F04 Chapter Contents
F04 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF04ABF

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

F04ABF calculates the accurate solution of a set of real symmetric positive definite linear equations with multiple right-hand sides, using a Cholesky factorization and iterative refinement.

## 2  Specification

 SUBROUTINE F04ABF ( A, LDA, B, LDB, N, M, C, LDC, WKSPCE, BB, LDBB, IFAIL)
 INTEGER LDA, LDB, N, M, LDC, LDBB, IFAIL REAL (KIND=nag_wp) A(LDA,*), B(LDB,*), C(LDC,M), WKSPCE(N), BB(LDBB,M)

## 3  Description

Given a set of real linear equations $AX=B$, where $A$ is symmetric positive definite, F04ABF first computes a Cholesky factorization of $A$ as $A=L{L}^{\mathrm{T}}$, where $L$ is lower 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 $L{L}^{\mathrm{T}}D=R$. $X$ 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:     A(LDA,$*$) – REAL (KIND=nag_wp) arrayInput/Output
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 upper triangle of the $n$ by $n$ positive definite symmetric matrix $A$. The elements of the array below the diagonal need not be set.
On exit: the elements of the array below the diagonal are overwritten; the upper triangle of $A$ is unchanged.
2:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F04ABF is called.
Constraint: ${\mathbf{LDA}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
3:     B(LDB,$*$) – REAL (KIND=nag_wp) arrayInput
Note: the second dimension of the array B must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{M}}\right)$.
On entry: the $n$ by $m$ right-hand side matrix $B$.
4:     LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F04ABF is called.
Constraint: ${\mathbf{LDB}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
5:     N – INTEGERInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
6:     M – INTEGERInput
On entry: $m$, the number of right-hand sides.
Constraint: ${\mathbf{M}}\ge 0$.
7:     C(LDC,${\mathbf{M}}$) – REAL (KIND=nag_wp) arrayOutput
Note: the second dimension of the array C must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{M}}\right)$.
On exit: the $n$ by $m$ solution matrix $X$.
8:     LDC – INTEGERInput
On entry: the first dimension of the array C as declared in the (sub)program from which F04ABF is called.
Constraint: ${\mathbf{LDC}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
9:     WKSPCE(N) – REAL (KIND=nag_wp) arrayWorkspace
10:   BB(LDBB,${\mathbf{M}}$) – REAL (KIND=nag_wp) arrayOutput
Note: the second dimension of the array BB must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{M}}\right)$.
On exit: the final $n$ by $m$ residual matrix $R=B-AX$.
11:   LDBB – INTEGERInput
On entry: the first dimension of the array BB as declared in the (sub)program from which F04ABF is called.
Constraint: ${\mathbf{LDBB}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
12:   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 not positive definite, 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 ${\mathbf{M}}<0$, or ${\mathbf{LDA}}<\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$, or ${\mathbf{LDB}}<\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$, or ${\mathbf{LDC}}<\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$, or ${\mathbf{LDBB}}<\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.

## 7  Accuracy

The computed solutions should be correct to full machine accuracy. For a detailed error analysis see page 39 of Wilkinson and Reinsch (1971).

The time taken by F04ABF is approximately proportional to ${n}^{3}$.
If there is only one right-hand side, it is simpler to use F04ASF.

## 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 (f04abfe.f90)

### 9.2  Program Data

Program Data (f04abfe.d)

### 9.3  Program Results

Program Results (f04abfe.r)