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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zsprfs (f07qv)

## Purpose

nag_lapack_zsprfs (f07qv) returns error bounds for the solution of a complex symmetric system of linear equations with multiple right-hand sides, $AX=B$, using packed storage. It improves the solution by iterative refinement, in order to reduce the backward error as much as possible.

## Syntax

[x, ferr, berr, info] = f07qv(uplo, ap, afp, ipiv, b, x, 'n', n, 'nrhs_p', nrhs_p)
[x, ferr, berr, info] = nag_lapack_zsprfs(uplo, ap, afp, ipiv, b, x, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_zsprfs (f07qv) returns the backward errors and estimated bounds on the forward errors for the solution of a complex symmetric system of linear equations with multiple right-hand sides $AX=B$, using packed storage. 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_zsprfs (f07qv) 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 $PUD{U}^{\mathrm{T}}{P}^{\mathrm{T}}$, where $U$ is upper triangular.
${\mathbf{uplo}}=\text{'L'}$
The lower triangular part of $A$ is stored and $A$ is factorized as $PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}$, where $L$ is lower triangular.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
2:     $\mathrm{ap}\left(:\right)$ – complex array
The dimension of the array ap must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$
The $n$ by $n$ original symmetric matrix $A$ as supplied to nag_lapack_zsptrf (f07qr).
3:     $\mathrm{afp}\left(:\right)$ – complex array
The dimension of the array afp must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$
The factorization of $A$ stored in packed form, as returned by nag_lapack_zsptrf (f07qr).
4:     $\mathrm{ipiv}\left(:\right)$int64int32nag_int array
The dimension of the array ipiv must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
Details of the interchanges and the block structure of $D$, as returned by nag_lapack_zsptrf (f07qr).
5:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – complex 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 $B$.
6:     $\mathrm{x}\left(\mathit{ldx},:\right)$ – complex 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_zsptrs (f07qs).

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the first dimension of the arrays b, x and the dimension of the array ipiv.
$n$, 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:     $\mathrm{x}\left(\mathit{ldx},:\right)$ – complex 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 $16{n}^{2}$ real floating-point operations. Each step of iterative refinement involves an additional $24{n}^{2}$ real 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 $5$ and never more than $11$. Each solution involves approximately $8{n}^{2}$ real operations.
The real analogue of this function is nag_lapack_dsprfs (f07ph).

## Example

This example solves the system of equations $AX=B$ using iterative refinement and to compute the forward and backward error bounds, where
 $A= -0.39-0.71i 5.14-0.64i -7.86-2.96i 3.80+0.92i 5.14-0.64i 8.86+1.81i -3.52+0.58i 5.32-1.59i -7.86-2.96i -3.52+0.58i -2.83-0.03i -1.54-2.86i 3.80+0.92i 5.32-1.59i -1.54-2.86i -0.56+0.12i$
and
 $B= -55.64+41.22i -19.09-35.97i -48.18+66.00i -12.08-27.02i -0.49-01.47i 6.95+20.49i -6.43+19.24i -4.59-35.53i .$
Here $A$ is symmetric, stored in packed form, and must first be factorized by nag_lapack_zsptrf (f07qr).
```function f07qv_example

fprintf('f07qv example results\n\n');

% Solve Ax = B, where A is complex symmetric matrix such that the
% lower triangular part is stored in packed format
uplo = 'L';
n = int64(4);
ap = [ -0.39 -  0.71i,   5.14 -  0.64i,  -7.86 - 2.96i,   3.80 + 0.92i, ...
8.86 +  1.81i,  -3.52 +  0.58i,   5.32 - 1.59i,   ...
-2.83 -  0.03i,  -1.54 -  2.86i,  ...
-0.56 + 0.12i];
b  = [-55.64 + 41.22i, -19.09 - 35.97i;
-48.18 + 66.00i, -12.08 - 27.02i;
-0.49 -  1.47i,   6.95 + 20.49i;
-6.43 + 19.24i,  -4.59 - 35.53i];

% Factorize
[apf, ipiv, info] = f07qr( ...
uplo, n, ap);
% Solve
[x, info] = f07qs(uplo, apf, ipiv, b);

% Refine
[x, ferr, berr, info] = f07qv( ...
uplo, ap, apf, ipiv, b, x);

disp('Solution(s)');
disp(x);

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

```
```f07qv example results

Solution(s)
1.0000 - 1.0000i  -2.0000 - 1.0000i
-2.0000 + 5.0000i   1.0000 - 3.0000i
3.0000 - 2.0000i   3.0000 + 2.0000i
-4.0000 + 3.0000i  -1.0000 + 1.0000i

Backward errors (machine-dependent)
4.0e-17    7.3e-17
Estimated forward error bounds (machine-dependent)
1.2e-14    1.2e-14
```

Chapter Contents
Chapter Introduction
NAG Toolbox

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