NAG Library Routine Document
F07AQF (ZCGESV)
1 Purpose
F07AQF (ZCGESV) 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
| SUBROUTINE F07AQF ( | N, NRHS, A, LDA, IPIV, B, LDB, X, LDX, WORK, SWORK, ITER, INFO) |
| INTEGER | N, NRHS, LDA, IPIV(N), LDB, LDX, ITER, INFO |
| complex*16 | A(LDA,N), B(LDB,NRHS), X(LDX,NRHS), WORK(N*NRHS) |
| complex | SWORK(N*(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
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
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 significiant 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/lugButtari 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:
N≥0.
- 2: NRHS – INTEGERInput
-
On entry:
r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint:
NRHS≥0.
- 3: A(LDA,N) – complex*16 arrayInput/Output
-
On entry: the n by n coefficient matrix A.
On exit: if iterative refinement has been successfully used (i.e., if INFO=0 and ITER≥0), 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:
LDA≥max1,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 ITER≥0) or to the double precision factorization (if INFO=0 and ITER<0).
- 6: B(LDB,NRHS) – complex*16 arrayInput
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:
LDB≥max1,N.
- 8: X(LDX,NRHS) – complex*16 arrayOutput
-
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:
LDX≥max1,N.
- 10: WORK(N*NRHS) – complex*16 arrayWorkspace
-
- 11: SWORK(N×N+NRHS) – complex arrayWorkspace
-
- 12: 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.
- 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.
- 13: 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:
- INFO<0
If INFO=-i, the ith argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
- INFO>0
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
where
and
ε
is the
machine precision. An approximate error bound for the computed solution is given by
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
where
A is the general matrix
9.1 Program Text
Program Text (f07aqfe.f)
9.2 Program Data
Program Data (f07aqfe.d)
9.3 Program Results
Program Results (f07aqfe.r)