Numerical Analysis 2021

Class Schedule: Tuesday and Friday ( at 12 PM), Thursday ( at 9:00 AM).

Class Strength: 78

Tutors:

Lectures

1. 17/08 rounding & truncation error, absolute and relative error, Discussion on the loss of significant digits. (38-39, 55-56 [1]).
   19/08 Closed Holiday

2. 20/08 The rate of convergence of order α at least, worse than linear, linear rate, superlinear rate. Big oh and little oh notations. Finite dimensional normed linear spaces, subordinate matrix norms. Notion of Condition Number, Derivation of the condition number of a non-singular matrix, Cond(A) depends on the choice of the norm. (17-18, 66-68, 191 [1])

3. 24/08 Matrix Norms and live discussions. (188--190 [1])

4. 26/08 Calculation of infinity norm. Bisection Method to solve a nonlinear equation f(x)=0, limitations, stopping criterion, proof of convergence, discussion about the rate of convergence. An algorithm of the bisection method. (74-79 [1])

5. 27/08 Newton's method for finding zero of a function of several variables, The iterative scheme, A set of sufficient conditions for convergence of the scheme, Proof of convergence of the scheme for simple zero cases, Proof of rate of convergence of order two at least (quadratic convergence). (81-85 [1]). An extension
   - An iterative method to find the fixed point of a contraction, Banach fixed point theorem, illustration with an example. (100-105 [1]) will be covered on 14th September

6. 31/08 Existence & uniqueness of interpolating polynomial p_n of degree at most n for given n+1nodes. A recursive formula for p_n in terms of p_n-1 and c_n, an expression for c_n. Newton's form, i.e., expression of p_n in terms of c_i for i= 0, ..., n. Horner's algorithm. Lagrange's form of the interpolating polynomial. A direct method to find coefficients of interpolating polynomial in canonical form by inverting the Vandermonde matrix. (309-314 [1])

7. 02/09 Error estimate of polynomial interpolation of a smooth function, the role of Chebyshev's nodes in minimizing the error. Remark on non-convergence of higher order interpolation. (315-320 [1])

8. 03/09 Two different methods to calculate c_i, the coefficients of interpolating polynomial of Newton's form. Method-1 involves inversion of a lower triangular matrix and Method-2 involves calculation of divided differences in tabular form. Proof of the algorithm to calculate the divided differences. Rewriting error estimates using a divided difference. (327-333 [1])

9. 07/09 Definition of Spline function of degree k, the existence of cubic splines, and uniqueness of natural cubic spline, Gershgorin circle theorem. (349-354, 268-269 [1])

10. 09/09 Optimality Theorem of natural cubic spline. Tutorial (Theoretical and computational) on cubic spline. (355 [1])
    10/09 Closed Holiday

11. 14/09 An iterative method to find the fixed point of a contraction, Banach fixed point theorem, illustration with an example. (100-105 [1])

12. 16/09 Live Tutorial (Computational)

13. 17/09 Tutorial (Computational).

14. 21/09 Quiz 1

15. 23/09 Numerical integration: Newton-Cotes; composite trapezoidal, error estimate. (478-481 [1])

16. 24/09 Composite Simpson rule of numerical integration and error estimate. (482-484 [1])

17. 28/09 Numerical Differentiation, Richardson extrapolation of 1st step, the order of convergence. (465-474 [1])

18. 30/09 Power method for finding principal eigenvalue of a square matrix. Inverse power method (257-259, 261 [1])

19. 01/10 Tutorial on Quiz1 solutions and Power method.
    -Aitken's acceleration and its limitations. (260 [1])
    9/10 Midsem 10-12 in Code Tantra 4th to 20th October: Teaching break for the Midsem and Festival holidays

20. 21/10 Some special type of linear systems requiring fewer computations to solve. LTM, UTM, PLTM, PUTM. An algorithm to find the permutation. Numerical approaches for solving a system of linear equations, discussion on computational complexity. Use of LU decomposition. (149-152 [1])

21. 22/10 The inverse of UTM is UTM. Sufficient condition for the existence of LU decomposition, proof of the theorem is left for self-study. An algorithm of LU decomposition. Sufficient condition for unique Cholesky decomposition, proof of the theorem. (153-157 [1])

22. 26/10 Live interaction for clarifying doubts regarding midsem paper grading. Tutorial: Demonstration of some spreadsheet computation for Matrix factorization.

23. 28/10 Basic Gaussian elimination without pivoting. Gaussian elimination as a special case of LU factorization. A pseudo code of the algorithm. (163-167 [1])

24. 29/10 The need of selecting pivot rows different from the successor, explained with some examples. Gaussian elimination algorithm with scaled row pivoting. This is an LU factorization of PA, where P is a permutation matrix. Illustration of pivoting with a numerical example. (167-172 [1])

25. 02/11 Revisiting the Gaussian elimination algorithms in a live session. A discussion on the limitation of the elimination algorithms for the large systems. The concepts and examples of various iterative methods to solve large systems of linear equations. Richardson, Jacobi and Gauss-Seidel iterative methods. (207-209 [1])
    04/11 Holiday

26. 05/11 Comparison of computational complexities between Gaussian elimination and iterative algorithms.

27. 09/11 Live session: theoretical tutorial on Assignments 4 and 5. Iterative algorithms for solving linear systems. (213 [1])

28. 11/11 Spectral radius of a square matrix, the relation between spectral radius and subordinate matrix norm. A necessary and sufficient condition for convergence of an iterative method with an initial vector. (198-199, 210, 213-215 [1])

29. 12/11 Sufficient condition for the convergence of Jacobi and Gauss-Seidel iterations. (212, 216-217 [1])

30. 16/11 Quiz2 at 12:00 (during the live class). Syllabus: The topics covered in Assignments 4 & 5.

31. 18/11 Conversion of higher order ODE to a 1st order system. Theorems on existence and uniqueness of the solution of 1st order ordinary differential equations. The numerical approach, one step method, Euler's, Taylor series, Huen's method. (565-567, 524-527, 530-535, 540 [1])
    19/11 Holiday

32. 23/11 General second order Runge-Kutta Method, Modified Euler's method, some comments on other approaches. Second-order boundary value problem. (541-543, 572-577 [1])

33. 25/11 Shooting method for the numerical solution to the boundary value problem. (581-584)

34. 26/11 Finite difference method for the numerical solution to the boundary value problem. (589-592 [1])

35. 30/11 2nd order Parabolic PDE, heat equation in one dimension. Comment on the Explicit method for numerical solution of the heat equation. (615-622 [1])

36. 02/11 Crank-Nicolson implicit method, stability. (623-628 [1])

37. 03/12 A multigrid method for differential equations. (667-669 [1])
    or Tutorial

38. 8/12 Endsem Exam