MATRIX COMPUTATIONS

Code: MA223 | L-T-P-C: 3-0-2-8

Floating point computations, IEEE floating point arithmetic, analysis of roundoff errors; Sensitivity analysis and condition numbers; Linear systems, LU decompositions, Gaussian elimination with partial pivoting; Banded systems, positive definite systems, Cholesky decomposition - sensitivity analysis; Gram-Schmidt orthonormal process, Householder transformation, Givens rotations; QR factorization, 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 canonical forms of matrices; Sensitivity of eigenvalues and eigenvectors.

Reduction to Hessenberg and tridiagonal forms; Power, inverse power and Rayleigh quotient iterations; Explicit and implicit QR algorithms for symmetric and non-symmetric matrices; Reduction to bidiagonal form; Golub-Kahan algorithm for computing SVD; Sensitivity analysis of singular values and singular vectors; Overview of iterative methods: Jacobi, Gauss-Seidel and successive over relaxation methods; Krylov subspace methods, Arnoldi and Lanczos methods, conjugate gradient method.

Texts:

1. G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd Edition, John Hopkins University Press, 1996.
2. C. T. Kelley, Iterative Methods for Linear and Nonlinear equations, SIAM, 1995.
3. D. S. Watkins, Fundamentals of Matrix Computations, John Wiley, 1991.
4. L. N. Trefethen and D. Bau III, Numerical Linear Algebra, SIAM, 1997.
5. J. W. Demmel, Applied Numerical Linear Algebra, SIAM, 1997.