”) 16 disp(”A1=−A2”) 17 disp(” −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 18 // f o r f i r s t mode 19 disp(”For second mode : ”) 20 disp(Tp2 ,” Period of o s c i l l a t i o n : ”) 21 disp(”A1=A2”) Scilab code Exa 27.5 Axially Loaded column 1 clc; 2 clear; 3 E=10*10^9; //Pa 4 I=1.25*10^ -5; //mˆ4 175
  • 177. 5 L=3; //m 6 for i=1:8 7 p=i*%pi/L; 8 P=i^2*( %pi)^2*E*I/(L^2*1000); 9 disp(i,”n=”) 10 disp(”mˆ−2”,p,”p=”) 11 disp(”kN”,P,”P=”) 12 disp(” −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 13 end Scilab code Exa 27.6 Polynomial Method 1 clc; 2 clear; 3 E=10*10^9; //Pa 4 I=1.25*10^ -5; //mˆ4 5 L=3; //m 6 true =[1.0472 2.0944 3.1416 4.1888]; 7 // part a 8 h1 =3/2; 9 p=poly(0,” s ”) 10 a=2-h1^2*p^2; 11 x=roots(a); 12 e=abs(abs(x(1))-true (1))*100/ true (1); 13 disp(x,”p=”) 14 disp(e,” e r r o r=”) 15 disp(” −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−” ) 16 // part b 17 h2 =3/3; 18 p=poly(0,” s ”) 19 a=(2-h2^2*p^2)^2 - 1; 176
  • 178. 20 x=roots(a); 21 e(1)=abs(abs(x(1))-true (2))*100/ true (2); 22 e(2)=abs(abs(x(2))-true (1))*100/ true (1); 23 disp(x,”p=”) 24 disp(e,” e r r o r=”) 25 disp(” −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 26 // part c 27 h3 =3/4; 28 p=poly(0,” s ”) 29 a=(2-h3^2*p^2)^3 - 2*(2-h3^2*p^2); 30 x=roots(a); 31 e(1)=abs(abs(x(1))-true (3))*100/ true (3); 32 e(2)=abs(abs(x(2))-true (2))*100/ true (2); 33 e(3)=abs(abs(x(3))-true (1))*100/ true (1); 34 disp(x,”p=”) 35 disp(e,” e r r o r=”) 36 disp(” −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 37 38 39 // part d 40 h4 =3/5; 41 p=poly(0,” s ”) 42 a=(2-h4^2*p^2)^4 - 3*(2-h4^2*p^2)^2 + 1; 43 x=roots(a); 44 e(1)=abs(abs(x(1))-true (4))*100/ true (4); 45 e(2)=abs(abs(x(2))-true (3))*100/ true (3); 46 e(3)=abs(abs(x(3))-true (2))*100/ true (2); 47 e(4)=abs(abs(x(4))-true (1))*100/ true (1); 48 disp(x,”p=”) 49 disp(e,” e r r o r=”) 50 disp(” −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ”) 177
  • 179. Scilab code Exa 27.7 Power Method Highest Eigenvalue 1 clc; 2 clear; 3 a=[3.556 -1.668 0; -1.778 3.556 -1.778;0 -1.778 3.556]; 4 b=[1.778;0;1.778]; 5 ea =100; 6 count =1; 7 while ea >0.1 8 maxim=b(1); 9 for i=2:3 10 if abs(b(i))>abs(maxim) then 11 maxim=b(i); 12 end 13 end 14 eigen(count)=maxim; 15 b=a*(b./ maxim); 16 if count ==1 then 17 ea =20; 18 count=count +1; 19 20 else 21 ea=abs(eigen(count)-eigen(count -1))*100/ abs( eigen(count)); 22 count=count +1; 23 end 24 end 25 disp(eigen(count -1),”The l a r g e s t e i g e n value ”) Scilab code Exa 27.8 Power Method Lowest Eigenvalue 178
