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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zgesv (f07an)

## Purpose

nag_lapack_zgesv (f07an) computes the solution to a complex system of linear equations
 $AX=B ,$
where $A$ is an $n$ by $n$ matrix and $X$ and $B$ are $n$ by $r$ matrices.

## Syntax

[a, ipiv, b, info] = f07an(a, b, 'n', n, 'nrhs_p', nrhs_p)
[a, ipiv, b, info] = nag_lapack_zgesv(a, b, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_zgesv (f07an) uses the $LU$ decomposition with partial pivoting and row interchanges to factor $A$ as
 $A=PLU ,$
where $P$ is a permutation matrix, $L$ is unit lower triangular, and $U$ is upper triangular. 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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex array
The first dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The $n$ by $n$ coefficient matrix $A$.
2:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – 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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex array
The first dimension of the array a will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array a will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The factors $L$ and $U$ from the factorization $A=PLU$; the unit diagonal elements of $L$ are not stored.
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 the 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
 $x^ - x 1 x 1 ≤ κA E 1 A 1$
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_zgesv (f07an), nag_lapack_zgecon (f07au) can be used to estimate the condition number of $A$ and nag_lapack_zgerfs (f07av) can be used to obtain approximate error bounds. Alternatives to nag_lapack_zgesv (f07an), which return condition and error estimates directly are nag_linsys_complex_square_solve (f04ca) and nag_lapack_zgesvx (f07ap).

The total number of floating-point operations is approximately $\frac{8}{3}{n}^{3}+8{n}^{2}r$, where $r$ is the number of right-hand sides.
The real analogue of this function is nag_lapack_dgesv (f07aa).

## Example

This example solves the equations
 $Ax = b ,$
where $A$ is the general matrix
 $A = -1.34+2.55i 0.28+3.17i -6.39-2.20i 0.72-0.92i -0.17-1.41i 3.31-0.15i -0.15+1.34i 1.29+1.38i -3.29-2.39i -1.91+4.42i -0.14-1.35i 1.72+1.35i 2.41+0.39i -0.56+1.47i -0.83-0.69i -1.96+0.67i and b = 26.26+51.78i 6.43-08.68i -5.75+25.31i 1.16+02.57i .$
Details of the $LU$ factorization of $A$ are also output.
```function f07an_example

fprintf('f07an example results\n\n');

% Linear system Ax = B
a = [ -1.34 + 2.55i,  0.28 + 3.17i,  -6.39 - 2.2i,   0.72 - 0.92i;
-0.17 - 1.41i,  3.31 - 0.15i,  -0.15 + 1.34i,  1.29 + 1.38i;
-3.29 - 2.39i, -1.91 + 4.42i,  -0.14 - 1.35i,  1.72 + 1.35i;
2.41 + 0.39i, -0.56 + 1.47i,  -0.83 - 0.69i, -1.96 + 0.67i];
b = [ 26.26 + 51.78i;
6.43 -  8.68i;
-5.75 + 25.31i;
1.16 +  2.57i];

[LU, ipiv, x, info] = f07an(a, b);

disp('Solution');
disp(x');
disp('Details of factorization');
disp(LU);
disp('Pivot indices');
disp(double(ipiv'));

```
```f07an example results

Solution
1.0000 - 1.0000i   2.0000 + 3.0000i  -4.0000 + 5.0000i   0.0000 - 6.0000i

Details of factorization
-3.2900 - 2.3900i  -1.9100 + 4.4200i  -0.1400 - 1.3500i   1.7200 + 1.3500i
0.2376 + 0.2560i   4.8952 - 0.7114i  -0.4623 + 1.6966i   1.2269 + 0.6190i
-0.1020 - 0.7010i  -0.6691 + 0.3689i  -5.1414 - 1.1300i   0.9983 + 0.3850i
-0.5359 + 0.2707i  -0.2040 + 0.8601i   0.0082 + 0.1211i   0.1482 - 0.1252i

Pivot indices
3     2     3     4

```