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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zgbtrs (f07bs)

## Purpose

nag_lapack_zgbtrs (f07bs) solves a complex band system of linear equations with multiple right-hand sides,
 $AX=B , ATX=B or AHX=B ,$
where $A$ has been factorized by nag_lapack_zgbtrf (f07br).

## Syntax

[b, info] = f07bs(trans, kl, ku, ab, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)
[b, info] = nag_lapack_zgbtrs(trans, kl, ku, ab, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_zgbtrs (f07bs) is used to solve a complex band system of linear equations $AX=B$, ${A}^{\mathrm{T}}X=B$ or ${A}^{\mathrm{H}}X=B$, the function must be preceded by a call to nag_lapack_zgbtrf (f07br) which computes the $LU$ factorization of $A$ as $A=PLU$. The solution is computed by forward and backward substitution.
If ${\mathbf{trans}}=\text{'N'}$, the solution is computed by solving $PLY=B$ and then $UX=Y$.
If ${\mathbf{trans}}=\text{'T'}$, the solution is computed by solving ${U}^{\mathrm{T}}Y=B$ and then ${L}^{\mathrm{T}}{P}^{\mathrm{T}}X=Y$.
If ${\mathbf{trans}}=\text{'C'}$, the solution is computed by solving ${U}^{\mathrm{H}}Y=B$ and then ${L}^{\mathrm{H}}{P}^{\mathrm{T}}X=Y$.

## 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 equations.
${\mathbf{trans}}=\text{'N'}$
$AX=B$ is solved for $X$.
${\mathbf{trans}}=\text{'T'}$
${A}^{\mathrm{T}}X=B$ is solved for $X$.
${\mathbf{trans}}=\text{'C'}$
${A}^{\mathrm{H}}X=B$ is solved for $X$.
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 $2×{\mathbf{kl}}+{\mathbf{ku}}+1$.
The second dimension of the array ab must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The $LU$ factorization of $A$, as returned by nag_lapack_zgbtrf (f07br).
5:     $\mathrm{ipiv}\left(:\right)$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).
6:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – 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 $B$.

### 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 array b.
$r$, the number of right-hand sides.
Constraint: ${\mathbf{nrhs_p}}\ge 0$.

### Output Parameters

1:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – complex array
The first dimension of the array b will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array b will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs_p}}\right)$.
The $n$ by $r$ solution matrix $X$.
2:     $\mathrm{info}$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

For each right-hand side vector $b$, the computed solution $x$ is the exact solution of a perturbed system of equations $\left(A+E\right)x=b$, where
 $E≤ckεLU ,$
$c\left(k\right)$ is a modest linear function of $k={k}_{l}+{k}_{u}+1$, and $\epsilon$ is the machine precision. This assumes $k\ll n$.
If $\stackrel{^}{x}$ is the true solution, then the computed solution $x$ satisfies a forward error bound of the form
 $x-x^∞ x∞ ≤ckcondA,xε$
where $\mathrm{cond}\left(A,x\right)={‖\left|{A}^{-1}\right|\left|A\right|\left|x\right|‖}_{\infty }/{‖x‖}_{\infty }\le \mathrm{cond}\left(A\right)={‖\left|{A}^{-1}\right|\left|A\right|‖}_{\infty }\le {\kappa }_{\infty }\left(A\right)$.
Note that $\mathrm{cond}\left(A,x\right)$ can be much smaller than $\mathrm{cond}\left(A\right)$, and $\mathrm{cond}\left({A}^{\mathrm{H}}\right)$ (which is the same as $\mathrm{cond}\left({A}^{\mathrm{T}}\right)$) can be much larger (or smaller) than $\mathrm{cond}\left(A\right)$.
Forward and backward error bounds can be computed by calling nag_lapack_zgbrfs (f07bv), and an estimate for ${\kappa }_{\infty }\left(A\right)$ can be obtained by calling nag_lapack_zgbcon (f07bu) with ${\mathbf{norm_p}}=\text{'I'}$.

The total number of real floating-point operations is approximately $8n\left(2{k}_{l}+{k}_{u}\right)r$, assuming $n\gg {k}_{l}$ and $n\gg {k}_{u}$.
This function may be followed by a call to nag_lapack_zgbrfs (f07bv) to refine the solution and return an error estimate.
The real analogue of this function is nag_lapack_dgbtrs (f07be).

## Example

This example solves the system of equations $AX=B$, 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
 $B= -1.06+21.50i 12.85+02.84i -22.72-53.90i -70.22+21.57i 28.24-38.60i -20.70-31.23i -34.56+16.73i 26.01+31.97i .$
Here $A$ is nonsymmetric and is treated as a band matrix, which must first be factorized by nag_lapack_zgbtrf (f07br).
```function f07bs_example

fprintf('f07bs example results\n\n');

m = int64(4);
kl = int64(1);
ku = int64(2);
ab = [ 0    + 0i,      0    +  0i,     0    + 0i,     0    + 0i;
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];

b = [ -1.06 + 21.5i,  12.85 +  2.84i;
-22.72 - 53.9i, -70.22 + 21.57i;
28.24 - 38.6i, -20.73 -  1.23i;
-34.56 + 16.73i, 26.01 + 31.97i];

% Factorize
[abf, ipiv, info] = f07br( ...
m, kl, ku, ab);

%Solve
trans = 'N';
[x, info] = f07bs( ...
trans, kl, ku, abf, ipiv, b);

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

```
```f07bs example results

Solution(s)
-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

```