# NAG Library Routine Document

## 1Purpose

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.

## 2Specification

Fortran Interface
 Subroutine f04abf ( a, lda, b, ldb, n, m, c, ldc, bb, ldbb,
 Integer, Intent (In) :: lda, ldb, n, m, ldc, ldbb Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: b(ldb,*) Real (Kind=nag_wp), Intent (Inout) :: a(lda,*), c(ldc,m), bb(ldbb,m) Real (Kind=nag_wp), Intent (Out) :: wkspce(n)
#include nagmk26.h
 void f04abf_ (double a[], const Integer *lda, const double b[], const Integer *ldb, const Integer *n, const Integer *m, double c[], const Integer *ldc, double wkspce[], double bb[], const Integer *ldbb, Integer *ifail)

## 3Description

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

## 5Arguments

1:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – 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:     $\mathbf{lda}$ – 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:     $\mathbf{b}\left({\mathbf{ldb}},*\right)$ – 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:     $\mathbf{ldb}$ – 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:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
6:     $\mathbf{m}$ – IntegerInput
On entry: $m$, the number of right-hand sides.
Constraint: ${\mathbf{m}}\ge 0$.
7:     $\mathbf{c}\left({\mathbf{ldc}},{\mathbf{m}}\right)$ – 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:     $\mathbf{ldc}$ – 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:     $\mathbf{wkspce}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
10:   $\mathbf{bb}\left({\mathbf{ldbb}},{\mathbf{m}}\right)$ – 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:   $\mathbf{ldbb}$ – 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:   $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, 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 ${\mathbf{ldbb}}<\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

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

## 8Parallelism and Performance

f04abf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f04abf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

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.

## 10Example

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

### 10.1Program Text

Program Text (f04abfe.f90)

### 10.2Program Data

Program Data (f04abfe.d)

### 10.3Program Results

Program Results (f04abfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017