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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 , $AX=B ,$
where A$A$ is an n$n$ by n$n$ matrix and X$X$ and B$B$ are n$n$ by r$r$ 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$LU$ decomposition with partial pivoting and row interchanges to factor A$A$ as
 A = PLU , $A=PLU ,$
where P$P$ is a permutation matrix, L$L$ is unit lower triangular, and U$U$ is upper triangular. The factored form of A$A$ is then used to solve the system of equations AX = B$AX=B$.

## 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:     a(lda, : $:$) – complex array
The first dimension of the array a must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The n$n$ by n$n$ coefficient matrix A$A$.
2:     b(ldb, : $:$) – complex array
The first dimension of the array b must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array must be at least max (1,nrhs)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$
The n$n$ by r$r$ right-hand side matrix B$B$.

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the array b.
n$n$, the number of linear equations, i.e., the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.
2:     nrhs_p – int64int32nag_int scalar
Default: The second dimension of the array b.
r$r$, the number of right-hand sides, i.e., the number of columns of the matrix B$B$.
Constraint: nrhs0${\mathbf{nrhs}}\ge 0$.

lda ldb

### Output Parameters

1:     a(lda, : $:$) – complex array
The first dimension of the array a will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
ldamax (1,n)$\mathit{lda}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The factors L$L$ and U$U$ from the factorization A = PLU$A=PLU$; the unit diagonal elements of L$L$ are not stored.
2:     ipiv(n) – int64int32nag_int array
If no constraints are violated, the pivot indices that define the permutation matrix P$P$; at the i$i$th step row i$i$ of the matrix was interchanged with row ipiv(i)${\mathbf{ipiv}}\left(i\right)$. ipiv(i) = i${\mathbf{ipiv}}\left(i\right)=i$ indicates a row interchange was not required.
3:     b(ldb, : $:$) – complex array
The first dimension of the array b will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array will be max (1,nrhs)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$
ldbmax (1,n)$\mathit{ldb}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If ${\mathbf{INFO}}={\mathbf{0}}$, the n$n$ by r$r$ solution matrix X$X$.
4:     info – int64int32nag_int scalar
info = 0${\mathbf{info}}=0$ unless the function detects an error (see Section [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.

info = i${\mathbf{info}}=-i$
If info = i${\mathbf{info}}=-i$, parameter i$i$ had an illegal value on entry. The parameters are numbered as follows:
1: n, 2: nrhs_p, 3: a, 4: lda, 5: ipiv, 6: b, 7: ldb, 8: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.
W INFO > 0${\mathbf{INFO}}>0$
If info = i${\mathbf{info}}=i$, uii${u}_{ii}$ is exactly zero. The factorization has been completed, but the factor U$U$ 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) x̂ = b , $(A+E) x^=b ,$
where
 ‖E‖1 = O(ε) ‖A‖1 $‖E‖1 = O(ε) ‖A‖1$
and ε $\epsilon$ is the machine precision. An approximate error bound for the computed solution is given by
 (‖x̂ − x‖1)/(‖x‖1) ≤ κ(A) (‖E‖1)/(‖A‖1) $‖ x^ - x ‖1 ‖ x ‖1 ≤ κ(A) ‖ E ‖1 ‖ A ‖1$
where κ(A) = A11 A1 $\kappa \left(A\right)={‖{A}^{-1}‖}_{1}{‖A‖}_{1}$, the condition number of A $A$ 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 $A$ 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 (8/3) n3 + 8n2 r $\frac{8}{3}{n}^{3}+8{n}^{2}r$, where r $r$ is the number of right-hand sides.
The real analogue of this function is nag_lapack_dgesv (f07aa).

## Example

```function nag_lapack_zgesv_example
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];
[aOut, ipiv, bOut, info] = nag_lapack_zgesv(a, b)
```
```

aOut =

-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

ipiv =

3
2
3
4

bOut =

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

info =

0

```
```function f07an_example
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];
[aOut, ipiv, bOut, info] = f07an(a, b)
```
```

aOut =

-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

ipiv =

3
2
3
4

bOut =

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

info =

0

```