Subject Code:  MA5L018 Subject Name:  Numerical Linear Algebra L-T-P: 3-0-0 Credit:3
Pre-requisite(s): Linear Algebra (MA5L001)
Fundamentals. Linear systems, LU decompositions, Gaussian elimination with partial pivoting, Banded systems, Positive definite systems, Cholesky decomposition. Vector and matrix norms, Perturbation theory of linear systems, Condition numbers, Estimating condition numbers, IEEE floating point arithmetic, Analysis of round off errors. Gram-Schmidt orthonormal process, Orthogonal matrices, Householder transformation, Givens rotations, QR factorization, Roundoff
 error analysis of orthogonal matrices, Stability of QR factorization. Solution of linear least squares problems, Normal equations, Singular Value Decomposition(SVD), Polar decomposition, Moore-Penrose inverse, Rank deficient least squares problems, Sensitivity analysis of least-squares problems. Review of eigenvalues and canonical forms of matrices, Sensitivity of eigenvalues and eigenvectors, Reduction to Hessenberg and tridiagonal forms, Power and inverse power methods, Rayleigh quotient iteration, Explicit and implicit QR algorithms for symmetric and non-symmetric matrices, Implementation of implicit QR algorithm. Computing the SVD, Sensitivity analysis of singular values and singular vectors. Overview of iterative methods: Jacobi, Gauss-Seidel and successive over relaxation methods, Krylov subspace method, The Arnoldi and the Lanczos iterations.
Software Support: MATLAB.
Text Books:
  1. Trefethen  L. N. and Bau D. Numerical Linear Algebra, SIAM
  2. Watkins D. S. Fundamentals of Matrix Computation, Wiley
Reference Books:
  1. Golub  G. H. and Van Loan C.F. Matrix Computation,  John Hopkins U. Press, Baltimore
  2. Stewart G. W.  Introduction to Matrix Computations, Academic Press
  3. Demmel  J.W.  Applied numerical linear algebra, SIAM, Philadelphia