HP MLIB User's Guide Vol. 2 7th Ed.

Chapter 13 Sparse Linear Equations 875
Overview
13 Sparse Linear Equations
Overview
This chapter describes state-of-the-art software for the direct solution of sparse
systems of linear equations with symmetric coefficient matrices or
unsymmetric coefficient matrices that are structurally symmetric. Throughout
this chapter, a system of linear equations is called “symmetric” if its coefficient
matrix is both symmetric in structure and in value. “Symmetric in value” or
simply “symmetric” means that a(i,j) = a(j,i) for every i and j. The term
“unsymmetric” means that the coefficient matrix is symmetric in structure, but
not symmetric in value. “Symmetric in structure” or “structurally symmetric”
means that if element (i,j) of the matrix is in the set of possible nonzeros, then
element (j,i) also is in the set.
This package of subprograms provides efficient use of your computer’s
architecture in conjunction with powerful techniques for using the sparsity of a
problem to reduce the cost of solution. Accuracy is assured through appropriate
numerical techniques.
This chapter explains how to use the sparse linear equation subprograms to
solve systems of sparse linear equations where the coefficient matrix is
symmetric or unsymmetric but with symmetric structure. For a symmetric
coefficient matrix, the solution is based on the LDL
T
factorization with an
option for pivoting. For coefficient matrices with symmetric structure but
unsymmetric values, the solution is based on LU factorization without pivoting.
Subprograms are provided to:
• Solve a single sparse linear system
• Estimate condition number of the coefficient matrix
• Efficiently solve multiple systems of equations