F07AQF (ZCGESV) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

NAG Library Routine Document


Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

F07AQF (ZCGESV) 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.

2  Specification

REAL (KIND=nag_wp)  RWORK(N)
COMPLEX (KIND=nag_wp)  A(LDA,*), B(LDB,*), X(LDX,*), WORK(N*NRHS)
The routine may be called by its LAPACK name zcgesv.

3  Description

F07AQF (ZCGESV) 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 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 resid is the -norm of the residual, x is the -norm of the solution, A is the -operator-norm of the matrix A and ε is the machine precision returned by X02AJF.
The iterative refinement strategy used by F07AQF (ZCGESV) 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 (ZCGESV) always attempts the iterative refinement strategy first; you are advised to compare the performance of F07AQF (ZCGESV) with that of its full precision counterpart F07ANF (ZGESV) 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 http://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  Parameters

1:     N – INTEGERInput
On entry: n, the number of linear equations, i.e., the order of the matrix A.
Constraint: N0.
2:     NRHS – INTEGERInput
On entry: r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint: NRHS0.
3:     A(LDA,*) – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array A must be at least max1,N.
On entry: the n by n coefficient matrix A.
On exit: if iterative refinement has been successfully used (i.e., if INFO=0 and ITER0), then A is unchanged. If double precision factorization has been used (when INFO=0 and ITER<0), A contains the factors L and U from the factorization A=PLU; the unit diagonal elements of L are not stored.
4:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F07AQF (ZCGESV) is called.
Constraint: LDAmax1,N.
5:     IPIV(N) – INTEGER arrayOutput
On exit: if no constraints are violated, the pivot indices that define the permutation matrix P; at the ith step row i of the matrix was interchanged with row IPIVi. IPIVi=i indicates a row interchange was not required. IPIV corresponds either to the single precision factorization (if INFO=0 and ITER0) or to the double precision factorization (if INFO=0 and ITER<0).
6:     B(LDB,*) – COMPLEX (KIND=nag_wp) arrayInput
Note: the second dimension of the array B must be at least max1,NRHS.
On entry: the n by r right-hand side matrix B.
7:     LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F07AQF (ZCGESV) is called.
Constraint: LDBmax1,N.
8:     X(LDX,*) – COMPLEX (KIND=nag_wp) arrayOutput
Note: the second dimension of the array X must be at least max1,NRHS.
On exit: if INFO=0, the n by r solution matrix X.
9:     LDX – INTEGERInput
On entry: the first dimension of the array X as declared in the (sub)program from which F07AQF (ZCGESV) is called.
Constraint: LDXmax1,N.
10:   WORK(N*NRHS) – COMPLEX (KIND=nag_wp) arrayWorkspace
11:   SWORK(N×N+NRHS) – COMPLEX (KIND=nag_rp) arrayWorkspace
Note: this array is utilized in the reduced precision computation, consequently its type nag_rp reflects this usage.
12:   RWORK(N) – REAL (KIND=nag_wp) arrayWorkspace
13:   ITER – INTEGEROutput
On exit: if ITER>0, iterative refinement has been successfully used and ITER is the number of iterations carried out.
If ITER<0, iterative refinement has failed for one of the reasons given below and double precision factorization has been carried out instead.
Taking into account machine parameters, and the values of N and NRHS, it is not worth working in single precision.
Overflow of an entry occurred when moving from double to single precision.
An intermediate single precision factorization failed.
The maximum permitted number of iterations was exceeded.
14:   INFO – INTEGEROutput
On exit: INFO=0 unless the routine detects an error (see Section 6).

6  Error Indicators and Warnings

Errors or warnings detected by the routine:
If INFO=-i, the ith argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
If INFO=i, uii 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 right-hand side, x^ , satisfies the equation of the form
A+E x^=b ,
E1 = Oε A1
and ε  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 κA = 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.

8  Further Comments

The real analogue of this routine is F07ACF (DSGESV).

9  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 .

9.1  Program Text

Program Text (f07aqfe.f90)

9.2  Program Data

Program Data (f07aqfe.d)

9.3  Program Results

Program Results (f07aqfe.r)

F07AQF (ZCGESV) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012