NAG FL Interface
f07aqf (zcgesv)
1
Purpose
f07aqf computes the solution to a complex system of linear equations
where
$A$ is an
$n$ by
$n$ matrix and
$X$ and
$B$ are
$n$ by
$r$ matrices.
2
Specification
Fortran Interface
Subroutine f07aqf ( 
n, nrhs, a, lda, ipiv, b, ldb, x, ldx, work, swork, rwork, 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)) 

C Header Interface
#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) 

C++ Header Interface
#include <nag.h> extern "C" {
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.
3
Description
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
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 righthand sides we have
where
$\Vert \mathit{resid}\Vert $ is the
$\infty $norm of the residual,
$\Vert x\Vert $ is the
$\infty $norm of the solution,
$\Vert A\Vert $ is the
$\infty $operatornorm 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 righthand side, any efficiency gains will reduce as the number of righthand 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 righthand sides and for large matrix sizes. The cutoff values for the number of righthand 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 backsubstitution. 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.
4
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
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
5
Arguments

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 righthand 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 $n$ by $n$ 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 $n$ by $r$ righthand 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 $n$ by $r$ 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:
$\mathbf{swork}\left({\mathbf{n}}\times \left({\mathbf{n}}+{\mathbf{nrhs}}\right)\right)$ – 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).
6
Error 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 $\u2329\mathit{\text{value}}\u232a$ 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.
7
Accuracy
The computed solution for a single righthand side,
$\hat{x}$, satisfies the equation of the form
where
and
$\epsilon $ is the
machine precision. An approximate error bound for the computed solution is given by
where
$\kappa \left(A\right)={\Vert {A}^{1}\Vert}_{1}{\Vert A\Vert}_{1}$, 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.
8
Parallelism 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 implementationspecific information.
The real analogue of this routine is
f07acf.
10
Example
This example solves the equations
where
$A$ is the general matrix
10.1
Program Text
10.2
Program Data
10.3
Program Results