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# NAG Toolbox: nag_lapack_zgbtrf (f07br)

## Purpose

nag_lapack_zgbtrf (f07br) computes the $LU$ factorization of a complex $m$ by $n$ band matrix.

## Syntax

[ab, ipiv, info] = f07br(m, kl, ku, ab, 'n', n)
[ab, ipiv, info] = nag_lapack_zgbtrf(m, kl, ku, ab, 'n', n)

## Description

nag_lapack_zgbtrf (f07br) forms the $LU$ factorization of a complex $m$ by $n$ band matrix $A$ using partial pivoting, with row interchanges. Usually $m=n$, and then, if $A$ has ${k}_{l}$ nonzero subdiagonals and ${k}_{u}$ nonzero superdiagonals, the factorization has the form $A=PLU$, where $P$ is a permutation matrix, $L$ is a lower triangular matrix with unit diagonal elements and at most ${k}_{l}$ nonzero elements in each column, and $U$ is an upper triangular band matrix with ${k}_{l}+{k}_{u}$ superdiagonals.
Note that $L$ is not a band matrix, but the nonzero elements of $L$ can be stored in the same space as the subdiagonal elements of $A$. $U$ is a band matrix but with ${k}_{l}$ additional superdiagonals compared with $A$. These additional superdiagonals are created by the row interchanges.

## References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{m}$int64int32nag_int scalar
$m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
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 $m$ by $n$ 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
 $abkl+ku+1+i-jj=Aij for ​max1,j-ku≤i≤minm,j+kl.$
See Further Comments in nag_lapack_zgbsv (f07bn) for further details.

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the second dimension of the array ab.
$n$, the number of columns of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.

### 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:     $\mathrm{ipiv}\left(\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$int64int32nag_int array
The pivot indices that define the permutation matrix. At the $\mathit{i}$th step, if ${\mathbf{ipiv}}\left(\mathit{i}\right)>\mathit{i}$ then row $\mathit{i}$ of the matrix $A$ was interchanged with row ${\mathbf{ipiv}}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$. ${\mathbf{ipiv}}\left(i\right)\le i$ indicates that, at the $i$th step, a row interchange was not required.
3:     $\mathrm{info}$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, and division by zero will occur if it is used to solve a system of equations.

## Accuracy

The computed factors $L$ and $U$ are the exact factors of a perturbed matrix $A+E$, where
 $E≤ckεPLU ,$
$c\left(k\right)$ is a modest linear function of $k={k}_{l}+{k}_{u}+1$, and $\epsilon$ is the machine precision. This assumes $k\ll \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$.

The total number of real floating-point operations varies between approximately $8n{k}_{l}\left({k}_{u}+1\right)$ and $8n{k}_{l}\left({k}_{l}+{k}_{u}+1\right)$, depending on the interchanges, assuming $m=n\gg {k}_{l}$ and $n\gg {k}_{u}$.
A call to nag_lapack_zgbtrf (f07br) may be followed by calls to the functions:
• nag_lapack_zgbtrs (f07bs) to solve $AX=B$, ${A}^{\mathrm{T}}X=B$ or ${A}^{\mathrm{H}}X=B$;
• nag_lapack_zgbcon (f07bu) to estimate the condition number of $A$.
The real analogue of this function is nag_lapack_dgbtrf (f07bd).

## Example

This example computes the $LU$ factorization of the matrix $A$, where
 $A= -1.65+2.26i -2.05-0.85i 0.97-2.84i 0.00+0.00i 0.00+6.30i -1.48-1.75i -3.99+4.01i 0.59-0.48i 0.00+0.00i -0.77+2.83i -1.06+1.94i 3.33-1.04i 0.00+0.00i 0.00+0.00i 4.48-1.09i -0.46-1.72i .$
Here $A$ is treated as a band matrix with one subdiagonal and two superdiagonals.
```function f07br_example

fprintf('f07br 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];

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

mtitle = 'Details of factorization';
[ifail] = x04de( ...
m, m, kl, kl+ku, abf, mtitle);

disp('Pivot indices');
disp(double(ipiv'));

```
```f07br example results

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

```

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