Slides & Notes


no of

classes

TOPICS

2

Fundamental Concepts

• Matrix-Vector operations

• Orthogonal vectors and matrices

• Matrix and vector norms

• Singular value decomposition (SVD)

lect1  lect2  lect3  matlab1  matlab2  matlab3

lect4  svd2 imagefile   svd_test

6

QR Factorization

• Projectors and QR factorization

• Gram-Schmidt orthogonalization process

• Householder triangularization

• Least-squares problems

qr    pagerank   least_square

6

Conditioning and Stability

• Condition numbers

• Floating point Arithmetic

• Stability

• Conditioning of Householder triangularization

• Conditioning of Back substitution

• Conditioning of Least-squares problems

 

6

Systems of Equations

• Gaussian elimination and LU factorization

• Pivoting and LUP factorization

• Stability of Gaussian elimination

• Cholesky Factorization 

 

6

Eigenvalue Problems

• Overview of eigenvalue problems

• Reduction to upper-Hessenberg/Tridiagonal form

• Power and inverse power iteration

• QR algorithm

• Computing SVD

QR_algorithm  Computing_SVD

9

Iterative Methods

• Arnoldi iteration

• GMRES method

• Lanczos iteration

• Conjugate gradient (CG) method

• Jacobi, SOR, and SSOR

• Multigrid method

Iterative_method

Krylov_subspace_methods1

CG_error_analysis.ppt

Arnoldi iterations  Arnoldi.htm

gmres_example  gmres_example.m

Lanczos.ppt

Krylov_subspace_method2

fine_mesh Multigrid_method

10

Preconditioning Techniques

• Introduction

• Jacobi, SOR, and SSOR Preconditioners

• ILU Factorization Preconditioners

• Block Preconditioners

preconditioning

rev