  • 180. 1 clc; 2 clear; 3 a=[3.556 -1.668 0; -1.778 3.556 -1.778;0 -1.778 3.556]; 4 b=[1.778;0;1.778]; 5 ea =100; 6 count =1; 7 ai=inv(a); 8 while ea >4 9 maxim=b(1); 10 for i=2:3 11 if abs(b(i))>abs(maxim) then 12 maxim=b(i); 13 end 14 end 15 eigen(count)=maxim; 16 b=ai*(b./ maxim); 17 if count ==1 then 18 ea=20; 19 count=count +1; 20 21 else 22 ea=abs(eigen(count)-eigen(count -1))*100/ abs( eigen(count)); 23 count=count +1; 24 end 25 end 26 disp ((1/ eigen(count -1))^0.5,”The s m a l l e s t e i g e n value ”) Scilab code Exa 27.9 Eigenvalues and ODEs 1 clc; 2 clear; 3 function yp=predprey(t,y) 179
  • 181. 4 yp =[1.2*y(1) -0.6*y(1)*y(2) ; -0.8*y(2) +0.3*y(1)*y (2)]; 5 endfunction 6 t=0:0.1:20; 7 y0=[2 1]; 8 sol=ode(y0 ,0,t,predprey); 9 count =1; 10 for i=1:2:401 11 x(count)=sol(i); 12 z(count)=sol(i+1); 13 count=count +1; 14 end 15 16 plot(t,x) 17 plot(t,z) 18 xtitle(”y vs t ”, ” t ”,”y”) 19 clf(); 20 plot(x,z) 21 xtitle(” space−space p l o t ( y1 vs y2 ) ”, ”y1”,”y2”) Scilab code Exa 27.10.a Stiff ODEs 1 clc; 2 clear; 3 function yp=vanderpol(t,y) 4 yp=[y(2) ;1*(1 -y(1) ^2)*y(2)-y(1)] 5 endfunction 6 y0=[1 1]; 7 t=0:0.1:20; 180
  • 182. Figure 27.2: Eigenvalues and ODEs 181
  • 183. Figure 27.3: Eigenvalues and ODEs 182
  • 184. Figure 27.4: Stiff ODEs 183
  • 185. Figure 27.5: Stiff ODEs 8 sol=ode(y0 ,0,t,vanderpol); 9 count =1; 10 for i=1:2:401 11 x(count)=sol(i); 12 count=count +1; 13 end 14 plot(t,x) 15 xtitle(”y vs t ”,” t ”,”y”) Scilab code Exa 27.10.b Stiff ODEs 184
  • 186. 1 clc; 2 clear; 3 clf(); 4 function yp=vanderpol1(t,y) 5 yp=[y(2) ;1000*(1 -y(1) ^2)*y(2)-y(1)] 6 endfunction 7 t=0:3000; 8 yo=[1 1]; 9 sol=ode(yo ,0,t,vanderpol1); 10 count =1; 11 12 for i=1:2:6001 13 z(count)=sol(i)*-1; 14 count=count +1; 15 end 16 17 plot(t,z) 18 xtitle(”y vs t ”,” t ”,”y”) Scilab code Exa 27.11 Solving ODEs 1 clc; 2 clear; 3 function yp=predprey(t,y) 4 yp =[1.2*y(1) -0.6*y(1)*y(2) ; -0.8*y(2) +0.3*y(1)*y (2)]; 5 endfunction 6 t=0:10; 7 y0=[2 1]; 8 sol=ode(y0 ,0,t,predprey); 9 count =0; 10 for i=1:2:22 11 disp(count ,” i s t e p=”) 12 disp(count ,” time=”) 13 disp(sol(i),”y1=”) 185
  • 188. Chapter 29 Finite Difference Elliptic Equations Scilab code Exa 29.1 Temperature of a heated plate with fixed boundary conditions 1 // c l c ( ) 2 wf = 1.5; 3 for i = 1:5 4 for j = 1:5 5 T(i,j) = 0; 6 if j == 1 then 7 T(i,j) = 0; //C 8 else 9 if j == 5 then 10 T(i,j) = 100; //C 11 end 12 end 13 if i == 1 then 14 T(i,j) = 75; //C 15 else 16 if i == 5 then 17 T(i,j) = 50; //C 18 end 19 end 187
