F04 Chapter Contents
F04 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF04AFF

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

F04AFF calculates the accurate solution of a set of real symmetric positive definite linear equations with multiple right-hand sides, $AX=B$, with iterative refinement, where $A$ has been factorized by F03AEF.

## 2  Specification

 SUBROUTINE F04AFF ( N, IR, A, LDA, P, B, LDB, EPS, X, LDX, BB, LDBB, K, IFAIL)
 INTEGER N, IR, LDA, LDB, LDX, LDBB, K, IFAIL REAL (KIND=nag_wp) A(LDA,N), P(N), B(LDB,IR), EPS, X(LDX,IR), BB(LDBB,IR)

## 3  Description

To solve a set of real linear equations $AX=B$ where $A$ is symmetric positive definite, F04AFF must be preceded by a call to F03AEF which computes a Cholesky factorization of $A$ as $A=L{L}^{\mathrm{T}}$, where $L$ is lower triangular. An approximation to $X$ is then 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 $L{L}^{\mathrm{T}}D=R$. $X$ 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 upper triangle of the $n$ by $n$ positive definite symmetric matrix $A$, and the subdiagonal elements of its Cholesky factor $L$, as returned by F03AEF.
On exit: is used as internal workspace, but is restored on exit.
4:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F04AFF is called.
Constraint: ${\mathbf{LDA}}\ge {\mathbf{N}}$.
5:     P(N) – REAL (KIND=nag_wp) arrayInput
On entry: the reciprocals of the diagonal elements of $L$, as returned by F03AEF.
6:     B(LDB,IR) – REAL (KIND=nag_wp) arrayInput
On entry: the $n$ by $r$ right-hand side matrix $B$.
7:     LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F04AFF is called.
Constraint: ${\mathbf{LDB}}\ge {\mathbf{N}}$.
8:     EPS – REAL (KIND=nag_wp)Input
On entry: must be set to the value of the machine precision.
9:     X(LDX,IR) – REAL (KIND=nag_wp) arrayOutput
On exit: the $n$ by $r$ solution matrix $X$.
10:   LDX – INTEGERInput
On entry: the first dimension of the array X as declared in the (sub)program from which F04AFF is called.
Constraint: ${\mathbf{LDX}}\ge {\mathbf{N}}$.
11:   BB(LDBB,IR) – REAL (KIND=nag_wp) arrayOutput
On exit: the final $n$ by $r$ residual matrix $R=B-AX$.
12:   LDBB – INTEGERInput
On entry: the first dimension of the array BB as declared in the (sub)program from which F04AFF is called.
Constraint: ${\mathbf{LDBB}}\ge {\mathbf{N}}$.
13:   K – INTEGEROutput
On exit: the number of iterations needed in the refinement process.
14:   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 39 of Wilkinson and Reinsch (1971).

The time taken by F04AFF is approximately proportional to ${n}^{2}r$.

## 9  Example

This example solves the set of linear equations $AX=B$ where
 $A= 5 7 6 5 7 10 8 7 6 8 10 9 5 7 9 10 and B= 23 32 33 31 .$

### 9.1  Program Text

Program Text (f04affe.f90)

### 9.2  Program Data

Program Data (f04affe.d)

### 9.3  Program Results

Program Results (f04affe.r)