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Numerical Analysis

Overview

Thenumerical analysis/ methodis a very important and common topic for computational mathematics and hence studied by the students from many disciplines like mathematics, computer science, physics, statistics and other subjects of physical sciences and engineering. The numerical analysis / method is an interdisciplinary course used by the students/ teachers/ researchers from several branches of science and technology, particularly frommathematics, statistics, computer science, physics, chemistry, electronics,etc. This subject is also known as computational mathematics. To design several functions of a computer and to solve a problem by computer numerical method is essential. It is not possible to solve any large scale problem without help of numerical methods. Numerical methods also simplify the conventional methods to solve problems, like definite integration, solution of equations, solution of differential equations, interpolation from the known to the unknown, etc. To explore complex systems, mathematicians, engineers, physicists require computational methods since mathematical models are only rarely solvable algebraically. The numerical methods based on computational mathematics are the basic algorithms underpinning computer predictions in modern systems science. After completion of the course, the students can design algorithms and program codes to solve the real life problems. In each module, an exercise is provided to test the performance of the students. Also, some more references are added in the learn more section to investigate the subject more thoroughly and to learn more topics of numerical analysis. The entire course is divided into nine chapters and thirty six modules. It is a15 weeksone semester course including assignment, discussion and evaluation. This course is offered by almost all Indian universities as acore course.

Syllabus

Week 1:Errors in Numerical Computations
1. Ch 1 Mod 1:Error in Numerical Computations.
2. Ch 1 Mod 2:Propagation of Errors and Computer Arithmetic.

Week 2:Interpolation - I
3. Ch 1 Mod 3:Operators in Numerical Analysis.
4. Ch 2 Mod 1: Lagrange’s. Interpolation.
5. Ch 2 Mod 2: Newton’s Interpolation Methods.
6. Ch 2 Mod 3: Central Deference Interpolation Formulae.

Week 3:Interpolation - II
7. Ch 2 Mod 4:Aitken’s and Hermite’s Interpolation Methods.
8. Ch 2 Mod 5: Spline Interpolation.
9. Ch 2 Mod 6:Inverse Interpolation.
10. Ch 2 Mod 7:Bivariate Interpolation.

Week 4:Approximation of Functions
11. Ch 3 Mod 1: Least Squares Method.
12. Ch 3 Mod 2:Approximation of Function by Least Squares Method.
13. Ch 3 Mod 3: Approximation of Function by Chebyshev Polynomials.

Week 5:Solution of Algebraic andTranscendental Equation
14. Ch 4 Mod 1:Newton’s Method to Solve Transcendental Equation.
15. Ch 4 Mod 2: Roots of a Polynomial Equation.
16. Ch 4 Mod 3: Solution of System of Non-linear Equations.

Week 6:Solution of System of Linear Equations-I
17. Ch 5 Mod 1:Matrix Inverse Method.
18. Ch 5 Mod 2:Iteration Methods to Solve System of Linear Equations.
19. Ch 5 Mod 3: Methods of Matrix Factorization.

Week 7:Solution of System of Linear Equations-II
20. Ch 5 Mod 4:Gauss Elimination Method and Tri-diagonal Equations.
21. Ch 5 Mod 5: Generalized Inverse of Matrix.
22. Ch 5 Mod 6: Solution of Inconsistent and Ill Conditioned Systems.

Week 8:Assessment

Week 9:Eigenvalues and Eigenvectors of Matrices
23. Ch 6 Mod 1:Construction of Characteristic Equation of a Matrix.
24. Ch 6 Mod 2:Eigenvalue and Eigenvector of Arbitrary Matrices.
25. Ch 6 Mod 3: Eigenvalues and Eigenvectors of Symmetric Matrices.

Week 10:Differentiation and Integration-I
26. Ch 7 Mod 1:Numerical Differentiation.
27. Ch 7 Mod 2:Newton-Cotes Quadrature.

Week 11:Differentiation and Integration-II
28. Ch 7 Mod 3:Gaussian Quadrature.
29. Ch 7 Mod 4: Monte-Carlo Method and Double Integration.

Week 12:Ordinary Differential Equations-I
30. Ch 8 Mod 1:Runge-Kutta Methods.
31. Ch 8 Mod 2:Predictor-Corrector Methods.

Week 13:Ordinary Differential Equations-II
32. Ch 8 Mod 3:Finite Difference Method and its Stability.
33. Ch 8 Mod 4: Shooting Method and Stability Analysis.

Week 14:Partial Differential Equations
34. Ch 9 Mod 1:Partial Differential Equation: Parabolic.
35. Ch 9 Mod 2:Partial Differential Equations: Hyperbolic.
36. Ch 9 Mod 3:Partial Differential Equations: Elliptic

Week 15:Final examination