  • 189. 20 end 21 end 22 e = 100; 23 while e>1 24 for i=1:5 25 for j = 1:5 26 if i>1 & j>1 & i<5 & j<5 then 27 Tn(i,j) = (T(i + 1,j) + T(i-1,j) + T(i,j+1) + T(i,j-1))/4; 28 Tn(i,j) = wf * Tn(i,j) + (1-wf)*T(i,j); 29 if i==2 & j==2 then 30 e = abs((Tn(i,j) - T(i,j)) * 100/ (Tn(i, j))); 31 end 32 T(i,j) = Tn(i,j); 33 end 34 end 35 end 36 end 37 disp(T(2:4 ,2:4) ,” f o r e r r o r < 1 , the temperatures are ”) Scilab code Exa 29.2 Flux distribution for a heated plate 1 // c l c ( ) 2 k = 0.49; // c a l /( s . cm .C) 3 T1 = 33.29755; //C 4 T2 = 75; //C 5 T3 = 63.21152; //C 6 T4 = 0; //C 7 qx = -k * (T1 - T2) / (2*10); 8 qy = -k * (T3 - T4) / (2*10); 9 disp(qx ,”qx = ”) 10 disp(qy ,”qy = ”) 11 q = (qx^2 + qy^2) ^0.5; 188
  • 190. 12 t = (atan(qy/qx)); 13 r = t * 180/( %pi); 14 disp(r,”Thus , the f l u x i s d i r e c e d at the angle ”) Scilab code Exa 29.3 Heated plate with a insulated edge 1 // c l c ( ) 2 // since , i n s u l a t i o n i s done at one end , the g e n e r a l equation becomes ( j =0) , 3 //T( i +1 ,0) + T( i −1 ,0) + 2∗T( i , 1 ) −2∗y ∗(dT/dy ) − 4∗T( i , 0 ) = 0 4 // the d e r i v a t i v e i s zero and we get , T( i +1 ,0) + T( i −1 ,0) + 2∗T( i , 1 ) − 4∗T( i , 0 ) = 0 5 // the simultaneous equations obtained can be w r i t t e n in matrix form as f o l l o w s , 6 A = [4,-1,0,-2,0,0,0,0,0,0,0,0;-1,4,-1,0,-2,0,0,0,0,0,0,0;0,-1,4,0,0,- 7 disp(A,”A = ”) 8 B = [75;0;50;75;0;50;75;0;50;175;100;150]; 9 disp(B,”B = ”) 10 T = inv(A)*B; 11 T10 = det(T(1,1)); 12 T20 = det(T(2,1)); 13 T30 = det(T(3,1)); 14 T11 = det(T(4,1)); 15 T21 = det(T(5,1)); 16 T31 = det(T(6,1)); 17 T12 = det(T(7,1)); 18 T22 = det(T(8,1)); 19 T32 = det(T(9,1)); 20 T13 = det(T(10 ,1)); 21 T23 = det(T(11 ,1)); 22 T33 = det(T(12 ,1)); 23 disp(T10 ,”T10 = ”) 189
  • 191. 24 disp(T20 ,”T20 = ”) 25 disp(T30 ,”T30 = ”) 26 disp(T11 ,”T11 = ”) 27 disp(T21 ,”T21 = ”) 28 disp(T31 ,”T31 = ”) 29 disp(T12 ,”T12 = ”) 30 disp(T22 ,”T22 = ”) 31 disp(T32 ,”T32 = ”) 32 disp(T13 ,”T13 = ”) 33 disp(T23 ,”T23 = ”) 34 disp(T33 ,”T33 = ”) Scilab code Exa 29.4 Heated plate with an irregular boundary 1 // c l c ( ) 2 // f o r i r r e g u l a r boundary , 3 //x = y 4 alpha1 = 0.732; 5 beta1 = 0.732; 6 alpha2 = 1; 7 beta2 = 1; 8 // s u b s t i t u t i n g the above value to obtain simultaneous equation we get the f o l l o w i n g matrix , 9 A = [4,-0.845,0,-0.845,0,0,0,0,0;-1,4,-1,0,-1,0,0,0,0;0,-1,4,0,0,-1,0, 10 disp(A,”A = ”) 11 B = [173.2;75;125;75;0;50;175;100;150]; 12 disp(B,”B = ”) 13 T = inv(A)*B; 14 T11 = det(T(1,1)); 15 T21 = det(T(2,1)); 16 T31 = det(T(3,1)); 17 T12 = det(T(4,1)); 190
