F04 Chapter Contents
F04 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF04ATF

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

F04ATF calculates the accurate solution of a set of real linear equations with a single right-hand side, using an $LU$ factorization with partial pivoting, and iterative refinement.

## 2  Specification

 SUBROUTINE F04ATF ( A, LDA, B, N, C, AA, LDAA, WKS1, WKS2, IFAIL)
 INTEGER LDA, N, LDAA, IFAIL REAL (KIND=nag_wp) A(LDA,*), B(*), C(N), AA(LDAA,N), WKS1(N), WKS2(N)

## 3  Description

Given a set of real linear equations, $Ax=b$, the routine first 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 in $Ly=Pb$ and $Ux=y$. The residual vector $r=b-Ax$ is then calculated using additional precision, and a correction $d$ to $x$ is found by solving $LUd=r$. $x$ is replaced by $x+d$, and this iterative refinement of the solution is repeated until full machine accuracy is obtained.

## 4  References

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

## 5  Parameters

1:     A(LDA,$*$) – REAL (KIND=nag_wp) arrayInput
Note: the second dimension of the array A must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: the $n$ by $n$ matrix $A$.
2:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F04ATF is called.
Constraint: ${\mathbf{LDA}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
3:     B($*$) – REAL (KIND=nag_wp) arrayInput
Note: the dimension of the array B must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: the right-hand side vector $b$.
4:     N – INTEGERInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
5:     C(N) – REAL (KIND=nag_wp) arrayOutput
On exit: the solution vector $x$.
6:     AA(LDAA,${\mathbf{N}}$) – REAL (KIND=nag_wp) arrayOutput
Note: the second dimension of the array AA must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On exit: the triangular factors $L$ and $U$, except that the unit diagonal elements of $U$ are not stored.
7:     LDAA – INTEGERInput
On entry: the first dimension of the array AA as declared in the (sub)program from which F04ATF is called.
Constraint: ${\mathbf{LDAA}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
8:     WKS1(N) – REAL (KIND=nag_wp) arrayWorkspace
9:     WKS2(N) – REAL (KIND=nag_wp) arrayWorkspace
10:   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 singular, possibly due to rounding errors.
${\mathbf{IFAIL}}=2$
Iterative refinement fails to improve the solution, i.e., the matrix $A$ is too ill-conditioned.
${\mathbf{IFAIL}}=3$
 On entry, ${\mathbf{N}}<0$, or ${\mathbf{LDA}}<\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$, or ${\mathbf{LDAA}}<\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.

## 7  Accuracy

The computed solutions should be correct to full machine accuracy. For a detailed error analysis see page 107 of Wilkinson and Reinsch (1971).

The time taken by F04ATF is approximately proportional to ${n}^{3}$.
The routine must not be called with the same name for parameters B and C.

## 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 (f04atfe.f90)

### 9.2  Program Data

Program Data (f04atfe.d)

### 9.3  Program Results

Program Results (f04atfe.r)