nag_dgbrfs (f07bhc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_dgbrfs (f07bhc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dgbrfs (f07bhc) returns error bounds for the solution of a real band system of linear equations with multiple right-hand sides, AX=B or ATX=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_dgbrfs (Nag_OrderType order, Nag_TransType trans, Integer n, Integer kl, Integer ku, Integer nrhs, const double ab[], Integer pdab, const double afb[], Integer pdafb, const Integer ipiv[], const double b[], Integer pdb, double x[], Integer pdx, double ferr[], double berr[], NagError *fail)

3  Description

nag_dgbrfs (f07bhc) returns the backward errors and estimated bounds on the forward errors for the solution of a real band system of linear equations with multiple right-hand sides AX=B or ATX=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_dgbrfs (f07bhc) 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:
maxi xi - x^i / maxi xi
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:     transNag_TransTypeInput
On entry: indicates the form of the linear equations for which X is the computed solution.
trans=Nag_NoTrans
The linear equations are of the form AX=B.
trans=Nag_Trans or Nag_ConjTrans
The linear equations are of the form ATX=B.
Constraint: trans=Nag_NoTrans, Nag_Trans or Nag_ConjTrans.
3:     nIntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
4:     klIntegerInput
On entry: kl, the number of subdiagonals within the band of the matrix A.
Constraint: kl0.
5:     kuIntegerInput
On entry: ku, the number of superdiagonals within the band of the matrix A.
Constraint: ku0.
6:     nrhsIntegerInput
On entry: r, the number of right-hand sides.
Constraint: nrhs0.
7:     ab[dim]const doubleInput
Note: the dimension, dim, of the array ab must be at least max1,pdab×n.
On entry: the original n by n band matrix A as supplied to nag_dgbtrf (f07bdc) but with reduced requirements since the matrix is not factorized.
This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements Aij, for row i=1,,n and column j=max1,i-kl,,minn,i+ku, depends on the order argument as follows:
  • if order=Nag_ColMajor, Aij is stored as ab[j-1×pdab+ku+i-j];
  • if order=Nag_RowMajor, Aij is stored as ab[i-1×pdab+kl+j-i].
See Section 8 in nag_dgbsv (f07bac) for further details.
8:     pdabIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix A in the array ab.
Constraint: pdabkl+ku+1.
9:     afb[dim]const doubleInput
Note: the dimension, dim, of the array afb must be at least max1,pdafb×n.
On entry: the LU factorization of A, as returned by nag_dgbtrf (f07bdc).
10:   pdafbIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix in the array afb.
Constraint: pdafb2×kl+ku+1.
11:   ipiv[dim]const IntegerInput
Note: the dimension, dim, of the array ipiv must be at least max1,n.
On entry: the pivot indices, as returned by nag_dgbtrf (f07bdc).
12:   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.
13:   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.
14:   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_dgbtrs (f07bec).
On exit: the improved solution matrix X.
15:   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.
16:   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.
17:   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.
18:   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, kl=value.
Constraint: kl0.
On entry, ku=value.
Constraint: ku0.
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, 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_INT_3
On entry, pdab=value, kl=value and ku=value.
Constraint: pdabkl+ku+1.
On entry, pdafb=value, kl=value and ku=value.
Constraint: pdafb2×kl+ku+1.
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 4nkl+ku floating point operations. Each step of iterative refinement involves an additional 2n4kl+3ku operations. This assumes nkl and nku. 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 or ATx=b; the number is usually 4 or 5 and never more than 11. Each solution involves approximately 2n2kl+ku operations.
The complex analogue of this function is nag_zgbrfs (f07bvc).

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= -0.23 2.54 -3.66 0.00 -6.98 2.46 -2.73 -2.13 0.00 2.56 2.46 4.07 0.00 0.00 -4.78 -3.82   and   B= 4.42 -36.01 27.13 -31.67 -6.14 -1.16 10.50 -25.82 .
Here A is nonsymmetric and is treated as a band matrix, which must first be factorized by nag_dgbtrf (f07bdc).

9.1  Program Text

Program Text (f07bhce.c)

9.2  Program Data

Program Data (f07bhce.d)

9.3  Program Results

Program Results (f07bhce.r)


nag_dgbrfs (f07bhc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

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