# NAG Library Routine Document

## 1Purpose

f04asf calculates the accurate solution of a set of real symmetric positive definite linear equations with a single right-hand side, $Ax=b$, using a Cholesky factorization and iterative refinement.

## 2Specification

Fortran Interface
 Subroutine f04asf ( a, lda, b, n, c, wk1, wk2,
 Integer, Intent (In) :: lda, n Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: b(max(1,n)) Real (Kind=nag_wp), Intent (Inout) :: a(lda,*) Real (Kind=nag_wp), Intent (Out) :: c(max(1,n)), wk1(max(1,n)), wk2(max(1,n))
#include <nagmk26.h>
 void f04asf_ (double a[], const Integer *lda, const double b[], const Integer *n, double c[], double wk1[], double wk2[], Integer *ifail)

## 3Description

Given a set of real linear equations $Ax=b$, where $A$ is a symmetric positive definite matrix, f04asf 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 vector $r=b-Ax$ is then calculated using additional precision and a correction $d$ to $x$ is found by solving $L{L}^{\mathrm{T}}d=r$. $x$ is then replaced by $x+d$, and this iterative refinement of the solution is repeated until machine accuracy is obtained.

## 4References

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 ${\mathbf{a}}$ is unchanged.
2:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f04asf is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
3:     $\mathbf{b}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)\right)$ – 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:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
5:     $\mathbf{c}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the solution vector $x$.
6:     $\mathbf{wk1}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)\right)$ – Real (Kind=nag_wp) arrayWorkspace
7:     $\mathbf{wk2}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)\right)$ – Real (Kind=nag_wp) arrayWorkspace
8:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . 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  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  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:
${\mathbf{ifail}}=1$
Matrix $A$ is not positive definite.
${\mathbf{ifail}}=2$
Matrix $A$ is too ill-conditioned; iterative refinement fails to improve the solution.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{lda}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
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

f04asf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f04asf 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 f04asf is approximately proportional to ${n}^{3}$.
The routine must not be called with the same name for arguments b and c.

## 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 (f04asfe.f90)

### 10.2Program Data

Program Data (f04asfe.d)

### 10.3Program Results

Program Results (f04asfe.r)