nag_dsprfs (f07phc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_dsprfs (f07phc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dsprfs (f07phc) returns error bounds for the solution of a real symmetric indefinite system of linear equations with multiple right-hand sides, AX=B, 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_dsprfs (Nag_OrderType order, Nag_UploType uplo, Integer n, Integer nrhs, const double ap[], const double afp[], const Integer ipiv[], const double b[], Integer pdb, double x[], Integer pdx, double ferr[], double berr[], NagError *fail)

3  Description

nag_dsprfs (f07phc) returns the backward errors and estimated bounds on the forward errors for the solution of a real symmetric indefinite system of linear equations with multiple right-hand sides AX=B, using packed storage. The function handles each right-hand side vector (stored as a column of the matrix B) independently, so we describe the function of nag_dsprfs (f07phc) in terms of a single right-hand side b and solution x.
Given a computed solution x, the function computes the component-wise backward error β. This is the size of the smallest relative perturbation in each element of A and b such that x is the exact solution of a perturbed system
A+δAx=b+δb δaijβaij   and   δbiβbi .
Then the function estimates a bound for the component-wise forward error in the computed solution, defined by:
maxixi-x^i/maxixi
where x^ 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:     orderNag_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 order=Nag_RowMajor. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     uploNag_UploTypeInput
On entry: specifies whether the upper or lower triangular part of A is stored and how A is to be factorized.
uplo=Nag_Upper
The upper triangular part of A is stored and A is factorized as PUDUTPT, where U is upper triangular.
uplo=Nag_Lower
The lower triangular part of A is stored and A is factorized as PLDLTPT, where L is lower triangular.
Constraint: uplo=Nag_Upper or Nag_Lower.
3:     nIntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
4:     nrhsIntegerInput
On entry: r, the number of right-hand sides.
Constraint: nrhs0.
5:     ap[dim]const doubleInput
Note: the dimension, dim, of the array ap must be at least max1,n×n+1/2.
On entry: the n by n original symmetric matrix A as supplied to nag_dsptrf (f07pdc).
6:     afp[dim]const doubleInput
Note: the dimension, dim, of the array afp must be at least max1,n×n+1/2.
On entry: the factorization of A stored in packed form, as returned by nag_dsptrf (f07pdc).
7:     ipiv[dim]const IntegerInput
Note: the dimension, dim, of the array ipiv must be at least max1,n.
On entry: details of the interchanges and the block structure of D, as returned by nag_dsptrf (f07pdc).
8:     b[dim]const doubleInput
Note: the dimension, dim, of the array b must be at least
  • max1,pdb×nrhs when order=Nag_ColMajor;
  • max1,n×pdb when order=Nag_RowMajor.
The i,jth 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 by r right-hand side matrix B.
9:     pdbIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
  • if order=Nag_ColMajor, pdbmax1,n;
  • if order=Nag_RowMajor, pdbmax1,nrhs.
10:   x[dim]doubleInput/Output
Note: the dimension, dim, of the array x must be at least
  • max1,pdx×nrhs when order=Nag_ColMajor;
  • max1,n×pdx when order=Nag_RowMajor.
The i,jth 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 entry: the n by r solution matrix X, as returned by nag_dsptrs (f07pec).
On exit: the improved solution matrix X.
11:   pdxIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array x.
Constraints:
  • if order=Nag_ColMajor, pdxmax1,n;
  • if order=Nag_RowMajor, pdxmax1,nrhs.
12:   ferr[nrhs]doubleOutput
On exit: ferr[j-1] contains an estimated error bound for the jth solution vector, that is, the jth column of X, for j=1,2,,r.
13:   berr[nrhs]doubleOutput
On exit: berr[j-1] contains the component-wise backward error bound β for the jth solution vector, that is, the jth column of X, for j=1,2,,r.
14:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, n=value.
Constraint: n0.
On entry, nrhs=value.
Constraint: nrhs0.
On entry, pdb=value.
Constraint: pdb>0.
On entry, pdx=value.
Constraint: pdx>0.
NE_INT_2
On entry, pdb=value and n=value.
Constraint: pdbmax1,n.
On entry, pdb=value and nrhs=value.
Constraint: pdbmax1,nrhs.
On entry, pdx=value and n=value.
Constraint: pdxmax1,n.
On entry, pdx=value and nrhs=value.
Constraint: pdxmax1,nrhs.
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.

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_dsprfs (f07phc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dsprfs (f07phc) 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 Users' Note for your implementation for any additional implementation-specific information.

9  Further Comments

For each right-hand side, computation of the backward error involves a minimum of 4n2 floating-point operations. Each step of iterative refinement involves an additional 6n2 operations. At most five steps of iterative refinement are performed, but usually only 1 or 2 steps are required.
Estimating the forward error involves solving a number of systems of linear equations of the form Ax=b; the number is usually 4 or 5 and never more than 11. Each solution involves approximately 2n2 operations.
The complex analogues of this function are nag_zhprfs (f07pvc) for Hermitian matrices and nag_zsprfs (f07qvc) for symmetric matrices.

10  Example

This example solves the system of equations AX=B using iterative refinement and to compute the forward and backward error bounds, where
A= 2.07 3.87 4.20 -1.15 3.87 -0.21 1.87 0.63 4.20 1.87 1.15 2.06 -1.15 0.63 2.06 -1.81   and   B= -9.50 27.85 -8.38 9.90 -6.07 19.25 -0.96 3.93 .
Here A is symmetric indefinite, stored in packed form, and must first be factorized by nag_dsptrf (f07pdc).

10.1  Program Text

Program Text (f07phce.c)

10.2  Program Data

Program Data (f07phce.d)

10.3  Program Results

Program Results (f07phce.r)


nag_dsprfs (f07phc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014