# 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
• HHbad.m, driveHHB.m, Illustrates loss of orthogonality and accuracy when wrong sign choice is made
• HHgood.m, driveHHG.m, Illustrates correct choice of sign cures numerical problems
• 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
• Illustrates convergence history ||G^k|| non-normal G
• powersR.m
• Illustrates convergence Jacobi,G-S, SOR history ||G^k(x0 -x)|| non-normal G
• Hump.m
• Illustrates effects of non-normal G on convergence history showing "the Hump"
Krylov Projection and Basic Arnoldi Factorization
• krylov.m
• Illustrates when [V,R] = qr(K,0) is computed where K = [b, Ab,..., A^(k-1)b] is obtained directly from the power sequence fails numerically. H = V'AV is not Hessenberg and AV .ne. VH + fe_k'.
• Arnoldi.m, ArnoldiC.m
• Illustrates loss orthogonality with CGS (driveA) and full orthogonaltiy with CGS followed by iterative refinement (driveAC)
• driveA.m, driveAC.m,
GMRES method and comparison
• Gmres0.m
• Basic GMRES code
• Gmres.m
• GMRES code using Givens rotations to update LS soln and residual
• driveGmres_Fom.m, Gmres_Fom.m
• Illustrates convergence of GMRES and FOM on various normal and non-normal cases
• Shows GMRES polynomial surface related to spectrum of A
• normal_vs_non.m
• Compares Eigenvalue Perturbations Normal vs Non-normal
• compLinSlvrs.m
• Compares performance of various Kryov based solvers
Preconditioned Gmres
• GmresP.m
• GMRES code that applies left pre-conditioning with M = LU in factored form
• driveGMLU.m
• Demonstrates performance of pre-conditioned vs unconditioned 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