Numerical solution of matrix eigenvalue problems
This page contains material from a minicourse at the Zurich summer school 2008
on eigenvalue problems.
Part 0: Introduction
lecture0.pdf - slides from the presentation [more printable 2p1 version]
Part 1: Power method and friends
lecture1.pdf - Slides from the presentation [more printable 2p1 version]
transient.m - Matlab script demonstrating that the initial convergence of
the power method may be erratic for nonsymmetric matrices.
[Inspired by the Trefethen/Embree book on Pseudospectra.]
ssiter.m - Matlab script demonstrating the convergence of Ritz values during the subspace iteration.
rqidomain.m - Global convergence of the Rayleigh quotient iteration for a 3-by-3 matrix. [Inspired by similar examples from
Pierre-Antoine Absil's PhD thesis.]
Part 2: Krylov subspace methods
lecture2.pdf - slides from the presentation [more printable 2p1 version]
loss.m - Loss of orthogonality in the Arnoldi method.
noloss.m - No loss of orthogonality in the Arnoldi method.
arngo.m - Convergence of Ritz values produced by the Arnoldi method. [Written by David Watkins.]
arngo2.m - Convergence of Ritz values produced by the shift-and-invert Arnoldi method. [Written by David Watkins.]
losslan.m - Loss of orthogonality in the Lanczos method.
nolosslan.m - No loss of orthogonality in the Lanczos method.
Part 3: Preconditioned Eigensolvers
lecture3.pdf - slides from the presentation [more printable 2p1 version]
ppd.m - Convergence of the preconditioned inverse iteration for three different preconditioners [Based on an implementation from the Numerics for CSE lecture notes.]
runlopgc.m - Convergence of the locally optimal preconditioned conjugate gradient method for three different preconditioners [Implementation of LOPCG based on Peter Arbenz' course notes.]
runjdqr.m - Convergence of Jacobi-Davidson for three different preconditioners [Requires the file jdqr.m.gz from G.L.G. Sleijpen.]
Additional Resources
Eigenvalue Templates [For C++ implementations of some of the algorithms, see
the The Iterative Eigensolver Template Library.]
Jack Dongarra's list on freely available software for linear algebra
Matlab programs for David Watkins' book The Matrix Eigenvalue Problem: GR and Krylov Subspace Methods.
Matlab programs for Jim Demmel's book Applied Numerical Linear Algebra.
ARPACK
SLEPc
Trilinos
Jacobi-Davidson gateway
For the class room presentation of Matlab snippets it can be helpful to increase the font size and line widths
of a figure, see makelarge.m.
Contact
Daniel Kressner