NAG Library Function Document
nag_dgerfs (f07ahc)
1 Purpose
nag_dgerfs (f07ahc) returns error bounds for the solution of a real system of linear equations with multiple right-hand sides, or . It improves the solution by iterative refinement, in order to reduce the backward error as much as possible.
2 Specification
#include <nag.h> |
#include <nagf07.h> |
void |
nag_dgerfs (Nag_OrderType order,
Nag_TransType trans,
Integer n,
Integer nrhs,
const double a[],
Integer pda,
const double af[],
Integer pdaf,
const Integer ipiv[],
const double b[],
Integer pdb,
double x[],
Integer pdx,
double ferr[],
double berr[],
NagError *fail) |
|
3 Description
nag_dgerfs (f07ahc) returns the backward errors and estimated bounds on the forward errors for the solution of a real system of linear equations with multiple right-hand sides or . The function handles each right-hand side vector (stored as a column of the matrix ) independently, so we describe the function of nag_dgerfs (f07ahc) in terms of a single right-hand side and solution .
Given a computed solution
, the function computes the
component-wise backward error
. This is the size of the smallest relative perturbation in each element of
and
such that
is the exact solution of a perturbed system
Then the function estimates a bound for the
component-wise forward error in the computed solution, defined by:
where
is the true solution.
For details of the method, see the
f07 Chapter Introduction.
4 References
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 2.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint:
or .
- 2:
– Nag_TransTypeInput
-
On entry: indicates the form of the linear equations for which
is the computed solution.
- The linear equations are of the form .
- or
- The linear equations are of the form .
Constraint:
, or .
- 3:
– IntegerInput
-
On entry: , the order of the matrix .
Constraint:
.
- 4:
– IntegerInput
-
On entry: , the number of right-hand sides.
Constraint:
.
- 5:
– const doubleInput
-
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
original matrix
as supplied to
nag_dgetrf (f07adc).
- 6:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
a.
Constraint:
.
- 7:
– const doubleInput
-
Note: the dimension,
dim, of the array
af
must be at least
.
The
th element of the matrix is stored in
- when ;
- when .
On entry: the
factorization of
, as returned by
nag_dgetrf (f07adc).
- 8:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
af.
Constraint:
.
- 9:
– const IntegerInput
-
Note: the dimension,
dim, of the array
ipiv
must be at least
.
On entry: the pivot indices, as returned by
nag_dgetrf (f07adc).
- 10:
– 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 .
- 11:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
b.
Constraints:
- if ,
;
- if , .
- 12:
– doubleInput/Output
-
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 entry: the
by
solution matrix
, as returned by
nag_dgetrs (f07aec).
On exit: the improved solution matrix .
- 13:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
x.
Constraints:
- if ,
;
- if , .
- 14:
– doubleOutput
-
On exit: contains an estimated error bound for the th solution vector, that is, the th column of , for .
- 15:
– doubleOutput
-
On exit: contains the component-wise backward error bound for the th solution vector, that is, the th column of , for .
- 16:
– NagError *Input/Output
-
The NAG error argument (see
Section 2.7 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: .
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: .
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.
An unexpected error has been triggered by this function. Please contact
NAG.
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.
7 Accuracy
The bounds returned in
ferr are not rigorous, because they are estimated, not computed exactly; but in practice they almost always overestimate the actual error.
8 Parallelism and Performance
nag_dgerfs (f07ahc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dgerfs (f07ahc) 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.
For each right-hand side, computation of the backward error involves a minimum of floating-point operations. Each step of iterative refinement involves an additional operations. At most five steps of iterative refinement are performed, but usually only one or two steps are required.
Estimating the forward error involves solving a number of systems of linear equations of the form or ; the number is usually or and never more than . Each solution involves approximately operations.
The complex analogue of this function is
nag_zgerfs (f07avc).
10 Example
This example solves the system of equations
using iterative refinement and to compute the forward and backward error bounds, where
Here
is nonsymmetric and must first be factorized by
nag_dgetrf (f07adc).
10.1 Program Text
Program Text (f07ahce.c)
10.2 Program Data
Program Data (f07ahce.d)
10.3 Program Results
Program Results (f07ahce.r)