We use the standard notation for a system of simultaneous
linear
equations :
where A is the coefficient matrix ,
b is the right-hand side ,
and x is the solution .
In (3.2) A is assumed to be a square matrix of order n,
but some of the individual routines allow A to be rectangular.
If there are several right-hand sides
we write
where the columns of B are the individual right-hand sides,
and the columns of
X are the corresponding solutions.
The basic task is to compute X, given A and B.
If A is upper or lower triangular, (3.2) can be solved by a straightforward process of backward or forward substitution. Otherwise, the solution is obtained after first factorizing A as a product of triangular matrices (and possibly also a diagonal matrix or permutation matrix).
The form of the factorization depends on the properties of the matrix A. ScaLAPACK provides routines for the following types of matrices, based on the stated factorizations:
If A is m-by-n with bwl subdiagonals and bwu superdiagonals,
the factorization is
where P and Q are permutation matrices and L and U are banded
lower and upper triangular matrices, respectively.
A diagonally dominant-like matrix
is one for which it is known
a priori that pivoting for stability is NOT required in the LU
factorization of the matrix. Diagonally dominant matrices themselves are
examples of diagonally dominant-like matrices.
If A is m-by-n with bwl subdiagonals and bwu superdiagonals,
the factorization is
where P is a permutation matrix and L and U are banded lower
and upper triangular matrices respectively.
Note: In the banded and tridiagonal factorizations (PxDBTRF, PxDTTRF, PxGBTRF, PxPBTRF, and PxPTTRF), the resulting factorization is not the same factorization as returned from LAPACK. Additional permutations are performed on the matrix for the sake of parallelism. Further details of the algorithmic implementations can be found in [32].
The factorization for a general diagonally dominant-like tridiagonal matrix is like that for a general diagonally dominant-like band matrix with bwl = 1 and bwu = 1. Band matrices use the band storage scheme described in section 4.4.3.
While the primary use of a matrix factorization is to solve a system of equations, other related tasks are provided as well. Wherever possible, ScaLAPACK provides routines to perform each of these tasks for each type of matrix and storage scheme (see table 3.6). The following list relates the tasks to the last three characters of the name of the corresponding computational routine:
Note that some of the above routines depend on the output of others:
The RFS (``refine solution'') routines perform iterative refinement and compute backward and forward error bounds for the solution. Iterative refinement is done in the same precision as the input data. In particular, the residual is not computed with extra precision, as has been traditionally done. The benefit of this procedure is discussed in section 6.5.
Table 3.6: Computational routines for linear equations