NAG Library Function Document
nag_dpprfs (f07ghc)
1
Purpose
nag_dpprfs (f07ghc) returns error bounds for the solution of a real symmetric positive definite system of linear equations with multiple right-hand sides, , using packed storage. 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_dpprfs (Nag_OrderType order,
Nag_UploType uplo,
Integer n,
Integer nrhs,
const double ap[],
const double afp[],
const double b[],
Integer pdb,
double x[],
Integer pdx,
double ferr[],
double berr[],
NagError *fail) |
|
3
Description
nag_dpprfs (f07ghc) returns the backward errors and estimated bounds on the forward errors for the solution of a real symmetric positive definite system of linear equations with multiple right-hand sides , using packed storage. The function handles each right-hand side vector (stored as a column of the matrix ) independently, so we describe the function of nag_dpprfs (f07ghc) 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 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:
– Nag_UploTypeInput
-
On entry: specifies whether the upper or lower triangular part of
is stored and how
is to be factorized.
- The upper triangular part of is stored and is factorized as , where is upper triangular.
- The lower triangular part of is stored and is factorized as , where is lower triangular.
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
ap
must be at least
.
On entry: the
by
original symmetric positive definite matrix
as supplied to
nag_dpptrf (f07gdc).
- 6:
– const doubleInput
-
Note: the dimension,
dim, of the array
afp
must be at least
.
On entry: the Cholesky factor of
stored in packed form, as returned by
nag_dpptrf (f07gdc).
- 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:
– 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_dpptrs (f07gec).
On exit: the improved 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:
– doubleOutput
-
On exit: contains an estimated error bound for the th solution vector, that is, the th column of , for .
- 12:
– doubleOutput
-
On exit: contains the component-wise backward error bound for the th solution vector, that is, the th column of , for .
- 13:
– 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: .
- NE_INT_2
-
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.
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_dpprfs (f07ghc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dpprfs (f07ghc) 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 or steps are required.
Estimating the forward error involves solving a number of systems of linear equations of the form ; the number is usually or and never more than . Each solution involves approximately operations.
The complex analogue of this function is
nag_zpprfs (f07gvc).
10
Example
This example solves the system of equations
using iterative refinement and to compute the forward and backward error bounds, where
Here
is symmetric positive definite, stored in packed form, and must first be factorized by
nag_dpptrf (f07gdc).
10.1
Program Text
Program Text (f07ghce.c)
10.2
Program Data
Program Data (f07ghce.d)
10.3
Program Results
Program Results (f07ghce.r)