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# NAG Toolbox: nag_lapack_zgerfs (f07av)

## Purpose

nag_lapack_zgerfs (f07av) returns error bounds for the solution of a complex system of linear equations with multiple right-hand sides, $AX=B$, ${A}^{\mathrm{T}}X=B$ or ${A}^{\mathrm{H}}X=B$. It improves the solution by iterative refinement, in order to reduce the backward error as much as possible.

## Syntax

[x, ferr, berr, info] = f07av(trans, a, af, ipiv, b, x, 'n', n, 'nrhs_p', nrhs_p)
[x, ferr, berr, info] = nag_lapack_zgerfs(trans, a, af, ipiv, b, x, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_zgerfs (f07av) returns the backward errors and estimated bounds on the forward errors for the solution of a complex system of linear equations with multiple right-hand sides $AX=B$, ${A}^{\mathrm{T}}X=B$ or ${A}^{\mathrm{H}}X=B$. 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_zgerfs (f07av) 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:
 $maxi xi - x^i / maxi xi$
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:     $\mathrm{trans}$ – string (length ≥ 1)
Indicates the form of the linear equations for which $X$ is the computed solution as follows:
${\mathbf{trans}}=\text{'N'}$
The linear equations are of the form $AX=B$.
${\mathbf{trans}}=\text{'T'}$
The linear equations are of the form ${A}^{\mathrm{T}}X=B$.
${\mathbf{trans}}=\text{'C'}$
The linear equations are of the form ${A}^{\mathrm{H}}X=B$.
Constraint: ${\mathbf{trans}}=\text{'N'}$, $\text{'T'}$ or $\text{'C'}$.
2:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex 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 matrix $A$ as supplied to nag_lapack_zgetrf (f07ar).
3:     $\mathrm{af}\left(\mathit{ldaf},:\right)$ – complex 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 $LU$ factorization of $A$, as returned by nag_lapack_zgetrf (f07ar).
4:     $\mathrm{ipiv}\left(:\right)$int64int32nag_int array
The dimension of the array ipiv must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The pivot indices, as returned by nag_lapack_zgetrf (f07ar).
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_zgetrs (f07as).

### 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, 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.

## Further Comments

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 $Ax=b$ or ${A}^{\mathrm{H}}x=b$; 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_dgerfs (f07ah).

## Example

This example solves the system of equations $AX=B$ using iterative refinement and to compute the forward and backward error bounds, where
 $A= -1.34+2.55i 0.28+3.17i -6.39-2.20i 0.72-0.92i -0.17-1.41i 3.31-0.15i -0.15+1.34i 1.29+1.38i -3.29-2.39i -1.91+4.42i -0.14-1.35i 1.72+1.35i 2.41+0.39i -0.56+1.47i -0.83-0.69i -1.96+0.67i$
and
 $B= 26.26+51.78i 31.32-06.70i 6.43-08.68i 15.86-01.42i -5.75+25.31i -2.15+30.19i 1.16+02.57i -2.56+07.55i .$
Here $A$ is nonsymmetric and must first be factorized by nag_lapack_zgetrf (f07ar).
```function f07av_example

fprintf('f07av example results\n\n');

a = [-1.34 + 2.55i,  0.28 + 3.17i, -6.39 - 2.20i,  0.72 - 0.92i;
-0.17 - 1.41i,  3.31 - 0.15i, -0.15 + 1.34i,  1.29 + 1.38i;
-3.29 - 2.39i, -1.91 + 4.42i, -0.14 - 1.35i,  1.72 + 1.35i;
2.41 + 0.39i, -0.56 + 1.47i, -0.83 - 0.69i, -1.96 + 0.67i];
b = [26.26 + 51.78i, 31.32 -  6.70i;
6.43 -  8.68i, 15.86 -  1.42i;
-5.75 + 25.31i, -2.15 + 30.19i;
1.16 +  2.57i, -2.56 +  7.55i];

% Factorize a and place results in af
[af, ipiv, info] = f07ar(a);

% Solve Ax = B
trans = 'N';
[x, info] = f07as( ...
trans, a, ipiv, b);

% Improve solution, and compute backward errors and
% estimated bounds on forward errors
[x, ferr, berr, info] = f07av( ...
trans, a, af, ipiv, b, x);

fprintf('Solution(s)\n ');
fprintf('%10d         ',[1:size(b,2)]);
fprintf('\n');
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');

```
```f07av example results

Solution(s)
1                  2
1.0000 + 1.0000i  -1.0000 - 2.0000i
2.0000 - 3.0000i   5.0000 + 1.0000i
-4.0000 - 5.0000i  -3.0000 + 4.0000i
0.0000 + 6.0000i   2.0000 - 3.0000i

Backward errors (machine-dependent)
4.1e-17    5.1e-17
Estimated forward error bounds (machine-dependent)
5.7e-14    7.4e-14
```

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Chapter Contents
Chapter Introduction
NAG Toolbox

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