NAG Library Function Document
nag_dsgesv (f07acc)
1
Purpose
nag_dsgesv (f07acc) computes the solution to a real system of linear equations
where
is an
by
matrix and
and
are
by
matrices.
2
Specification
#include <nag.h> |
#include <nagf07.h> |
void |
nag_dsgesv (Nag_OrderType order,
Integer n,
Integer nrhs,
double a[],
Integer pda,
Integer ipiv[],
const double b[],
Integer pdb,
double x[],
Integer pdx,
Integer *iter,
NagError *fail) |
|
3
Description
nag_dsgesv (f07acc) first attempts to factorize the matrix in single precision and use this factorization within an iterative refinement procedure to produce a solution with full 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
is the maximum number of iterations allowed, which is fixed at
iterations. The process is also stopped if for all right-hand sides we have
where
is the
-norm of the residual,
is the
-norm of the solution,
is the
-operator-norm of the matrix
and
is the
machine precision returned by
nag_machine_precision (X02AJC).
The iterative refinement strategy used by
nag_dsgesv (f07acc) 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_dsgesv (f07acc) always attempts the iterative refinement strategy first; you are advised to compare the performance of
nag_dsgesv (f07acc) with that of its full precision counterpart
nag_dgesv (f07aac) 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:
– Nag_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
. See
Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint:
or .
- 2:
– IntegerInput
-
On entry: , the number of linear equations, i.e., the order of the matrix .
Constraint:
.
- 3:
– IntegerInput
-
On entry: , the number of right-hand sides, i.e., the number of columns of the matrix .
Constraint:
.
- 4:
– doubleInput/Output
-
Note: the dimension,
dim, of the array
a
must be at least
.
The
th element of the matrix
is stored in
- when ;
- when .
On entry: the by coefficient matrix .
On exit: if iterative refinement has been successfully used (i.e., if NE_NOERROR and ), then is unchanged. If double precision factorization has been used (when NE_NOERROR and ), contains the factors and from the factorization ; the unit diagonal elements of are not stored.
- 5:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
a.
Constraint:
.
- 6:
– IntegerOutput
-
On exit: if no constraints are violated, the pivot indices that define the permutation matrix ; at the th step row of the matrix was interchanged with row . indicates a row interchange was not required. corresponds either to the single precision factorization (if NE_NOERROR and ) or to the double precision factorization (if NE_NOERROR and ).
- 7:
– const doubleInput
-
Note: the dimension,
dim, of the array
b
must be at least
- when
;
- when
.
The
th element of the matrix
is stored in
- when ;
- when .
On entry: the by right-hand side matrix .
- 8:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
b.
Constraints:
- if ,
;
- if , .
- 9:
– doubleOutput
-
Note: the dimension,
dim, of the array
x
must be at least
- when
;
- when
.
The
th element of the matrix
is stored in
- when ;
- when .
On exit: if NE_NOERROR, the by solution matrix .
- 10:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
x.
Constraints:
- if ,
;
- if , .
- 11:
– Integer *Output
-
On exit: if
, iterative refinement has been successfully used and
iter is the number of iterations carried out.
If , 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:
– NagError *Input/Output
-
The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_INT
-
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_INT_2
-
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
- 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.
See
Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
- NE_SINGULAR
-
Element of the diagonal is exactly zero.
The factorization has been completed, but the factor is exactly singular, so the solution could not be computed.
7
Accuracy
The computed solution for a single right-hand side,
, satisfies the equation of the form
where
and
is the
machine precision. An approximate error bound for the computed solution is given by
where
, the condition number of
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
nag_dsgesv (f07acc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dsgesv (f07acc) 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.
The complex analogue of this function is
nag_zcgesv (f07aqc).
10
Example
This example solves the equations
where
is the general matrix
10.1
Program Text
Program Text (f07acce.c)
10.2
Program Data
Program Data (f07acce.d)
10.3
Program Results
Program Results (f07acce.r)