NAG CL Interface
f07aqc (zcgesv)

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1 Purpose

f07aqc 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.

2 Specification

#include <nag.h>
void  f07aqc (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)
The function may be called by the names: f07aqc, nag_lapacklin_zcgesv or nag_zcgesv.

3 Description

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 X02AJC.
The iterative refinement strategy used by 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, f07aqc always attempts the iterative refinement strategy first; you are advised to compare the performance of f07aqc with that of its full precision counterpart 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
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: order Nag_OrderType Input
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 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2: n Integer Input
On entry: n, the number of linear equations, i.e., the order of the matrix A.
Constraint: n0.
3: nrhs Integer Input
On entry: r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint: nrhs0.
4: a[dim] Complex Input/Output
Note: the dimension, dim, of the array a must be at least max(1,pda×n).
The (i,j)th 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×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: pda Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: pdamax(1,n).
6: ipiv[n] Integer Output
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 Complex Input
Note: the dimension, dim, of the array b must be at least
  • max(1,pdb×nrhs) when order=Nag_ColMajor;
  • max(1,n×pdb) when order=Nag_RowMajor.
The (i,j)th 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×r right-hand side matrix B.
8: pdb Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
  • if order=Nag_ColMajor, pdbmax(1,n);
  • if order=Nag_RowMajor, pdbmax(1,nrhs).
9: x[dim] Complex Output
Note: the dimension, dim, of the array x must be at least
  • max(1,pdx×nrhs) when order=Nag_ColMajor;
  • max(1,n×pdx) when order=Nag_RowMajor.
The (i,j)th 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×r solution matrix X.
10: pdx Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array x.
  • if order=Nag_ColMajor, pdxmax(1,n);
  • if order=Nag_RowMajor, pdxmax(1,nrhs).
11: iter Integer * 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.
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.
12: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument value had an illegal value.
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.
On entry, pda=value and n=value.
Constraint: pdamax(1,n).
On entry, pdb=value and n=value.
Constraint: pdbmax(1,n).
On entry, pdb=value and nrhs=value.
Constraint: pdbmax(1,nrhs).
On entry, pdx=value and n=value.
Constraint: pdxmax(1,n).
On entry, pdx=value and nrhs=value.
Constraint: pdxmax(1,nrhs).
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
Element 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.

7 Accuracy

The computed solution for a single right-hand side, x^ , satisfies the equation of the form
(A+E) x^=b ,  
E1 = O(ε) A1  
and ε is the machine precision. An approximate error bound for the computed solution is given by
x^-x1 x1 κ(A) E1 A1  
where κ(A) = A-11 A1 , 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

f07aqc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07aqc 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The real analogue of this function is f07acc.

10 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 ) .  

10.1 Program Text

Program Text (f07aqce.c)

10.2 Program Data

Program Data (f07aqce.d)

10.3 Program Results

Program Results (f07aqce.r)