nag_dpbrfs (f07hhc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_dpbrfs (f07hhc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dpbrfs (f07hhc) returns error bounds for the solution of a real symmetric positive definite band system of linear equations with multiple right-hand sides, AX=B. 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_dpbrfs (Nag_OrderType order, Nag_UploType uplo, Integer n, Integer kd, Integer nrhs, const double ab[], Integer pdab, const double afb[], Integer pdafb, const double b[], Integer pdb, double x[], Integer pdx, double ferr[], double berr[], NagError *fail)

3  Description

nag_dpbrfs (f07hhc) returns the backward errors and estimated bounds on the forward errors for the solution of a real symmetric positive definite band system of linear equations with multiple right-hand sides AX=B. The function handles each right-hand side vector (stored as a column of the matrix B) independently, so we describe the function of nag_dpbrfs (f07hhc) 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 UTU, where U is upper triangular.
uplo=Nag_Lower
The lower triangular part of A is stored and A is factorized as LLT, 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:     kdIntegerInput
On entry: kd, the number of superdiagonals or subdiagonals of the matrix A.
Constraint: kd0.
5:     nrhsIntegerInput
On entry: r, the number of right-hand sides.
Constraint: nrhs0.
6:     ab[dim]const doubleInput
Note: the dimension, dim, of the array ab must be at least max1,pdab×n.
On entry: the n by n original symmetric positive definite band matrix A as supplied to nag_dpbtrf (f07hdc).
7:     pdabIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix in the array ab.
Constraint: pdabkd+1.
8:     afb[dim]const doubleInput
Note: the dimension, dim, of the array afb must be at least max1,pdafb×n.
On entry: the Cholesky factor of A, as returned by nag_dpbtrf (f07hdc).
9:     pdafbIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix in the array afb.
Constraint: pdafbkd+1.
10:   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.
11:   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.
12:   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_dpbtrs (f07hec).
On exit: the improved solution matrix X.
13:   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.
14:   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.
15:   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.
16:   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, kd=value.
Constraint: kd0.
On entry, n=value.
Constraint: n0.
On entry, nrhs=value.
Constraint: nrhs0.
On entry, pdab=value.
Constraint: pdab>0.
On entry, pdafb=value.
Constraint: pdafb>0.
On entry, pdb=value.
Constraint: pdb>0.
On entry, pdx=value.
Constraint: pdx>0.
NE_INT_2
On entry, pdab=value and kd=value.
Constraint: pdabkd+1.
On entry, pdafb=value and kd=value.
Constraint: pdafbkd+1.
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  Further Comments

For each right-hand side, computation of the backward error involves a minimum of 8nk floating point operations. Each step of iterative refinement involves an additional 12nk operations. This assumes nk. 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 Ax=b; the number is usually 4 or 5 and never more than 11. Each solution involves approximately 4nk operations.
The complex analogue of this function is nag_zpbrfs (f07hvc).

9  Example

This example solves the system of equations AX=B using iterative refinement and to compute the forward and backward error bounds, where
A= 5.49 2.68 0.00 0.00 2.68 5.63 -2.39 0.00 0.00 -2.39 2.60 -2.22 0.00 0.00 -2.22 5.17   and   B= 22.09 5.10 9.31 30.81 -5.24 -25.82 11.83 22.90 .
Here A is symmetric and positive definite, and is treated as a band matrix, which must first be factorized by nag_dpbtrf (f07hdc).

9.1  Program Text

Program Text (f07hhce.c)

9.2  Program Data

Program Data (f07hhce.d)

9.3  Program Results

Program Results (f07hhce.r)


nag_dpbrfs (f07hhc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012