- AUTHORS:
C. C. Paige,
M. A. Saunders.
- CONTRIBUTORS: Sou-Cheng Choi
- CONTENTS: Implementation of a conjugate-gradient type method
for solving sparse linear equations:
Solve Ax = b or (A - sI)x = b.
The matrix A - sI must be symmetric
but it may be definite or indefinite or singular.
The scalar s is a shifting parameter – it may be any number.
The method is based on Lanczos tridiagonalization.
You may provide a preconditioner, but it must be positive definite.
MINRES is really solving one of the least-squares problems
min ||Ax - b|| or min ||(A - sI)x - b||.
If A is singular (and s = 0), MINRES returns a least-squares solution
with small ||Ar|| (where r = b - Ax), but in general it is not the
minimum-length solution. To get the min-length solution, use MINRESQLP
(see PhD dissertation of Sou-Cheng Choi).
Similarly if (A - sI) is singular.
If A is symmetric (and A - sI is nonsingular),
SYMMLQ may be slightly more reliable.
If A is unsymmetric,
use LSQR.
- REFERENCES:
C. C. Paige and M. A. Saunders (1975).
Solution of sparse indefinite systems of linear equations,
SIAM J. Numerical Analysis 12, 617-629.
S.-C. Choi (2006).
Iterative Methods for Singular
Linear Equations and Least-Squares Problems,
PhD thesis, Stanford University.
- RELEASE:
22 Jul 2003: f77 files derived from f77 version of SYMMLQ.
Preconditioning permitted; singular systems not.
17 Oct 2003: MATLAB files derived from f77 version.
Singular systems allowed.
11 Oct 2007: f90 files derived from f77 version.
04 Dec 2007: f90 version allows singular systems.
11 May 2009: Matlab version updated following comments from Jeffery Kline.