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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zpprfs (f07gv)

## Purpose

nag_lapack_zpprfs (f07gv) returns error bounds for the solution of a complex Hermitian positive definite 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] = f07gv(uplo, ap, afp, b, x, 'n', n, 'nrhs_p', nrhs_p)
[x, ferr, berr, info] = nag_lapack_zpprfs(uplo, ap, afp, b, x, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_zpprfs (f07gv) returns the backward errors and estimated bounds on the forward errors for the solution of a complex Hermitian positive definite 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_zpprfs (f07gv) 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 ${U}^{\mathrm{H}}U$, where $U$ is upper triangular.
${\mathbf{uplo}}=\text{'L'}$
The lower triangular part of $A$ is stored and $A$ is factorized as $L{L}^{\mathrm{H}}$, 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 Hermitian positive definite matrix $A$ as supplied to nag_lapack_zpptrf (f07gr).
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 Cholesky factor of $A$ stored in packed form, as returned by nag_lapack_zpptrf (f07gr).
4:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – 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 $B$.
5:     $\mathrm{x}\left(\mathit{ldx},:\right)$ – 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_zpptrs (f07gs).

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the first dimension of the arrays b, x.
$n$, 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_dpprfs (f07gh).

## Example

This example solves the system of equations $AX=B$ using iterative refinement and to compute the forward and backward error bounds, where
 $A= 3.23+0.00i 1.51-1.92i 1.90+0.84i 0.42+2.50i 1.51+1.92i 3.58+0.00i -0.23+1.11i -1.18+1.37i 1.90-0.84i -0.23-1.11i 4.09+0.00i 2.33-0.14i 0.42-2.50i -1.18-1.37i 2.33+0.14i 4.29+0.00i$
and
 $B= 3.93-06.14i 1.48+06.58i 6.17+09.42i 4.65-04.75i -7.17-21.83i -4.91+02.29i 1.99-14.38i 7.64-10.79i .$
Here $A$ is Hermitian positive definite, stored in packed form, and must first be factorized by nag_lapack_zpptrf (f07gr).
```function f07gv_example

fprintf('f07gv example results\n\n');

% Symmetric A in lower packed form
uplo = 'L';
n = int64(4);
ap = [3.23 + 0i    1.51 + 1.92i    1.90 - 0.84i    0.42 - 2.50i ...
3.58 + 0i      -0.23 - 1.11i   -1.18 - 1.37i ...
4.09 + 0.00i    2.33 + 0.14i ...
4.29 + 0.00i];

[L, info] = f07gr( ...
uplo, n, ap);

% RHS
b = [ 3.93 -  6.14i,  1.48 +  6.58i;
6.17 +  9.42i,  4.65 -  4.75i;
-7.17 - 21.83i, -4.91 +  2.29i;
1.99 - 14.38i,  7.64 - 10.79i];

% Solve AX = B

[x, info] = f07gs( ...
uplo, L, b);

% Refine X
[x, ferr, berr, info] = f07gv( ...
uplo, ap, L, b, x);

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

```
```f07gv example results

Solution(s)
1.0000 - 1.0000i  -1.0000 + 2.0000i
-0.0000 + 3.0000i   3.0000 - 4.0000i
-4.0000 - 5.0000i  -2.0000 + 3.0000i
2.0000 + 1.0000i   4.0000 - 5.0000i

Backward errors (machine-dependent)
8.4e-17    7.9e-17
Estimated forward error bounds (machine-dependent)
6.0e-14    7.4e-14
```