Four mutually orthogonal functions, defined in the plane. Each function, after multiplication by an auxillary function (not shown), is nonnegative definite (in the sense of Bochner). The matrices above were found with cvxopt and the iterative solvers below.

CVXOPT Solvers

You will find here several iterative solvers for the linear problem Ax=b. They are written in Python and require a working installation of cvxopt. Scroll down to get them.

Each file in the list below is (essentially) a line-for-line translation of a matlab original. The original matlab files, and a lot more information may be found at Stanford's SOL. Their software is released under a BSD-style license. Per the requirements of that license, it is displayed below. Note that the solvers here exist because people at SOL wrote them first.

Requirements: a working copy of cvxopt.

The Files

Minres was last updated in October 2009, Symmlq in May 2009, LSQR in November 2009. See the changelogs near the top of each file for details.

minres	Example using MINRES
symmlq	Example using SYMMLQ
lsqr	Example using LSQR

Comments

I added the routine 'SymOrtho' for numerical stability. This is recommended by S.-C. Choi in the dissertation, Iterative Methods for Singular Linear Equations and Least-Squares Problems. It removes the unpleasant potential of '1/eps' in some important places (see, for example text following "Compute the next plane rotation Qk" in minres_py).

The SciPy/Numpy versions of the same routines have no connection with the versions posted here.

The files are named '*_py' instead of '*.py' because the current web server tries to execute the '*.py' files instead of sending them.

The current version and archive management of these files is really poor. This will eventually change.

If you find bugs in these python versions, please tell me. jeffery.kline @ gmail com

License

    BSD License for SOL Numerical Software
    http://www.opensource.org/licenses/bsd-license.php

This notice applies to the software packages

    cgls, lsqr, lumod, minres, pdco, pdsco, symmlq

made available at
    http://www.stanford.edu/group/SOL/software.html
by the
    Systems Optimization Laboratory (SOL)
    Dept of Management Science and Engineering
    Stanford University,
    Stanford, CA 94305-4026, USA


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