  • 192. 18 T22 = det(T(5,1)); 19 T32 = det(T(6,1)); 20 T13 = det(T(7,1)); 21 T23 = det(T(8,1)); 22 T33 = det(T(9,1)); 23 disp(T11 ,”T11 = ”) 24 disp(T21 ,”T21 = ”) 25 disp(T31 ,”T31 = ”) 26 disp(T12 ,”T12 = ”) 27 disp(T22 ,”T22 = ”) 28 disp(T32 ,”T32 = ”) 29 disp(T13 ,”T13 = ”) 30 disp(T23 ,”T23 = ”) 31 disp(T33 ,”T33 = ”) 191
  • 193. Chapter 30 Finite Difference Parabolic Equations Scilab code Exa 30.1 Explicit solution of a one dimensional heat conduction equati 1 // c l c ( ) 2 l = 10; //cm 3 k1 = 0.49; // c a l /( s . cm .C) 4 x = 2; //cm 5 dt = 0.1; // sec 6 T0 = 100; //C 7 T10 = 50; //C 8 C = 0.2174; // c a l /( g .C) 9 rho = 2.7; //g/cmˆ3 10 k = k1/(C*rho); 11 L = k * dt / x^2; 12 disp(L,”L =”) 13 T(1,1) = 100; 14 T(1,2) = 0; 15 T(1,3) = 0; 16 T(1,4) = 0; 17 T(1,5) = 0; 18 T(1,6) = 50; 19 T(2,1) = 100; 192
  • 194. 20 T(2,6) = 50; 21 for i = 2:3 22 for j = 2:5 23 T(i,j) = T(i-1,j) + L * (T(i-1,j+1) - 2* T(i -1,j) + T(i-1,j-1)); 24 end 25 end 26 disp(T(2,2),”T11 = ”) 27 disp(T(2,3),”T12 = ”) 28 disp(T(2,4),”T13 = ”) 29 disp(T(2,5),”T14 = ”) 30 disp(T(3,2),”T21 = ”) 31 disp(T(3,3),”T22 = ”) 32 disp(T(3,4),”T23 = ”) 33 disp(T(3,5),”T24 = ”) Scilab code Exa 30.2 Simple implicit solution of a heat conduction equation 1 // c l c ( ) 2 l = 10; //cm 3 k1 = 0.49; // c a l /( s . cm .C) 4 x = 2; //cm 5 dt = 0.1; // sec 6 C = 0.2174; // c a l /( g .C) 7 rho = 2.7; //g/cmˆ3 8 k = k1/(C*rho); 9 L = k * dt / x^2; 10 disp(L,”L =”) 11 //now , at t = 0 , 1.04175 ∗T’ 1 + 0.020875 ∗T’ 2 = 0 + 0.020875∗100 12 // s i m i l a r l y g e t t i n g other simultaneous eqautions , we get the f o l l o w i n g matrix 13 A = [1.04175 , -0.020875 ,0 ,0; -0.020875 ,1.04175 , -0.020875 ,0;0 , -0.020875 , 193
  • 195. 14 B = [2.0875;0;0;1.04375]; 15 X = inv(A) * B; 16 T11 = det(X(1,1)); 17 T12 = det(X(2,1)); 18 T13 = det(X(3,1)); 19 T14 = det(X(4,1)); 20 disp(” f o r t = 0”) 21 disp(T11 ,”T11 = ”) 22 disp(T12 ,”T12 = ”) 23 disp(T13 ,”T13 = ”) 24 disp(T14 ,”T14 = ”) 25 // to s o l v e f o r t = 0 . 