Matlab如何解决大型，对称和稀疏线性系统

Question

也就是说，当我对一个非常大，对称且稀疏的A执行A\\b ，matlab使用哪种算法？

Answer 1

If the matrix is sparse and symmetric positive definite, but has a very narrow band, then a specialized band solver is used. Most matrices do not have a narrow enough band to trigger this condition. Typically, this comes up with 1-D splines in the Spline Toolbox. 2-D problems have a large "band" (they are not truly banded matrices).

More typically, MATLAB uses CHOLMOD for sparse symmetric positive definite matrices. If the matrix is sparse, symmetric, and indefinite, then it uses MA49 by Iain Duff.

You can get MATLAB to tell you what it is doing inside A\\b with this option:

spparms ('spumoni',3)

then turn it off again with

spparms ('spumoni',0)

For sparse square matrices it uses UMFPACK. For sparse rectangular matrices it uses SuiteSparseQR (or spqr for short).

For sparse matrices that are lower or upper triangular, or can be permuted into such, it uses a forward/back solver that exploits these properties.

MATLAB does not use the simplex method in backslash at all. If the matrix is rectangular (and short and fat, with more columns than rows) the A\\b returns a basic solution. If you want a minimum 2-norm solution, you need to factorize A' . That can be done in the spqr MATLAB interface but that option is not available in the MATLAB distribution. You would need to install spqr from the source code at suitesparse.com .

Answer 2

The answer depends on the some properties of A (diagonal/square/banded? etc.). CHOLMOD, UMFPACK and qr factorization are some of the options.

The documentation explains it. Here are links to online snapshots of the docs. This may be outdated. - http://amath.colorado.edu/computing/Matlab/OldTechDocs/ref/arithmeticoperators.html - http://www.maths.lth.se/na/courses/NUM115/NUM115-11/backslash.html