CAAM 551: Advanced Numerical Linear
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
- 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
Krylov Projection and Basic Arnoldi Factorization
- Illustrates convergence history ||G^k|| non-normal G
- Illustrates convergence Jacobi,G-S, SOR history ||G^k(x0 -x)|| non-normal G
- Illustrates effects of non-normal G on convergence history showing "the Hump"
GMRES method and comparison
- 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,
- Basic GMRES code
- GMRES code using Givens rotations to update LS soln and residual
- Illustrates convergence of GMRES and FOM on various normal and non-normal cases
- Shows GMRES polynomial surface related to spectrum of A
- Compares Eigenvalue Perturbations Normal vs Non-normal
- Compares performance of various Kryov based solvers