CAAM 551: Advanced Numerical Linear Algebra

Matlab Code

Codes and Descriptions Dense LU decompostions for Ax = b Dense Cholesky decompostion for Ax = b Matlab Sparse Cholesky Demo -- Compare Orderings Matlab Sparse Unsymmetric LU -- Compare Orderings QR_factorization: Compare MGS, CGS, CGS w/ refinement
  • QR_factorization and Least Squares via Row-wise Givens Method
  • Sparse QR_factorization motivating example
  • Numerical Effects of Sign Choice for Householder Transformation
  • QR_factorization via Householder's Method
  • Heat test problem for regularization exercise Illustration of perturbation bounds for Ax = b,
  • Shows || x1 - x||/||x|| < K(A)||b1 - b||/||b||.
  • Illustration of perturbation bounds for min ||b - Ax ||
  • Shows || x1 - x||/||x|| < (K(A)/cos(theta))||b1 - b||/||b||. Demonstrates that condition of Normal Equations (NEQ) is the square of the condition number of A
  • Shows || x1 - x||/||x|| < (K(A)^2)||A'(b1 - b)||/||A'b||.
  • Illustrations of best rank k SVD approximation
  • Basic Iterative Methods for Ax = b Krylov Projection and Basic Arnoldi Factorization GMRES method and comparison Preconditioned Gmres Basic Iterative Eigenvalue Methods Ritz Value Convergence Effect of Polynomial Restarting Gui: Graphic Illustration of Ritz Value Convergence Simple Lanczos Process: Breakdown Ritz value behavior in Lanczos Graphic Illustration of QR-convergence Implicitly Restarted Arnoldi (IRA) Graphic Illustration of IRA Hessenberg convergence to real schur form Graphic Illustration of IRA and Shift-Invert-IRA Ritz Value convergence Graphic Illustration of IRA exact shift selection
  • and resulting filter polynomial surface in relation
  • to the spectrum of the matrix