F04AXF calculates the approximate solution of a set of real sparse linear equations with a single right-hand side,
$Ax=b$ or
${A}^{\mathrm{T}}x=b$, where
$A$ has been factorized by
F01BRF or
F01BSF.
To solve a system of real linear equations
$Ax=b$ or
${A}^{\mathrm{T}}x=b$, where
$A$ is a general sparse matrix,
$A$ must first be factorized by
F01BRF or
F01BSF. F04AXF then computes
$x$ by block forward or backward substitution using simple forward and backward substitution within each diagonal block.
The method is fully described in
Duff (1977).
A more recent method is available through solver routine
F11MFF and factorization routine
F11MEF.
Duff I S (1977) MA28 – a set of Fortran subroutines for sparse unsymmetric linear equations AERE Report R8730 HMSO
If an error is detected in an input parameter F04AXF will act as if a soft noisy exit has been requested (see
Section 3.3.4 in the Essential Introduction).
The accuracy of the computed solution depends on the conditioning of the original matrix. Since F04AXF is always used with either
F01BRF or
F01BSF, you are recommended to set
${\mathbf{GROW}}=\mathrm{.TRUE.}$ on entry to these routines and to examine the value of
${\mathbf{W}}\left(1\right)$ on exit (see
F01BRF and
F01BSF). For a detailed error analysis see page 17 of
Duff (1977).
If storage for the original matrix is available then the error can be estimated by calculating the residual
and calling F04AXF again to find a correction
$\delta $ for
$x$ by solving
This example solves the set of linear equations
$Ax=b$ where
The nonzero elements of
$A$ and indexing information are read in by the program, as described in the document for
F01BRF.