nag_zcgesv (f07aqc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_zcgesv (f07aqc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_zcgesv (f07aqc) 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

#include <nag.h>
#include <nagf07.h>
void  nag_zcgesv (Nag_OrderType order, Integer n, Integer nrhs, Complex a[], Integer pda, Integer ipiv[], const Complex b[], Integer pdb, Complex x[], Integer pdx, Integer *iter, NagError *fail)

3  Description

nag_zcgesv (f07aqc) 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 nag_machine_precision (X02AJC).
The iterative refinement strategy used by nag_zcgesv (f07aqc) 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, nag_zcgesv (f07aqc) always attempts the iterative refinement strategy first; you are advised to compare the performance of nag_zcgesv (f07aqc) with that of its full precision counterpart nag_zgesv (f07anc) 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  Arguments

1:     orderNag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     nIntegerInput
On entry: n, the number of linear equations, i.e., the order of the matrix A.
Constraint: n0.
3:     nrhsIntegerInput
On entry: r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint: nrhs0.
4:     a[dim]ComplexInput/Output
Note: the dimension, dim, of the array a must be at least max1,pda×n.
The i,jth element of the matrix A is stored in
  • a[j-1×pda+i-1] when order=Nag_ColMajor;
  • a[i-1×pda+j-1] when order=Nag_RowMajor.
On entry: the n by n coefficient matrix A.
On exit: if iterative refinement has been successfully used (i.e., if fail.code= NE_NOERROR and iter0), then A is unchanged. If double precision factorization has been used (when fail.code= NE_NOERROR and iter<0), A contains the factors L and U from the factorization A=PLU; the unit diagonal elements of L are not stored.
5:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: pdamax1,n.
6:     ipiv[n]IntegerOutput
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 ipiv[i-1]. ipiv[i-1]=i indicates a row interchange was not required. ipiv corresponds either to the single precision factorization (if fail.code= NE_NOERROR and iter0) or to the double precision factorization (if fail.code= NE_NOERROR and iter<0).
7:     b[dim]const ComplexInput
Note: the dimension, dim, of the array b must be at least
  • max1,pdb×nrhs when order=Nag_ColMajor;
  • max1,n×pdb when order=Nag_RowMajor.
The i,jth element of the matrix B is stored in
  • b[j-1×pdb+i-1] when order=Nag_ColMajor;
  • b[i-1×pdb+j-1] when order=Nag_RowMajor.
On entry: the n by r right-hand side matrix B.
8:     pdbIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
  • if order=Nag_ColMajor, pdbmax1,n;
  • if order=Nag_RowMajor, pdbmax1,nrhs.
9:     x[dim]ComplexOutput
Note: the dimension, dim, of the array x must be at least
  • max1,pdx×nrhs when order=Nag_ColMajor;
  • max1,n×pdx when order=Nag_RowMajor.
The i,jth element of the matrix X is stored in
  • x[j-1×pdx+i-1] when order=Nag_ColMajor;
  • x[i-1×pdx+j-1] when order=Nag_RowMajor.
On exit: if fail.code= NE_NOERROR, the n by r solution matrix X.
10:   pdxIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array x.
Constraints:
  • if order=Nag_ColMajor, pdxmax1,n;
  • if order=Nag_RowMajor, pdxmax1,nrhs.
11:   iterInteger *Output
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.
iter=-1
Taking into account machine parameters, and the values of n and nrhs, it is not worth working in single precision.
iter=-2
Overflow of an entry occurred when moving from double to single precision.
iter=-3
An intermediate single precision factorization failed.
iter=-31
The maximum permitted number of iterations was exceeded.
12:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, n=value.
Constraint: n0.
On entry, nrhs=value.
Constraint: nrhs0.
On entry, pda=value.
Constraint: pda>0.
On entry, pdb=value.
Constraint: pdb>0.
On entry, pdx=value.
Constraint: pdx>0.
NE_INT_2
On entry, pda=value and n=value.
Constraint: pdamax1,n.
On entry, pdb=value and n=value.
Constraint: pdbmax1,n.
On entry, pdb=value and nrhs=value.
Constraint: pdbmax1,nrhs.
On entry, pdx=value and n=value.
Constraint: pdxmax1,n.
On entry, pdx=value and nrhs=value.
Constraint: pdxmax1,nrhs.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_SINGULAR
Uvalue,value 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 ,
where
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 function is nag_dsgesv (f07acc).

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 (f07aqce.c)

9.2  Program Data

Program Data (f07aqce.d)

9.3  Program Results

Program Results (f07aqce.r)


nag_zcgesv (f07aqc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

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