F04 Chapter Contents
F04 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF04AHF

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

F04AHF calculates the accurate solution of a set of real linear equations with multiple right-hand sides, $AX=B$, with iterative refinement, where $A$ has been factorized by F03AFF.

## 2  Specification

 SUBROUTINE F04AHF ( N, IR, A, LDA, AA, LDAA, P, B, LDB, EPS, X, LDX, BB, LDBB, K, IFAIL)
 INTEGER N, IR, LDA, LDAA, LDB, LDX, LDBB, K, IFAIL REAL (KIND=nag_wp) A(LDA,N), AA(LDAA,N), P(N), B(LDB,IR), EPS, X(LDX,IR), BB(LDBB,IR)

## 3  Description

To solve a set of real linear equations $AX=B$, F04AHF must be preceded by a call to F03AFF which computes an $LU$ factorization of $A$ with partial pivoting, $PA=LU$, where $P$ is a permutation matrix, $L$ is lower triangular and $U$ is unit upper triangular. An approximation to $X$ is found by forward and backward substitution. The residual matrix $R=B-AX$ is then calculated using additional precision, and a correction $D$ to $X$ is found by solving $LUD=PR$. $X$ is replaced by $X+D$, and this iterative refinement of the solution is repeated until full machine accuracy has been obtained.

## 4  References

Wilkinson J H and Reinsch C (1971) Handbook for Automatic Computation II, Linear Algebra Springer–Verlag

## 5  Parameters

1:     N – INTEGERInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
2:     IR – INTEGERInput
On entry: $r$, the number of right-hand sides.
3:     A(LDA,N) – REAL (KIND=nag_wp) arrayInput
On entry: the $n$ by $n$ matrix $A$.
4:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F04AHF is called.
Constraint: ${\mathbf{LDA}}\ge {\mathbf{N}}$.
5:     AA(LDAA,N) – REAL (KIND=nag_wp) arrayInput
On entry: details of the $LU$ factorization, as returned by F03AFF.
6:     LDAA – INTEGERInput
On entry: the first dimension of the array AA as declared in the (sub)program from which F04AHF is called.
Constraint: ${\mathbf{LDAA}}\ge {\mathbf{N}}$.
7:     P(N) – REAL (KIND=nag_wp) arrayInput
On entry: details of the row interchanges as returned by F03AFF.
8:     B(LDB,IR) – REAL (KIND=nag_wp) arrayInput
On entry: the $n$ by $r$ right-hand side matrix $B$.
9:     LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F04AHF is called.
Constraint: ${\mathbf{LDB}}\ge {\mathbf{N}}$.
10:   EPS – REAL (KIND=nag_wp)Input
On entry: must be set to the value of the machine precision.
11:   X(LDX,IR) – REAL (KIND=nag_wp) arrayOutput
On exit: the $n$ by $r$ solution matrix $X$.
12:   LDX – INTEGERInput
On entry: the first dimension of the array X as declared in the (sub)program from which F04AHF is called.
Constraint: ${\mathbf{LDX}}\ge {\mathbf{N}}$.
13:   BB(LDBB,IR) – REAL (KIND=nag_wp) arrayOutput
On exit: the final $n$ by $r$ residual matrix $R=B-AX$.
14:   LDBB – INTEGERInput
On entry: the first dimension of the array BB as declared in the (sub)program from which F04AHF is called.
Constraint: ${\mathbf{LDBB}}\ge {\mathbf{N}}$.
15:   K – INTEGEROutput
On exit: the number of iterations needed in the refinement process.
16:   IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit: ${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
${\mathbf{IFAIL}}=1$
The matrix $A$ is too ill-conditioned to produce a correctly rounded solution.

## 7  Accuracy

The computed solutions should be correct to full machine accuracy. For a detailed error analysis see page 106 of Wilkinson and Reinsch (1971).

The time taken by F04AHF is approximately proportional to ${n}^{2}r$.

## 9  Example

This example solves the set of linear equations $AX=B$ where
 $A= 33 16 72 -24 -10 -57 -8 -4 -17 and B= -359 281 85 .$

### 9.1  Program Text

Program Text (f04ahfe.f90)

### 9.2  Program Data

Program Data (f04ahfe.d)

### 9.3  Program Results

Program Results (f04ahfe.r)