2 , the r i g h t hand s i d e v e c t o r i s modified to D, 26 D = [4.09215;0.04059;0.02090;2.04069]; 27 Y = inv(A)*D; 28 T21 = det(Y(1,1)); 29 T22 = det(Y(2,1)); 30 T23 = det(Y(3,1)); 31 T24 = det(Y(4,1)); 32 disp(” f o r t = 0 .2 ”) 33 disp(T21 ,”T21 = ”) 34 disp(T22 ,”T22 = ”) 35 disp(T23 ,”T23 = ”) 36 disp(T24 ,”T24 = ”) Scilab code Exa 30.3 Crank Nicolson solution to the heat conduction equation 1 // c l c ( ) 2 // using crank n i c o l s o n method , we get simultaneous equations which can be s i m p l i f i e d to f o l l o w i n g matrics 3 A = [2.04175 , -0.020875 ,0 ,0; -0.020875 ,2.04175 , -0.020875 ,0;0 , -0.020875 , 4 B = [4.175;0;0;2.0875]; 194
  • 196. 5 X = inv(A)*B; 6 disp(” at t = 0. 1 s ”) 7 disp(det(X(1,1)),”T11 = ”) 8 disp(det(X(2,1)),”T12 = ”) 9 disp(det(X(3,1)),”T13 = ”) 10 disp(det(X(4,1)),”T14 = ”) 11 C = [8.1801;0.0841;0.0427;4.0901]; 12 Y = inv(A)*C; 13 disp(” at t = 0. 2 s ”) 14 disp(det(Y(1,1)),”T21 = ”) 15 disp(det(Y(2,1)),”T22 = ”) 16 disp(det(Y(3,1)),”T23 = ”) 17 disp(det(Y(4,1)),”T24 = ”) Scilab code Exa 30.4 Comparison of Analytical and Numerical solution 1 // c l c ( ) 2 x = 2; //cm 3 L = 10; //cm 4 k = 0.835; //cmˆ2/ s 5 t = 10; // s 6 n = 1:10000; 7 T = 324.* (x/L + sum (2.*(( -1)^n).*sin(n.*x/L).*exp(- n^2.* %pi ^2.* k.* t / L^2)/(n.*%pi))); 8 disp(T,”T[ 2 , 1 0 ] a n a l y t i c a l = ”) 9 Tex = 64.97; 10 disp(Tex ,”T[ 2 , 1 0 ] e x p l i c i t = ”) 11 Tim = 64.49; 12 disp(Tim ,”T[ 2 , 1 0 ] i m p l i c i t = ”) 13 Tcn = 64.73; 14 disp(Tcn ,”T[ 2 , 1 0 ] crank−n i c o l s o n = ”) Scilab code Exa 30.5 ADI Method 195
  • 197. 1 // c l c ( ) 2 x = 10; //cm 3 L = 0.0835; 4 t1 = 5; 5 // f o r f i r s t step t = 5 i s a p p l i e d to nodes ( 1 , 1 ) , ( 1 , 2 ) and ( 1 , 3 ) to y i e l d f o l l o w i n g matrices 6 A = [2.167 , -0.0835 ,0; -0.0835 ,2.167 , -0.0835;0 , -0.0835 ,2.167]; 7 B = [6.2625;6.2625;14.6125]; 8 X = inv(A)*B; 9 disp(”At t = 5 s ”) 10 disp(det(X(1,1)),”T11 = ”) 11 disp(det(X(2,1)),”T12 = ”) 12 disp(det(X(3,1)),”T13 = ”) 13 // s i m i l a r l y we get , 14 T21 = 0.1274; 15 T22 = 0.2900; 16 T23 = 4.1291; 17 T31 = 2.0181; 18 T32 = 2.2477; 19 T33 = 6.0256; 20 disp(T21 ,”T21 = ”) 21 disp(T22 ,”T22 = ”) 22 disp(T23 ,”T23 = ”) 23 disp(T31 ,”T31 = ”) 24 disp(T32 ,”T32 = ”) 25 disp(T33 ,”T33 = ”) 26 C = [13.0639;0.2577;8.0619]; 27 Y = inv(A)*C; 28 disp(”At t = 10 s ”) 29 disp(det(Y(1,1)),”T11 = ”) 30 disp(det(Y(2,1)),”T12 = ”) 31 disp(det(Y(3,1)),”T13 = ”) 32 // s i m i l a r l y we get , 33 T21 = 6.1683; 34 T22 = 0.8238; 35 T23 = 4.2359; 196
