Numerical Analysis Class1 Introduction to numerical analysis
- 1. Numerical Analysis ME – 263 Credit: 3.00 Contact hour: 3.00 Course Teacher: Lecturer Md. Fahim Faisal Contact Number: +8801521503917 Email: fahim_faisal@me.mist.ac.bd
- 2. Topic : Introduction Class: 1 Date: 2 December 2024 (Monday)
- 3. 3 Class Routine For section A + B: Probable Holiday no: Class time may be adjusted later. 16 December (Monday): Sec A + B 1hr each 25 December (Wednesday): Sec A 2hrs 26 December (Thursday): Sec B 1hr *CR Please share your class routine for free time slot.
- 4. 4 1. This course will emphasize the development of numerical algorithms to provide solutions to common problems formulated in science and engineering. 2. The primary objective of the course is to develop the basic understanding of the construction of numerical algorithms, and perhaps more importantly, the applicability and limits of their appropriate use. Objectives: Students will be able to……………….. CO 1: Demonstrate the ability to solve engineering problems with analytical methods. CO 2: Compare numerical algorithms and techniques in terms of their effectiveness in solving real life engineering problems. CO 3: Develop graphical representation of engineering data set generated from numerical algorithms. Course Outcomes (CO):
- 5. 5 1. Numerical Methods for Engineers (4th edition) - Steven C. Chapra, Raymond P. Carale 2. Applied Numerical Analysis (5th edition) - Curtis F. Gerald, Patrick O. wheatley 3. Numerical Methods: Using Matlab, Fourth Edition, 2004 John H. Mathews and Kurtis D. Fink 4. Numerical Methods - E. Balagurusamy Reference Books: Class Performance 5% Class Test/Assignment 20% Mid-Term Assessment (Exam/Project) 15% Final Examination (Section A & B) 60% Total 100% Evaluation:
- 6. 6 Approximations, Taylor’s Series, and Errors. Linear, Quadratic, Newton’s Divide Difference Interpolating Polynomials, ad Lagrange Interpolating Polynomials. Graphical Method, Bisection Method, False-Position Method. The trapezoidal rule, Simpson’s Rule. Simple Fixed-Point Iteration, Newton-Raphson Method, Secant Method, System of Nonlinear Equations. Numerical Differentiation, Richardson’s extrapolation, Forward, backward, and central divide difference formula. Solving ODE, Euler’s Method, Heun’s Method, Runge-Kutta Methods for lower and higher order, and Engineering Applications of Roots of Equations. Boundary Value Problems, Eigen Value Problems. Gauss Elimination, Gauss- Jordan, LU Decomposition, Matrix Inverse, Gauss-Seidel.) derivation of Laplace Equation, Laplacian Difference Equation, Liebman Method. Engineering Applications of Linear Algebraic Equations. Solving PDE for Derivative Boundary Conditions, Solution of first-order differential equations and 2nd order Partial Differential Equation. Detail Contents : *Underline contents will be taught by Lec Md. Fahim Faisal Subject to change
- 7. 7 Introduction Numerical Analysis is an area of mathematics that creates, analyzes and implements algorithms for obtaining numerical solutions to problems involving unknown variables. Such problems arise throughout the realm of natural sciences, engineering, medicine, social science, business etc. Areas where numerical analysis is required 1. Computing values of functions 2. Interpolation, extrapolation and regression 3. Solving equations and system of equations 4. Solving eigenvalue or singular value problems 5. Evaluating integrals 6. Differential equations 7. Optimization
- 8. 8 Analytical approach vs Numerical approach Analytical approach 3 𝑥3 +4 =28 3 𝑥3 =24 𝑥3 =8 𝑥=2 Numerical approach 𝐴𝑠𝑠𝑢𝑚𝑖𝑛𝑔 𝑥=1 3 ∗ 13 − 24=− 21 −21𝑎𝑛𝑑 0h𝑢𝑔𝑒𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 ; 3 ∗ 1.53 −24=− 13.875 7; 9.261 9; 3.426 99; 0.35 𝑣𝑒𝑟𝑦 𝑐𝑙𝑜𝑠𝑒
- 9. 9 Significant digits Observation 1: 6.2 2: 6.24 3: 6.246 4: 6.2468 Actual: 6.246893847 The concept of significant digit has been developed to formally designate the reliability of a numerical value Observation 1: Accurate up to 2 sig digits 2: 3 sig digits 3: 4 sig digits 4: 5 sig digits
- 10. 10 Accuracy vs Precision Accuracy: How close the measured values are to the exact value Precision: How close the values are to each other
- 11. 11 Errors Errors are formulated because of using approximations to represent exact values True values = 3.1415928 Approximation = 3.14 Error (True) = But in most cases, we don’t know the exact(true) value Numerical analysis involves iterations and the current approximation is better than the previous approximation(in most cases), we use percent relative error true percent relative error.
- 12. 12 Errors True fractional relative error = True percent relative error
- 13. 13 Significance of relative error Measured size of a screw = 9 cm Actual size of a screw = 10 cm Measured size of a screw = 9999 cm Actual size of a screw = 10000 cm An error of 10% is much more significant than an error of 0.01%
- 14. 14 Percent Relative error This is first iteration, so no relative error
- 15. 15 Truncation error 𝑒 𝑥 = 1 + 𝑥 + 𝑥2 2 ! + 𝑥 3 3 ! +…+ 𝑥 𝑛 𝑛 ! Terms Result 1 1 39.3 - 2 1.5 9.02 33.3 3 1.625 1.44 7.69 4 1.645833 0.175 1.27 5 1.6484375 0.0172 0.158 6 1.648697917 0.00142 0.0158 True value of Tylor series expansion
- 16. 16 Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century. Refn: *Taylor series - Wikipedia Round off error During calculation, because of limiting factors in hardware and software, computers often omit significant figures of a value. The error caused by this emission is called round off error