Numerical Analysis 2018

Class Schedule: Tuesday and Thursday ( at 2 PM in LHC 301 ), Friday ( at 8:30 AM in LHC 301/303).

Class Strength: 78 = 67 (Pune) + 11(Tirupati) (This is the largest among all elective courses, offered at IISER Pune in Fall 2018)

Office hour: 5:30 PM - 6:00 PM on Tue and Wed (A404/A423), Tirupati students may call at my office number during office hours.

Tutors: Ankur Mandal (Tirupati), Ravishankar Kapildev Yadav (Pune)

Tutorials: The tutorials would be of two types, theoretical and computational. The theoretical tutorials would be conducted in the NKN room LHC 301 and the computational would be in the computer lab LHC 303, unless otherwise specified. For computational tutorials, the codes and relevant teaching materials would be given to the tutors beforehand.

Lectures

1. 02/08 rounding & truncation error, absolute and relative error, the rate of convergence of order α at least, worse than linear, linear rate, superlinear rate. (17, 38-39 [1]).

2. 03/08 Big oh and little oh notations. Discussion on the loss of significant digits. Finite dimensional normed linear spaces, subordinate matrix norms, calculation of infinity norm. (17-18, 188-189 [1])

3. 07/08 Notion of Condition Number, Derivation of the condition number of a non-singular matrix, Cond(A) depends on the choice of the norm. Calculation of cond(A) with an example.(66-68, 190-191 [1])

4. 09/08 Bisection Method to solve a nonlinear equation f(x)=0, limitations, stopping criterion, proof of convergence, discussion about the rate of convergence. (74-79 [1])

5. 10/08 An algorithm of the bisection method, An iterative method to find the fixed point of a contraction, Banach fixed point theorem, illustration with an example. (100-105 [1])

6. 14/08 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

7. 17/08 Tutorial (Computational) There is no class on 16th August.

8. 21/08 Continuation of the computational tutorial on finding zero of a nonlinear function.

9. 23/08 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. (309-310 [1])

10. 24/08 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. A direct method to find coefficients of interpolating polynomial in canonical form by inverting the Vandermonde matrix. 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. (315-318, 327-329 [1])

11. 28/08 Proof of the algorithm to calculate the divided differences. Rewriting error estimates using a divided difference. Spreadsheet computation of divided difference table. (330-333 [1])

12. 30/08 Lagrange's form of the interpolating polynomial. Definition of Spline function of degree k, the existence of cubic splines, and uniqueness of natural cubic spline, Gershgorin circle theorem. (312-313, 349-354, 268-269 [1])

13. 31/08 Tutorial (Theoretical)

14. 04/09 Numerical integration: Newton-Cotes; composite trapezoidal, error estimate. (478-481 [1])

15. 06/09 Composite Simpson rule of numerical integration (482-483 [1])

16. 07/09 Quiz 1 (Max=10, Min=0, Mean=3.6, Median=3, STD=2.8)

17. 11/09 error estimate of Simpson rule. (483-484 [1])

18. 14/09 Tutorial (Computational). 13th September is a holiday.

19. 18/09 Numerical Differentiation, Richardson extrapolation of 1st step, the order of convergence. (465-474 [1])

20. 20/09 Tutorial (Theoretical), 21/09 is a holiday, 24th to 30th September is the Midsem week, 02/10 is a holiday
    25/09 Midsem 10-12 in LHC 203 & 301 (Max=35, Min=7, Mean=24, Median=24.5, STD=6)

21. 04/10 Discussion on Midsem answer papers. Some special type of linear systems requiring fewer computations to solve. LTM, UTM, PLTM, PUTM. An algorithm to find the permutation. (149-151 [1])

22. 05/10 Numerical approaches for solving a system of linear equations, discussion on computational complexity. Use of LU decomposition. (152 [1])
    09/10 Quiz 2 (Max=10, Min=0, Mean=8.5, Median=10, STD=2.14)

23. 11/10 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])

24. 12/10 Gaussian elimination with scaled row pivoting. Application of Gaussian elimination to factor A=BC, where B is a PLTM and C a PUTM. (163-172 [1])

25. 16/10 Iterative methods to solve large systems of linear equations. Richardson, Jacobi and Gauss-Seidel iterative methods. 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, 207-217 [1])
    18th and 19th have no teaching due to the festival break.

26. 23/10 Tutorial (in 301) Demonstration of some spreadsheet computation for Matrix factorization. The spreadsheet template is here.

27. 25/10 Power method for finding principal eigenvalue of a square matrix, Aitken's acceleration and its limitations. (257-262 [1])
    26/10 Quiz 3 (Max=10, Min=0, Mean=7.2, Median=7.5, STD=2.9)

28. 30/10 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. (524-540 [1])

29. 01/11 General second order Runge-Kutta Method, Modified Euler's method, some comments on other approaches. Second-order boundary value problem. (541-543, 572-578 [1])

30. 02/11 Shooting method for the numerical solution to the boundary value problem. (581-584)

31. 06/11 Finite difference method for the numerical solution to the boundary value problem. (589-592 [1])

32. 08/11 2nd order Parabolic PDE, heat equation in one dimension. Comment on the Explicit method for numerical solution of the heat equation. (615-622 [1])

33. 09/11 Crank-Nicolson implicit method, stability. (623-628 [1])

34. 10/11 Quiz 4 (Max=10.0, Min=0, Mean=6, Median=6, STD=2.8)

35. 13/11 A multigrid method for differential equations. (667-669 [1])

36. 15/11 Tutorial

37. 16/11 Tutorial (Computational)

38. 20/11 Tutorial
    30/11 Endsem Exam (Max=34, Min=0, Mean=18.8, Median=19, STD=7.3) To see the paper meet the TA on 11th December at 11:30 in seminar room 41.

Fortran Compiler - Some free Fortran compilers are downloadable from this link.