Numerical Analysis and Computer Applications

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Mujeeb UR Rahman
17CH106
Chemical Engineering Department
Assigned by: Sher Khan Awan
BSRS Mehran UET
NUMERICAL ANALYSY AND COMPUTER APPLICATIONS
Task: Compare the iterations of
Numerical Methods.
Mehran University of Engineering & Technology
Jamshoro, Pk
Date: 13Nov2020

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NUMERICAL ANALYSY AND COMPUTER APPLICATIONS. 2020
FOR THE SYSTEM OF NON-LINEAR EQUATIONS.
1. BISECTIONMETHOD:
This is the simplest and robust method for finding a root of an equation that applies to any
continuous functions for which one knows two values with opposite signs. The method is also
called the interval halving method, the binary search method, or the dichotomy method. It is
type of incremental search method in which the interval [a,b] is always divided into two equal
sub-intervals. It is very simplest but converges very slow.
BisectionMethodFormula:
𝑥 = (𝑎 + 𝑏)/2
2. REGULA FALSI METHOD:
Regula falsi method is also known as false position method. The false position method is an
alternative method to find the root of non-linear equation and more efficient than the bisection
method. It is the oldest method for finding the real rootof an equation f(x) = 0 and is somewhat
similar to the bisection method.
Regula FalsiMethod Formula:
𝑥 =
𝑎𝑓( 𝑏) − 𝑏𝑓(𝑎)
𝑓( 𝑏) − 𝑓(𝑎)
3. NEWTON-RAPSHON METHOD:
Instead of using two points on the graph of y = 𝑓(𝑥) as in bisection and regula falsi method,
Newton’s method uses only one instead of two initial guesses. But if the function 𝑓′(𝑥0) = 0
then the Newton’s method is divergence.
Regula FalsiMethod Formula:
𝑥 = 𝑥0
𝑓( 𝑥0)
𝑓′(𝑥0)

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4. FIXED POINT ITERATION :
Fixed point iteration method is open and simple method for finding real root of non-linear
equation by successive approximation. It requires only one initial guess to start. Since it is
open method its convergence is not guaranteed. This method is also known as Iterative.
Method Formula:
To find the root of nonlinear equation f(x)=0 by fixed point iteration method, we write given
equation f(x)=0 in the form of x = g(x).
Q.1)Solve the x2
+5x-4 by upto 10% accuracyby using:
1) BisectionMethod
2) Regula FalsiMethod
3) NewtonRapshonMethod
4) Fixed Point iteration.
S.N
o
Bisection Method Regula Falsi
Method
Newton Rapshon
Method
Fixed point iteration
- it
r
Root Error It
r
Root Error It
r
Root Error Itr Root Error
1 8 1.5273
4
0.25575
4
2 1.5236
2
0.26183
5
4 1.5228
1
0.49809
5
9 1.5245
4
0.28384
9
2 - - - - - - - - - - - -
Conclusion:
The above table show the result for approximate solutions of the function
𝑓( 𝑥) = 𝒙 𝟐
+ 𝟓𝐱 − 𝟒
in the interval [0 ,1] with error inaccuracy of 10% from above table its is cleared that
newton Rapshonmethod has fast conversion to the roots with better accuracy,
followed by usually fixed-point iteration method and sometimes Regula falsi method
while leaving behind.

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FOR THE SYSTEM OF LINEAR EQUATIONS:
5. JACOBI METHOD :
In numerical linear algebra, the Jacobimethod is an iterative algorithm for determining
the solutions ofa strictly diagonally dominant system oflinear equations. Each diagonal
element is solved for, and an approximate value is plugged in. The process is then
iterated until it converges. This algorithm is a stripped-down version of the Jacobi
transformation method of matrix diagonalization. The method is named after Carl
Gustav Jacob Jacobi.
Method Formula:
6. GAUSS–SEIDEL METHOD :
In numerical linear algebra, the Gauss–Seidel method, also known as the Liebman
method or the method of successive displacement, is an iterative method used to solve
a system of linear equations. Gauss–Seidel method is an improved form of Jacobi
method.
Method Formula:

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Q.2)Solve the by upto 10% accuracyby using:
1) JacobiMethod
2) Gauss SeidelMethod
42x1 + 6x2 – 8x3 = 316
-6x1 + 56x2 +17x3 = 645
8x1 -17x2 -25x3 = 767
Jacobi’s Method Gauss Seidel’s Method
No. of
iter
Root
No. of
iter
Root
14
-5.8024 26.1759 -50.334
9
-5.80319 26.1765 -50.337
X Y Z X Y Z
P.Error 0.03 0.009 0.003 P.Error 0.3 0.006 0.003

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NUMERICAL DIFFERENTIATION AND INTIGRATION:
7. Trapezoidal Rule:
The trapezoidal rule (also known as the trapezoid rule or trapezium rule) is a
technique for approximating the definite integral. . The trapezoidal rule works by
approximating the region under the graph of the function as a trapezoid and
calculating its area.
Method Formula:
The trapezoidal rule works by approximating the region under the graph of the
function f(x) as a trapezoid and calculating its area. It follows that
8. Simpson’s 1/3 Rule:
Simpson’s Rule is a numerical method that approximates the value of a definite
integral by using quadratic functions. Simpson’s Rule is based on the fact that given
three points, we can find the equation of a quadratic through those points.
To obtain an approximation of the definite integral b∫af(x)dx using Simpson’s Rule, we
partition the interval [a,b] into an even number n of subintervals, each of width
Method Formula:

