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# NAG Toolbox: nag_lapack_ztbtrs (f07vs)

## Purpose

nag_lapack_ztbtrs (f07vs) solves a complex triangular band system of linear equations with multiple right-hand sides, $AX=B$, ${A}^{\mathrm{T}}X=B$ or ${A}^{\mathrm{H}}X=B$.

## Syntax

[b, info] = f07vs(uplo, trans, diag, kd, ab, b, 'n', n, 'nrhs_p', nrhs_p)
[b, info] = nag_lapack_ztbtrs(uplo, trans, diag, kd, ab, b, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_ztbtrs (f07vs) solves a complex triangular band system of linear equations $AX=B$, ${A}^{\mathrm{T}}X=B$ or ${A}^{\mathrm{H}}X=B$.

## References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Higham N J (1989) The accuracy of solutions to triangular systems SIAM J. Numer. Anal. 26 1252–1265

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{uplo}$ – string (length ≥ 1)
Specifies whether $A$ is upper or lower triangular.
${\mathbf{uplo}}=\text{'U'}$
$A$ is upper triangular.
${\mathbf{uplo}}=\text{'L'}$
$A$ is lower triangular.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
2:     $\mathrm{trans}$ – string (length ≥ 1)
Indicates the form of the equations.
${\mathbf{trans}}=\text{'N'}$
The equations are of the form $AX=B$.
${\mathbf{trans}}=\text{'T'}$
The equations are of the form ${A}^{\mathrm{T}}X=B$.
${\mathbf{trans}}=\text{'C'}$
The equations are of the form ${A}^{\mathrm{H}}X=B$.
Constraint: ${\mathbf{trans}}=\text{'N'}$, $\text{'T'}$ or $\text{'C'}$.
3:     $\mathrm{diag}$ – string (length ≥ 1)
Indicates whether $A$ is a nonunit or unit triangular matrix.
${\mathbf{diag}}=\text{'N'}$
$A$ is a nonunit triangular matrix.
${\mathbf{diag}}=\text{'U'}$
$A$ is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be $1$.
Constraint: ${\mathbf{diag}}=\text{'N'}$ or $\text{'U'}$.
4:     $\mathrm{kd}$int64int32nag_int scalar
${k}_{d}$, the number of superdiagonals of the matrix $A$ if ${\mathbf{uplo}}=\text{'U'}$, or the number of subdiagonals if ${\mathbf{uplo}}=\text{'L'}$.
Constraint: ${\mathbf{kd}}\ge 0$.
5:     $\mathrm{ab}\left(\mathit{ldab},:\right)$ – complex array
The first dimension of the array ab must be at least ${\mathbf{kd}}+1$.
The second dimension of the array ab must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The $n$ by $n$ triangular band matrix $A$.
The matrix is stored in rows $1$ to ${k}_{d}+1$, more precisely,
• if ${\mathbf{uplo}}=\text{'U'}$, the elements of the upper triangle of $A$ within the band must be stored with element ${A}_{ij}$ in ${\mathbf{ab}}\left({k}_{d}+1+i-j,j\right)\text{​ for ​}\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j-{k}_{d}\right)\le i\le j$;
• if ${\mathbf{uplo}}=\text{'L'}$, the elements of the lower triangle of $A$ within the band must be stored with element ${A}_{ij}$ in ${\mathbf{ab}}\left(1+i-j,j\right)\text{​ for ​}j\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+{k}_{d}\right)\text{.}$
If ${\mathbf{diag}}=\text{'U'}$, the diagonal elements of $A$ are assumed to be $1$, and are not referenced.
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

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_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.
W  ${\mathbf{info}}>0$
Element $_$ of the diagonal is exactly zero. $A$ is singular and the solution has not been computed.

## Accuracy

The solutions of triangular systems of equations are usually computed to high accuracy. See Higham (1989).
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
 $E≤ckεA ,$
$c\left(k\right)$ is a modest linear function of $k$, and $\epsilon$ is the machine precision.
If $\stackrel{^}{x}$ is the true solution, then the computed solution $x$ satisfies a forward error bound of the form
 $x-x^∞ x∞ ≤ckcondA,xε , provided ckcondA,xε<1 ,$
where $\mathrm{cond}\left(A,x\right)={‖\left|{A}^{-1}\right|\left|A\right|\left|x\right|‖}_{\infty }/{‖x‖}_{\infty }$.
Note that $\mathrm{cond}\left(A,x\right)\le \mathrm{cond}\left(A\right)={‖\left|{A}^{-1}\right|\left|A\right|‖}_{\infty }\le {\kappa }_{\infty }\left(A\right)$; $\mathrm{cond}\left(A,x\right)$ can be much smaller than $\mathrm{cond}\left(A\right)$ and it is also possible for $\mathrm{cond}\left({A}^{\mathrm{H}}\right)$, which is the same as $\mathrm{cond}\left({A}^{\mathrm{T}}\right)$, to be much larger (or smaller) than $\mathrm{cond}\left(A\right)$.
Forward and backward error bounds can be computed by calling nag_lapack_ztbrfs (f07vv), and an estimate for ${\kappa }_{\infty }\left(A\right)$ can be obtained by calling nag_lapack_ztbcon (f07vu) with ${\mathbf{norm_p}}=\text{'I'}$.

## Further Comments

The total number of real floating-point operations is approximately $8nkr$ if $k\ll n$.
The real analogue of this function is nag_lapack_dtbtrs (f07ve).

## Example

This example solves the system of equations $AX=B$, where
 $A= -1.94+4.43i 0.00+0.00i 0.00+0.00i 0.00+0.00i -3.39+3.44i 4.12-4.27i 0.00+0.00i 0.00+0.00i 1.62+3.68i -1.84+5.53i 0.43-2.66i 0.00+0.00i 0.00+0.00i -2.77-1.93i 1.74-0.04i 0.44+0.10i$
and
 $B= -8.86-03.88i -24.09-05.27i -15.57-23.41i -57.97+08.14i -7.63+22.78i 19.09-29.51i -14.74-02.40i 19.17+21.33i .$
Here $A$ is treated as a lower triangular band matrix with two subdiagonals.
function f07vs_example

fprintf('f07vs example results\n\n');

% Solve AX=B, where A is complex lower triangular banded
% and stored in triangular/symmetric banded format
kd = int64(2);
ab = [-1.94 + 4.43i,   4.12 - 4.27i,  0.43 - 2.66i,  0.44 + 0.10i;
-3.39 + 3.44i,  -1.84 + 5.53i,  1.74 - 0.04i,  0    + 0i;
1.62 + 3.68i,  -2.77 - 1.93i,  0    + 0i,     0    + 0i];
b = [ -8.86 -  3.88i, -24.09 -  5.27i;
-15.57 - 23.41i, -57.97 +  8.14i;
-7.63 + 22.78i,  19.09 - 29.51i;
-14.74 -  2.40i,  19.17 + 21.33i];

% Solve
uplo  = 'L';
trans = 'N';
diag  = 'N';
[x, info] = f07vs( ...
uplo, trans, diag, kd, ab, b);

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

f07vs example results

Solution(s)
0.0000 + 2.0000i   1.0000 + 5.0000i
1.0000 - 3.0000i  -7.0000 - 2.0000i
-4.0000 - 5.0000i   3.0000 + 4.0000i
2.0000 - 1.0000i  -6.0000 - 9.0000i

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