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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dporfs (f07fh)

## Purpose

nag_lapack_dporfs (f07fh) returns error bounds for the solution of a real symmetric positive definite system of linear equations with multiple right-hand sides, $AX=B$. It improves the solution by iterative refinement, in order to reduce the backward error as much as possible.

## Syntax

[x, ferr, berr, info] = f07fh(uplo, a, af, b, x, 'n', n, 'nrhs_p', nrhs_p)
[x, ferr, berr, info] = nag_lapack_dporfs(uplo, a, af, b, x, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_dporfs (f07fh) returns the backward errors and estimated bounds on the forward errors for the solution of a real symmetric positive definite system of linear equations with multiple right-hand sides $AX=B$. The function handles each right-hand side vector (stored as a column of the matrix $B$) independently, so we describe the function of nag_lapack_dporfs (f07fh) in terms of a single right-hand side $b$ and solution $x$.
Given a computed solution $x$, the function computes the component-wise backward error $\beta$. This is the size of the smallest relative perturbation in each element of $A$ and $b$ such that $x$ is the exact solution of a perturbed system
 $A+δAx=b+δb δaij≤βaij and δbi≤βbi .$
Then the function estimates a bound for the component-wise forward error in the computed solution, defined by:
 $maxixi-x^i/maxixi$
where $\stackrel{^}{x}$ is the true solution.
For details of the method, see the F07 Chapter Introduction.

## References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{uplo}$ – string (length ≥ 1)
Specifies whether the upper or lower triangular part of $A$ is stored and how $A$ is to be factorized.
${\mathbf{uplo}}=\text{'U'}$
The upper triangular part of $A$ is stored and $A$ is factorized as ${U}^{\mathrm{T}}U$, where $U$ is upper triangular.
${\mathbf{uplo}}=\text{'L'}$
The lower triangular part of $A$ is stored and $A$ is factorized as $L{L}^{\mathrm{T}}$, where $L$ is lower triangular.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
2:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double array
The first dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The $n$ by $n$ original symmetric positive definite matrix $A$ as supplied to nag_lapack_dpotrf (f07fd).
3:     $\mathrm{af}\left(\mathit{ldaf},:\right)$ – double array
The first dimension of the array af must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array af must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The Cholesky factor of $A$, as returned by nag_lapack_dpotrf (f07fd).
4:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – double array
The first dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs_p}}\right)$.
The $n$ by $r$ right-hand side matrix $B$.
5:     $\mathrm{x}\left(\mathit{ldx},:\right)$ – double array
The first dimension of the array x must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array x must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs_p}}\right)$.
The $n$ by $r$ solution matrix $X$, as returned by nag_lapack_dpotrs (f07fe).

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the first dimension of the arrays a, af, b, x and the second dimension of the arrays a, af.
$n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
2:     $\mathrm{nrhs_p}$int64int32nag_int scalar
Default: the second dimension of the arrays b, x. (An error is raised if these dimensions are not equal.)
$r$, the number of right-hand sides.
Constraint: ${\mathbf{nrhs_p}}\ge 0$.

### Output Parameters

1:     $\mathrm{x}\left(\mathit{ldx},:\right)$ – double array
The first dimension of the array x will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array x will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs_p}}\right)$.
The improved solution matrix $X$.
2:     $\mathrm{ferr}\left({\mathbf{nrhs_p}}\right)$ – double array
${\mathbf{ferr}}\left(\mathit{j}\right)$ contains an estimated error bound for the $\mathit{j}$th solution vector, that is, the $\mathit{j}$th column of $X$, for $\mathit{j}=1,2,\dots ,r$.
3:     $\mathrm{berr}\left({\mathbf{nrhs_p}}\right)$ – double array
${\mathbf{berr}}\left(\mathit{j}\right)$ contains the component-wise backward error bound $\beta$ for the $\mathit{j}$th solution vector, that is, the $\mathit{j}$th column of $X$, for $\mathit{j}=1,2,\dots ,r$.
4:     $\mathrm{info}$int64int32nag_int scalar
${\mathbf{info}}=0$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

## Accuracy

The bounds returned in ferr are not rigorous, because they are estimated, not computed exactly; but in practice they almost always overestimate the actual error.

For each right-hand side, computation of the backward error involves a minimum of $4{n}^{2}$ floating-point operations. Each step of iterative refinement involves an additional $6{n}^{2}$ operations. At most five steps of iterative refinement are performed, but usually only one or two steps are required.
Estimating the forward error involves solving a number of systems of linear equations of the form $Ax=b$; the number is usually $4$ or $5$ and never more than $11$. Each solution involves approximately $2{n}^{2}$ operations.
The complex analogue of this function is nag_lapack_zporfs (f07fv).

## Example

This example solves the system of equations $AX=B$ using iterative refinement and to compute the forward and backward error bounds, where
 $A= 4.16 -3.12 0.56 -0.10 -3.12 5.03 -0.83 1.18 0.56 -0.83 0.76 0.34 -0.10 1.18 0.34 1.18 and B= 8.70 8.30 -13.35 2.13 1.89 1.61 -4.14 5.00 .$
Here $A$ is symmetric positive definite and must first be factorized by nag_lapack_dpotrf (f07fd).
```function f07fh_example

fprintf('f07fh example results\n\n');

% Lower triangular part of symmetric matrix A
uplo = 'Lower';
a = [ 4.16,  0,    0,    0;
-3.12,  5.03, 0,    0;
0.56, -0.83, 0.76, 0;
-0.10,  1.18, 0.34, 1.18];

[L, info] = f07fd( ...
uplo, a);

% Rhs
b = [  8.70, 8.30;
-13.35, 2.13;
1.89, 1.61;
-4.14, 5.00];

% Solve
[x, info]  = f07fe( ...
uplo, L, b);

% Refine
[x, ferr, berr, info] = f07fh( ...
uplo, a, L, b, x);

[ifail] = x04ca( ...
'General', 'N', x, 'Solution(s)');

fprintf('\nBackward errors (machine-dependent)\n   ')
fprintf('%11.1e', berr);
fprintf('\nEstimated forward error bounds (machine-dependent)\n   ')
fprintf('%11.1e', ferr);
fprintf('\n');

```
```f07fh example results

Solution(s)
1          2
1      1.0000     4.0000
2     -1.0000     3.0000
3      2.0000     2.0000
4     -3.0000     1.0000

Backward errors (machine-dependent)
1.0e-16    5.0e-17
Estimated forward error bounds (machine-dependent)
2.3e-14    2.3e-14
```