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9. Simpson’s 3/8 Rule:
Simpson's 3/8 rule is another method for numerical integration proposed by Thomas
Simpson. It is based upon a cubic interpolation rather than a quadratic interpolation.
Simpson's 3/8 rule is as follows:
Dividing the interval [a,b] into n subintervals of length h = (b-a)/n and introducing the
nodes xi = a + ih we have:
Method Formula:
Q.2)Evaluate ∫ 𝒔𝒊𝒏 𝒙 ∗ (𝟏 + 𝒄𝒐𝒔𝒙) 𝟐
𝒅𝒙
𝟐
𝟏
using:
1. TrapezoidalRule
2. Simpson’s 1/3 Rule
3. Simpson’s 3/8 Rule
Result and Compersionof N.I Rules:
Method Trapezoidal Rule
Simpson’s 1/3
Rule
Simpson’s 3/8
Rule
App. Sol. 1.15168 1.16901 1.15179
ExactSol 1.151796255 1.151796255 1.151796255
% Error 0.01 1.5 0.0005

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ORDINARY DIFFERENTIAL LINEAR EQUATIONS:
10. Euler Method:
Euler’s Method, is just another technique used to analyze a Differential Equation, which
uses the idea oflocal linearity or linear approximation, where we use small tangent lines
over a short distance to approximate the solution to an initial-value problem.
Method Formula:
𝒚 𝒏+ = 𝒚 𝒏 + 𝒉 𝒇(𝑥 𝑛, 𝑦𝑛)
11. Modified Euler Method:
In this method the average of the slops at (𝑥 𝑜, 𝑦𝑜) and (𝑥1, 𝑦1 = 1(1)
) is taken instead
of the slope at (𝑥 𝑜, 𝑦𝑜) where 𝑦1
(1)
= 𝑦1 + ℎ 𝑓(𝑥 𝑜, 𝑦𝑜).
Method Formula:
𝒚 𝒏+𝟏 = 𝒚 𝒏 + 𝟏/𝟐𝒉[ 𝒇( 𝒙 𝒏, 𝒚 𝒏) 𝒇(𝒙 𝒏 + 𝒉, 𝒚 𝒏 + 𝒉 𝒇(𝒙 𝒏, 𝒚 𝒏))
12. Improved Euler Method:
In this method points are averaged instead of slops.
Method Formula:
𝒚 𝒏+𝟏 = 𝒚 𝒏 + 𝒉𝒇 ( 𝒙 𝒏 +
𝒉
𝟐
, 𝒚 𝒏 +
𝒉
𝟐
𝒇(𝒙 𝒏, 𝒚 𝒏))
13. Runge-Kutta 4th
Order:
The Runge-Kutta method finds approximate value of y for a given x. Only first order
ordinary differential equations can be solved by using the Runge Kutta 4th order
method.
Runge-Kutta gives greater accuracy without the need to calculate higher derivates.
Method Formula:
𝒚 𝒏+𝟏 = 𝒚 𝒏 + ( 𝒌 𝟏 + 𝟐𝒌 𝟐+ 𝟐𝒌 𝟑 + 𝒌 𝟒 )

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where,
Q.3)Find an approximate value of 𝒚 𝒂𝒕 𝒙 = 𝟎. 𝟓using: given that
𝒅𝒚
𝒅𝒙
= 𝟐 −
𝒆−𝟒𝒙
− 𝟐𝒚 and y(0)=1.
1. Euler’s Method
2. M. Euler’s Method
3. R.K 4th order
Method
Euler’s
Method
M. Euler’s
Method
Imp.
Euler’s
Method
R.K 4th
order
App. Sol. 0.851677 0.8597930 0.8481210 0.859477
Exact Sol
𝟐𝒆 𝟐𝒙−𝟏
𝟐𝒆 𝟐𝒙
0.883727921 0.883727921 0.880505700 0.883727921
% Error 3.6 2.71 2.56 2.74

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ORDINARY DIFFERENTIAL LINEAR EQUATIONS:
14. Interpolation:
Interpolation refers to using the data in order to predict data within the dataset.
Interpolation is an estimation of a value within two known values in a sequence of
values. Polynomial interpolation is a method of estimating values between known data
points. When graphical data contains a gap, but data is available on either side of the
gap or at a few specific points within the gap, interpolation allows us to estimate the
values within the gap.
Method Formula:
y - y0 = ((y1 - y0)/(x1 - x0)) * (x - x0)
15. Extrapolation:
Extrapolation is the use of the data set to predict beyond the data set. Extrapolation is
an estimation of a value based onextending a known sequence ofvalues or facts beyond
the area that is certainly known. In a general sense, to extrapolate is to infer something
that is not explicitly stated from existing information.
Method Formula:
Y(x) = Y(1) + (x) – (x1) / (x2) – (x1) x {Y(2) – Y(1)}
Q.4)From some observation it is found that pressurerecorded at temperature 35°C is
5.6KPa and at 40°C is 7.4 KPa. Later it is required to use pressure at 37°C which is
not in observation table. So pressurevalue at 37°C need to be Interpolated.
S.No Interpolation Extrapolation
Error %
1 6.68 5.78 1.47

Numerical Analysis and Computer Applications

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