# NAG FL Interfacef07aqf (zcgesv)

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## 1Purpose

f07aqf computes the solution to a complex system of linear equations
 $AX=B ,$
where $A$ is an $n×n$ matrix and $X$ and $B$ are $n×r$ matrices.

## 2Specification

Fortran Interface
 Subroutine f07aqf ( n, nrhs, a, lda, ipiv, b, ldb, x, ldx, work, iter, info)
 Integer, Intent (In) :: n, nrhs, lda, ldb, ldx Integer, Intent (Out) :: ipiv(n), iter, info Real (Kind=nag_wp), Intent (Out) :: rwork(n) Complex (Kind=nag_wp), Intent (In) :: b(ldb,*) Complex (Kind=nag_wp), Intent (Inout) :: a(lda,*), x(ldx,*) Complex (Kind=nag_wp), Intent (Out) :: work(n*nrhs) Complex (Kind=nag_rp), Intent (Out) :: swork(n*(n+nrhs))
#include <nag.h>
 void f07aqf_ (const Integer *n, const Integer *nrhs, Complex a[], const Integer *lda, Integer ipiv[], const Complex b[], const Integer *ldb, Complex x[], const Integer *ldx, Complex work[], Complexf swork[], double rwork[], Integer *iter, Integer *info)
The routine may be called by the names f07aqf, nagf_lapacklin_zcgesv or its LAPACK name zcgesv.

## 3Description

f07aqf first attempts to factorize the matrix in single precision and use this factorization within an iterative refinement procedure to produce a solution with double precision accuracy. If the approach fails the method switches to a double precision factorization and solve.
The iterative refinement process is stopped if
 $iter>itermax ,$
where iter is the number of iterations carried out thus far and $\mathit{itermax}$ is the maximum number of iterations allowed, which is fixed at $30$ iterations. The process is also stopped if for all right-hand sides we have
 $‖resid‖ < n ‖x‖ ‖A‖ ε ,$
where $‖\mathit{resid}‖$ is the $\infty$-norm of the residual, $‖x‖$ is the $\infty$-norm of the solution, $‖A‖$ is the $\infty$-operator-norm of the matrix $A$ and $\epsilon$ is the machine precision returned by x02ajf.
The iterative refinement strategy used by f07aqf can be more efficient than the corresponding direct full precision algorithm. Since this strategy must perform iterative refinement on each right-hand side, any efficiency gains will reduce as the number of right-hand sides increases. Conversely, as the matrix size increases the cost of these iterative refinements become less significant relative to the cost of factorization. Thus, any efficiency gains will be greatest for a very small number of right-hand sides and for large matrix sizes. The cut-off values for the number of right-hand sides and matrix size, for which the iterative refinement strategy performs better, depends on the relative performance of the reduced and full precision factorization and back-substitution. For now, f07aqf always attempts the iterative refinement strategy first; you are advised to compare the performance of f07aqf with that of its full precision counterpart f07anf to determine whether this strategy is worthwhile for your particular problem dimensions.
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 https://www.netlib.org/lapack/lug
Buttari A, Dongarra J, Langou J, Langou J, Luszczek P and Kurzak J (2007) Mixed precision iterative refinement techniques for the solution of dense linear systems International Journal of High Performance Computing Applications
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5Arguments

1: $\mathbf{n}$Integer Input
On entry: $n$, the number of linear equations, i.e., the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
2: $\mathbf{nrhs}$Integer Input
On entry: $r$, the number of right-hand sides, i.e., the number of columns of the matrix $B$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
3: $\mathbf{a}\left({\mathbf{lda}},*\right)$Complex (Kind=nag_wp) array Input/Output
Note: the second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $n×n$ coefficient matrix $A$.
On exit: if iterative refinement has been successfully used (i.e., if ${\mathbf{info}}={\mathbf{0}}$ and ${\mathbf{iter}}\ge 0$), then $A$ is unchanged. If double precision factorization has been used (when ${\mathbf{info}}={\mathbf{0}}$ and ${\mathbf{iter}}<0$), $A$ contains the factors $L$ and $U$ from the factorization $A=PLU$; the unit diagonal elements of $L$ are not stored.
4: $\mathbf{lda}$Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f07aqf is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
5: $\mathbf{ipiv}\left({\mathbf{n}}\right)$Integer array Output
On exit: 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. ${\mathbf{ipiv}}$ corresponds either to the single precision factorization (if ${\mathbf{info}}={\mathbf{0}}$ and ${\mathbf{iter}}\ge 0$) or to the double precision factorization (if ${\mathbf{info}}={\mathbf{0}}$ and ${\mathbf{iter}}<0$).
6: $\mathbf{b}\left({\mathbf{ldb}},*\right)$Complex (Kind=nag_wp) array Input
Note: the second dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
On entry: the $n×r$ right-hand side matrix $B$.
7: $\mathbf{ldb}$Integer Input
On entry: the first dimension of the array b as declared in the (sub)program from which f07aqf is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
8: $\mathbf{x}\left({\mathbf{ldx}},*\right)$Complex (Kind=nag_wp) array Output
Note: the second dimension of the array x must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
On exit: if ${\mathbf{info}}={\mathbf{0}}$, the $n×r$ solution matrix $X$.
9: $\mathbf{ldx}$Integer Input
On entry: the first dimension of the array x as declared in the (sub)program from which f07aqf is called.
Constraint: ${\mathbf{ldx}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
10: $\mathbf{work}\left({\mathbf{n}}*{\mathbf{nrhs}}\right)$Complex (Kind=nag_wp) array Workspace
11: $\mathbf{swork}\left({\mathbf{n}}×\left({\mathbf{n}}+{\mathbf{nrhs}}\right)\right)$Complex (Kind=nag_rp) array Workspace
Note: this array is utilized in the reduced precision computation, consequently its type nag_rp reflects this usage.
12: $\mathbf{rwork}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Workspace
13: $\mathbf{iter}$Integer Output
On exit: if ${\mathbf{iter}}>0$, iterative refinement has been successfully used and iter is the number of iterations carried out.
If ${\mathbf{iter}}<0$, iterative refinement has failed for one of the reasons given below and double precision factorization has been carried out instead.
${\mathbf{iter}}=-1$
Taking into account machine parameters, and the values of n and nrhs, it is not worth working in single precision.
${\mathbf{iter}}=-2$
Overflow of an entry occurred when moving from double to single precision.
${\mathbf{iter}}=-3$
An intermediate single precision factorization failed.
${\mathbf{iter}}=-31$
The maximum permitted number of iterations was exceeded.
14: $\mathbf{info}$Integer Output
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error Indicators and 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.
${\mathbf{info}}>0$
Element $⟨\mathit{\text{value}}⟩$ 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.

## 7Accuracy

The computed solution for a single right-hand side, $\stackrel{^}{x}$, satisfies the equation of the form
 $(A+E) x^=b ,$
where
 $‖E‖1 = O(ε) ‖A‖1$
and $\epsilon$ is the machine precision. An approximate error bound for the computed solution is given by
 $‖x^-x‖1 ‖x‖1 ≤ κ(A) ‖E‖1 ‖A‖1$
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.

## 8Parallelism and Performance

f07aqf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07aqf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The real analogue of this routine is f07acf.

## 10Example

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

### 10.1Program Text

Program Text (f07aqfe.f90)

### 10.2Program Data

Program Data (f07aqfe.d)

### 10.3Program Results

Program Results (f07aqfe.r)