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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zgbsv (f07bn)

## Purpose

nag_lapack_zgbsv (f07bn) computes the solution to a complex system of linear equations
 $AX=B ,$
where $A$ is an $n$ by $n$ band matrix, with ${k}_{l}$ subdiagonals and ${k}_{u}$ superdiagonals, and $X$ and $B$ are $n$ by $r$ matrices.

## Syntax

[ab, ipiv, b, info] = f07bn(kl, ku, ab, b, 'n', n, 'nrhs_p', nrhs_p)
[ab, ipiv, b, info] = nag_lapack_zgbsv(kl, ku, ab, b, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_zgbsv (f07bn) uses the $LU$ decomposition with partial pivoting and row interchanges to factor $A$ as $A=PLU$, where $P$ is a permutation matrix, $L$ is a product of permutation and unit lower triangular matrices with ${k}_{l}$ subdiagonals, and $U$ is upper triangular with $\left({k}_{l}+{k}_{u}\right)$ superdiagonals. The factored form of $A$ is then used to solve the system of equations $AX=B$.

## References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{kl}$int64int32nag_int scalar
${k}_{l}$, the number of subdiagonals within the band of the matrix $A$.
Constraint: ${\mathbf{kl}}\ge 0$.
2:     $\mathrm{ku}$int64int32nag_int scalar
${k}_{u}$, the number of superdiagonals within the band of the matrix $A$.
Constraint: ${\mathbf{ku}}\ge 0$.
3:     $\mathrm{ab}\left(\mathit{ldab},:\right)$ – 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 $n$ by $n$ coefficient matrix $A$.
The matrix is stored in rows ${k}_{l}+1$ to $2{k}_{l}+{k}_{u}+1$; the first ${k}_{l}$ rows need not be set, more precisely, the element ${A}_{ij}$ must be stored in
 $abkl+ku+1+i-jj=Aij for ​max1,j-ku≤i≤minn,j+kl.$
See Further Comments for further details.
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 $B$.

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the first dimension of the array b.
$n$, the number of linear equations, i.e., 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, i.e., the number of columns of the matrix $B$.
Constraint: ${\mathbf{nrhs_p}}\ge 0$.

### Output Parameters

1:     $\mathrm{ab}\left(\mathit{ldab},:\right)$ – complex array
The first dimension of the array ab will be $2×{\mathbf{kl}}+{\mathbf{ku}}+1$.
The second dimension of the array ab will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If ${\mathbf{info}}\ge 0$, ab stores details of the factorization.
The upper triangular band matrix $U$, with ${k}_{l}+{k}_{u}$ superdiagonals, is stored in rows $1$ to ${k}_{l}+{k}_{u}+1$ of the array, and the multipliers used to form the matrix $L$ are stored in rows ${k}_{l}+{k}_{u}+2$ to $2{k}_{l}+{k}_{u}+1$.
2:     $\mathrm{ipiv}\left({\mathbf{n}}\right)$int64int32nag_int array
If no constraints are violated, the pivot indices that define the permutation matrix $P$; at the $i$th step row $i$ of the matrix was interchanged with row ${\mathbf{ipiv}}\left(i\right)$. ${\mathbf{ipiv}}\left(i\right)=i$ indicates a row interchange was not required.
3:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – 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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs_p}}\right)$.
If ${\mathbf{info}}={\mathbf{0}}$, the $n$ by $r$ solution matrix $X$.
4:     $\mathrm{info}$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. The factorization has been completed, but the factor $U$ is exactly singular, so the solution could not be computed.

## Accuracy

The computed solution for a single right-hand side, $\stackrel{^}{x}$, satisfies an equation of the form
 $A+E x^ = b ,$
where
 $E1 = Oε A1$
and $\epsilon$ is the machine precision. An approximate error bound for the computed solution is given by
 $x^-x 1 x1 ≤ κA E1 A1 ,$
where $\kappa \left(A\right)={‖{A}^{-1}‖}_{1}{‖A‖}_{1}$, the condition number of $A$ with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) for further details.
Following the use of nag_lapack_zgbsv (f07bn), nag_lapack_zgbcon (f07bu) can be used to estimate the condition number of $A$ and nag_lapack_zgbrfs (f07bv) can be used to obtain approximate error bounds. Alternatives to nag_lapack_zgbsv (f07bn), which return condition and error estimates directly are nag_linsys_complex_band_solve (f04cb) and nag_lapack_zgbsvx (f07bp).

The band storage scheme for the array ab is illustrated by the following example, when $n=6$, ${k}_{l}=1$, and ${k}_{u}=2$. Storage of the band matrix $A$ in the array ab:
 $* * * + + + * * a13 a24 a35 a46 * a12 a23 a34 a45 a56 a11 a22 a33 a44 a55 a66 a21 a32 a43 a54 a65 *$
Array elements marked $*$ need not be set and are not referenced by the function. Array elements marked $+$ need not be set, but are defined on exit from the function and contain the elements ${u}_{14}$, ${u}_{25}$ and ${u}_{36}$.
The total number of floating-point operations required to solve the equations $AX=B$ depends upon the pivoting required, but if $n\gg {k}_{l}+{k}_{u}$ then it is approximately bounded by $\mathit{O}\left(n{k}_{l}\left({k}_{l}+{k}_{u}\right)\right)$ for the factorization and $\mathit{O}\left(n\left(2{k}_{l}+{k}_{u}\right)r\right)$ for the solution following the factorization.
The real analogue of this function is nag_lapack_dgbsv (f07ba).

## Example

This example solves the equations
 $Ax=b ,$
where $A$ is the band matrix
 $A = -1.65+2.26i -2.05-0.85i 0.97-2.84i 0.00i+0.00 6.30i -1.48-1.75i -3.99+4.01i 0.59-0.48i 0.00i+0.00 -0.77+2.83i -1.06+1.94i 3.33-1.04i 0.00i+0.00 0.00i+0.00 4.48-1.09i -0.46-1.72i$
and
 $b = -1.06+21.50i -22.72-53.90i 28.24-38.60i -34.56+16.73i .$
Details of the $LU$ factorization of $A$ are also output.
```function f07bn_example

fprintf('f07bn 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;
-22.72 - 53.9i;
28.24 - 38.6i;
-34.56 + 16.73i];

% Solve Ax = B
[abf, ipiv, x, info] = f07bn( ...
kl, ku, ab, b);

disp('Solution');
disp(x');
mtitle = 'Details of factorization';
[ifail] = x04de( ...
m, m, kl, kl+ku, abf, mtitle);
fprintf('\n');
disp('Pivot indices');
disp(double(ipiv'));

```
```f07bn example results

Solution
-3.0000 - 2.0000i   1.0000 + 7.0000i  -5.0000 - 4.0000i   6.0000 + 8.0000i

Details of factorization
1          2          3          4
1      0.0000    -1.4800    -3.9900     0.5900
6.3000    -1.7500     4.0100    -0.4800

2      0.3587    -0.7700    -1.0600     3.3300
0.2619     2.8300     1.9400    -1.0400

3                 0.2314     4.9303    -1.7692
0.6358    -3.0086    -1.8587

4                            0.7604     0.4338
0.2429     0.1233

Pivot indices
2     3     3     4

```

Chapter Contents
Chapter Introduction
NAG Toolbox

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