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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zgbrfs (f07bv)

## Purpose

nag_lapack_zgbrfs (f07bv) returns error bounds for the solution of a complex band 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] = f07bv(trans, kl, ku, ab, afb, ipiv, b, x, 'n', n, 'nrhs_p', nrhs_p)
[x, ferr, berr, info] = nag_lapack_zgbrfs(trans, kl, ku, ab, afb, ipiv, b, x, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_zgbrfs (f07bv) returns the backward errors and estimated bounds on the forward errors for the solution of a complex band 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_zgbrfs (f07bv) 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:     $\mathrm{trans}$ – 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:     $\mathrm{kl}$int64int32nag_int scalar
${k}_{l}$, the number of subdiagonals within the band of the matrix $A$.
Constraint: ${\mathbf{kl}}\ge 0$.
3:     $\mathrm{ku}$int64int32nag_int scalar
${k}_{u}$, the number of superdiagonals within the band of the matrix $A$.
Constraint: ${\mathbf{ku}}\ge 0$.
4:     $\mathrm{ab}\left(\mathit{ldab},:\right)$ – complex array
The first dimension of the array ab must be at least ${\mathbf{kl}}+{\mathbf{ku}}+1$.
The second dimension of the array ab must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The original $n$ by $n$ band matrix $A$ as supplied to nag_lapack_zgbtrf (f07br).
The matrix is stored in rows $1$ to ${k}_{l}+{k}_{u}+1$, more precisely, the element ${A}_{ij}$ must be stored in
 $abku+1+i-jj for ​max1,j-ku≤i≤minn,j+kl.$
See Further Comments in nag_lapack_zgbsv (f07bn) for further details.
5:     $\mathrm{afb}\left(\mathit{ldafb},:\right)$ – complex array
The first dimension of the array afb must be at least $2×{\mathbf{kl}}+{\mathbf{ku}}+1$.
The second dimension of the array afb 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_zgbtrf (f07br).
6:     $\mathrm{ipiv}\left(:\right)$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_zgbtrf (f07br).
7:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – 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 $B$.
8:     $\mathrm{x}\left(\mathit{ldx},:\right)$ – 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_zgbtrs (f07bs).

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the second dimension of the array ab.
$n$, 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.
$r$, 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 $16n\left({k}_{l}+{k}_{u}\right)$ real floating-point operations. Each step of iterative refinement involves an additional $8n\left(4{k}_{l}+3{k}_{u}\right)$ real operations. This assumes $n\gg {k}_{l}$ and $n\gg {k}_{u}$. 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 $8n\left(2{k}_{l}+{k}_{u}\right)$ real operations.
The real analogue of this function is nag_lapack_dgbrfs (f07bh).

## Example

This example solves the system of equations $AX=B$ using iterative refinement and to compute the forward and backward error bounds, where
 $A= -1.65+2.26i -2.05-0.85i 0.97-2.84i 0.00+0.00i 0.00+6.30i -1.48-1.75i -3.99+4.01i 0.59-0.48i 0.00+0.00i -0.77+2.83i -1.06+1.94i 3.33-1.04i 0.00+0.00i 0.00+0.00i 4.48-1.09i -0.46-1.72i$
and
 $B= -1.06+21.50i 12.85+02.84i -22.72-53.90i -70.22+21.57i 28.24-38.60i -20.73-01.23i -34.56+16.73i 26.01+31.97i .$
Here $A$ is nonsymmetric and is treated as a band matrix, which must first be factorized by nag_lapack_zgbtrf (f07br).
```function f07bv_example

fprintf('f07bv example results\n\n');

m  = int64(4);
kl = int64(1);
ku = int64(2);
ab = [ 0    + 0i,     0    + 0i,     0.97 - 2.84i,  0.59 - 0.48i;
0    + 0i,    -2.05 - 0.85i, -3.99 + 4.01i,  3.33 - 1.04i;
-1.65 + 2.26i, -1.48 - 1.75i, -1.06 + 1.94i, -0.46 - 1.72i;
0    + 6.3i,  -0.77 + 2.83i,  4.48 - 1.09i,  0    + 0i];

% Convert ab to full representation a
[a, ab, ifail] = f01zd( ...
'u', kl, ku, complex(zeros(m, m)), ab);

% Exact Solution
nrhs = 2;
y = [ -3 + 2i,   1 + 6i;
1 - 7i,  -7 - 4i;
-5 + 4i,   3 + 5i;
6 - 8i,  -8 + 2i];

% Evaluate RHS
b = a*y;

% Factorize
afb = [complex(zeros(kl,m)); ab];
[afb, ipiv, info]     = f07br( ...
m, kl, ku, afb);
% Solve
trans = 'N';
[x, info]             = f07bs( ...
trans, kl, ku, afb, ipiv, b);

% Iterative refinement
[x, ferr, berr, info] = f07bv( ...
trans, kl, ku, ab, afb, ipiv, b, x);

fprintf('Refined solution:\n');
disp(x);

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

```
```f07bv example results

Refined solution:
-3.0000 + 2.0000i   1.0000 + 6.0000i
1.0000 - 7.0000i  -7.0000 - 4.0000i
-5.0000 + 4.0000i   3.0000 + 5.0000i
6.0000 - 8.0000i  -8.0000 + 2.0000i

Backward errors (machine dependent)
5.4e-17    8.4e-17
Estimated forward error bounds
3.6e-14    4.4e-14
```