  • 198. 36 T31 = 13.1120; 37 T32 = 8.3207; 38 T33 = 11.3606; 39 disp(T21 ,”T21 = ”) 40 disp(T22 ,”T22 = ”) 41 disp(T23 ,”T23 = ”) 42 disp(T31 ,”T31 = ”) 43 disp(T32 ,”T32 = ”) 44 disp(T33 ,”T33 = ”) 197
  • 199. Chapter 31 Finite Element Method Scilab code Exa 31.1 Analytical Solution for Heated Rod 1 clc; 2 clear; 3 //d2T/dx2=−10; equation to be s o l v e d 4 //T(0 , t ) =40; boundary c o n d i t i o n 5 //T(10 , t ) =200; boundary c o n d i t i o n 6 // f ( x ) =10; uniform heat source 7 //we assume a s o l u t i o n T=a∗Xˆ2 + b∗x +c 8 // d i f f e r e n t i a t i n g twice we get d2T/dx2=2∗a 9 a= -10/2; 10 // using f i r s t boundary c o n d i t i o n 11 c=40; 12 // using second boundary condtion 13 b=66; 14 // hence f i n a l s o l u t i o n T=−5∗xˆ2 + 66∗x + 40 15 function T=f(x) 16 T=-5*x^2 + 66*x + 40 17 endfunction 18 count =1; 19 for i=0:0.1:11 198
  • 200. Figure 31.1: Analytical Solution for Heated Rod 199
  • 201. 20 T(count)=f(i); 21 count=count +1; 22 end 23 x=0:0.1:11 24 plot(x,T) 25 xtitle(” Temperature (T) vs d i s t a n c e ( x ) ”,”x (cm) ”,”T ( u n i t s ) ”) Scilab code Exa 31.2 Element Equation for Heated Rod 1 clc; 2 clear; 3 xf=10; //cm 4 xe =2.5; //cm 5 //T(0 , t ) =40; boundary c o n d i t i o n 6 //T(10 , t ) =200; boundary c o n d i t i o n 7 // f ( x ) =10; uniform heat source 8 function y=f(x) 9 y=10*(xe -x)/xe; 10 endfunction 11 int1=intg(0,xe ,f) 12 function y=g(x) 13 y=10*(x-0)/xe; 14 endfunction 15 int2=intg(0,xe ,g) 16 disp(”The r e s u l t s are : ”) 17 disp(” 0.4∗ T1−0.4∗T2=−(dT/dx ) ∗x1 + c1 ”) 18 disp(int1 ,” where c1=”) 19 disp(”and”) 20 disp(” −0.4∗T1+0.4∗T2=−(dT/dx ) ∗x2 + c2 ”) 21 disp(int2 ,” where c2=”) Scilab code Exa 31.3 Temperature of a heated plate 200
  • 202. 1 // c l c ( ) 2 wf = 01.5; 3 for i = 1:41 4 for j = 1:41 5 T(i,j) = 0; 6 if j == 1 then 7 T(i,j) = 0; //C 8 else 9 if j == 41 then 10 T(i,j) = 100; //C 11 end 12 end 13 if i == 1 then 14 T(i,j) = 75; //C 15 else 16 if i == 41 then 17 T(i,j) = 50; //C 18 end 19 end 20 end 21 end 22 e = 100; 23 while e>1 24 for i=1:41 25 for j = 1:41 26 if i>1 & j>1 & i<41 & j<41 then 27 Tn(i,j) = (T(i + 1,j) + T(i-1,j) + T(i,j+1) + T(i,j-1))/4; 28 Tn(i,j) = wf * Tn(i,j) + (1-wf)*T(i,j); 29 if i==2 & j==2 then 30 e = abs((Tn(i,j) - T(i,j)) * 100/ (Tn(i, j))); 31 end 32 T(i,j) = Tn(i,j); 33 end 34 end 35 end 36 end 201
  • 203. 37 disp(T,” f o r e r r o r < 1 , the temperatures are ”) 202