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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
```