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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zgttrf (f07cr)

## Purpose

nag_lapack_zgttrf (f07cr) computes the $LU$ factorization of a complex $n$ by $n$ tridiagonal matrix $A$.

## Syntax

[dl, d, du, du2, ipiv, info] = f07cr(dl, d, du, 'n', n)
[dl, d, du, du2, ipiv, info] = nag_lapack_zgttrf(dl, d, du, 'n', n)

## Description

nag_lapack_zgttrf (f07cr) uses Gaussian elimination with partial pivoting and row interchanges to factorize the matrix $A$ as
 $A=PLU ,$
where $P$ is a permutation matrix, $L$ is unit lower triangular with at most one nonzero subdiagonal element in each column, and $U$ is an upper triangular band matrix, with two superdiagonals.

## 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

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{dl}\left(:\right)$ – complex array
The dimension of the array dl must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$
Must contain the $\left(n-1\right)$ subdiagonal elements of the matrix $A$.
2:     $\mathrm{d}\left(:\right)$ – complex array
The dimension of the array d must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
Must contain the $n$ diagonal elements of the matrix $A$.
3:     $\mathrm{du}\left(:\right)$ – complex array
The dimension of the array du must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$
Must contain the $\left(n-1\right)$ superdiagonal elements of the matrix $A$.

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the dimension of the array d.
$n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.

### Output Parameters

1:     $\mathrm{dl}\left(:\right)$ – complex array
The dimension of the array dl will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$
Stores the $\left(n-1\right)$ multipliers that define the matrix $L$ of the $LU$ factorization of $A$.
2:     $\mathrm{d}\left(:\right)$ – complex array
The dimension of the array d will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
Stores the $n$ diagonal elements of the upper triangular matrix $U$ from the $LU$ factorization of $A$.
3:     $\mathrm{du}\left(:\right)$ – complex array
The dimension of the array du will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$
Stores the $\left(n-1\right)$ elements of the first superdiagonal of $U$.
4:     $\mathrm{du2}\left({\mathbf{n}}-2\right)$ – complex array
Contains the $\left(n-2\right)$ elements of the second superdiagonal of $U$.
5:     $\mathrm{ipiv}\left({\mathbf{n}}\right)$int64int32nag_int array
Contains the $n$ pivot indices that define the permutation matrix $P$. At the $i$th step, row $i$ of the matrix was interchanged with row ${\mathbf{ipiv}}\left(i\right)$. ${\mathbf{ipiv}}\left(i\right)$ will always be either $i$ or $\left(i+1\right)$, ${\mathbf{ipiv}}\left(i\right)=i$ indicating that a row interchange was not performed.
6:     $\mathrm{info}$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 factorization satisfies an equation of the form
 $A+E=PLU ,$
where
 $E∞=OεA∞$
and $\epsilon$ is the machine precision.
Following the use of this function, nag_lapack_zgttrs (f07cs) can be used to solve systems of equations $AX=B$ or ${A}^{\mathrm{T}}X=B$ or ${A}^{\mathrm{H}}X=B$, and nag_lapack_zgtcon (f07cu) can be used to estimate the condition number of $A$.

The total number of floating-point operations required to factorize the matrix $A$ is proportional to $n$.
The real analogue of this function is nag_lapack_dgttrf (f07cd).

## Example

This example factorizes the tridiagonal matrix $A$ given by
 $A = -1.3+1.3i 2.0-1.0i 0.0i+0.0 0.0i+0.0 0.0i+0.0 1.0-2.0i -1.3+1.3i 2.0+1.0i 0.0i+0.0 0.0i+0.0 0.0i+0.0 1.0+1.0i -1.3+3.3i -1.0+1.0i 0.0i+0.0 0.0i+0.0 0.0i+0.0 2.0-3.0i -0.3+4.3i 1.0-1.0i 0.0i+0.0 0.0i+0.0 0.0i+0.0 1.0+1.0i -3.3+1.3i .$
```function f07cr_example

fprintf('f07cr example results\n\n');

% Tridiagonal matrix stored by diagonals
du = [              2   - 1i     2   + 1i    -1   + 1i     1   - 1i  ];
d  = [-1.3 + 1.3i  -1.3 + 1.3i  -1.3 + 3.3i  -0.3 + 4.3i  -3.3 + 1.3i];
dl = [ 1   - 2i     1   + 1i     2   - 3i     1   + 1i               ];

% Factorize.
[dlf, df, duf, du2, ipiv, info] = ...
f07cr(dl, d, du);

disp('Details of factorization');
fprintf('\n');
disp(' Second super-diagonal of U');
disp(du2');
disp(' First super-diagonal of U');
disp(duf);
disp(' Main diagonal of U');
disp(df(1:4));
disp(df(5:end));
disp(' Multipliers');
disp(dlf);
disp(' Vector of interchanges');
disp(double(ipiv)');

```
```f07cr example results

Details of factorization

Second super-diagonal of U
2.0000 - 1.0000i  -1.0000 - 1.0000i   1.0000 + 1.0000i

First super-diagonal of U
-1.3000 + 1.3000i  -1.3000 + 3.3000i  -0.3000 + 4.3000i  -3.3000 + 1.3000i

Main diagonal of U
1.0000 - 2.0000i   1.0000 + 1.0000i   2.0000 - 3.0000i   1.0000 + 1.0000i

-1.3399 + 0.2875i

Multipliers
-0.7800 - 0.2600i   0.1620 - 0.4860i  -0.0452 - 0.0010i  -0.3979 - 0.0562i

Vector of interchanges
2     3     4     5     5

```