F04 Chapter Contents
F04 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF04AXF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

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.

## 2  Specification

 SUBROUTINE F04AXF ( N, A, LICN, ICN, IKEEP, RHS, W, MTYPE, IDISP, RESID)
 INTEGER N, LICN, ICN(LICN), IKEEP(5*N), MTYPE, IDISP(2) REAL (KIND=nag_wp) A(LICN), RHS(N), W(N), RESID

## 3  Description

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.

## 4  References

Duff I S (1977) MA28 – a set of Fortran subroutines for sparse unsymmetric linear equations AERE Report R8730 HMSO

## 5  Parameters

1:     N – INTEGERInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
2:     A(LICN) – REAL (KIND=nag_wp) arrayInput
On entry: the nonzero elements in the factorization of the matrix $A$, as returned by F01BRF or F01BSF.
3:     LICN – INTEGERInput
On entry: the dimension of the arrays A and ICN as declared in the (sub)program from which F04AXF is called.
4:     ICN(LICN) – INTEGER arrayCommunication Array
On entry: the column indices of the nonzero elements of the factorization, as returned by F01BRF or F01BSF.
5:     IKEEP($5×{\mathbf{N}}$) – INTEGER arrayInput
IKEEP provides, on entry, indexing information about the factorization, as returned by F01BRF or F01BSF. Used as internal workspace prior to being restored and hence is unchanged.
6:     RHS(N) – REAL (KIND=nag_wp) arrayInput/Output
On entry: the right-hand side vector $b$.
On exit: RHS is overwritten by the solution vector $x$.
7:     W(N) – REAL (KIND=nag_wp) arrayWorkspace
8:     MTYPE – INTEGERInput
On entry: MTYPE specifies the task to be performed.
${\mathbf{MTYPE}}=1$
Solve $Ax=b$.
${\mathbf{MTYPE}}\ne 1$
Solve ${A}^{\mathrm{T}}x=b$.
9:     IDISP($2$) – INTEGER arrayCommunication Array
On entry: the values returned in IDISP by F01BRF.
10:   RESID – REAL (KIND=nag_wp)Output
On exit: the value of the maximum residual, $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(\left|{b}_{i}-\sum _{j}^{}{a}_{ij}{x}_{j}\right|\right)$, over all the unsatisfied equations, in case F01BRF or F01BSF has been used to factorize a singular or rectangular matrix.

## 6  Error Indicators and Warnings

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).

## 7  Accuracy

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
 $r=b-Ax or ​b-ATx$
and calling F04AXF again to find a correction $\delta$ for $x$ by solving
 $Aδ=r or ​ATδ=r.$

## 8  Further Comments

If the factorized form contains $\tau$ nonzeros (${\mathbf{IDISP}}\left(2\right)=\tau$) then the time taken is very approximately $2\tau$ times longer than the inner loop of full matrix code. Some advantage is taken of zeros in the right-hand side when solving ${A}^{\mathrm{T}}x=b$ (${\mathbf{MTYPE}}\ne 1$).

## 9  Example

This example solves the set of linear equations $Ax=b$ where
 $A= 5 0 0 0 0 0 0 2 -1 2 0 0 0 0 3 0 0 0 -2 0 0 1 1 0 -1 0 0 -1 2 -3 -1 -1 0 0 0 6 and b= 15 12 18 3 -6 0 .$
The nonzero elements of $A$ and indexing information are read in by the program, as described in the document for F01BRF.

### 9.1  Program Text

Program Text (f04axfe.f90)

### 9.2  Program Data

Program Data (f04axfe.d)

### 9.3  Program Results

Program Results (f04axfe.r)