NAG Library Routine Document

f07bhf (dgbrfs)

1
Purpose

f07bhf (dgbrfs) 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

Fortran Interface
Subroutine f07bhf ( trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv, b, ldb, x, ldx, ferr, berr, work, iwork, info)
Integer, Intent (In):: n, kl, ku, nrhs, ldab, ldafb, ipiv(*), ldb, ldx
Integer, Intent (Out):: iwork(n), info
Real (Kind=nag_wp), Intent (In):: ab(ldab,*), afb(ldafb,*), b(ldb,*)
Real (Kind=nag_wp), Intent (Inout):: x(ldx,*)
Real (Kind=nag_wp), Intent (Out):: ferr(nrhs), berr(nrhs), work(3*n)
Character (1), Intent (In):: trans
C Header Interface
#include <nagmk26.h>
void  f07bhf_ (const char *trans, const Integer *n, const Integer *kl, const Integer *ku, const Integer *nrhs, const double ab[], const Integer *ldab, const double afb[], const Integer *ldafb, const Integer ipiv[], const double b[], const Integer *ldb, double x[], const Integer *ldx, double ferr[], double berr[], double work[], Integer iwork[], Integer *info, const Charlen length_trans)
The routine may be called by its LAPACK name dgbrfs.

3
Description

f07bhf (dgbrfs) 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 routine handles each right-hand side vector (stored as a column of the matrix B) independently, so we describe the function of f07bhf (dgbrfs) in terms of a single right-hand side b and solution x.
Given a computed solution x, the routine 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 routine 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:     trans – Character(1)Input
On entry: indicates the form of the linear equations for which X is the computed solution.
trans='N'
The linear equations are of the form AX=B.
trans='T' or 'C'
The linear equations are of the form ATX=B.
Constraint: trans='N', 'T' or 'C'.
2:     n – IntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
3:     kl – IntegerInput
On entry: kl, the number of subdiagonals within the band of the matrix A.
Constraint: kl0.
4:     ku – IntegerInput
On entry: ku, the number of superdiagonals within the band of the matrix A.
Constraint: ku0.
5:     nrhs – IntegerInput
On entry: r, the number of right-hand sides.
Constraint: nrhs0.
6:     abldab* – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array ab must be at least max1,n.
On entry: the original n by n band matrix A as supplied to f07bdf (dgbtrf).
The matrix is stored in rows 1 to kl+ku+1, more precisely, the element Aij must be stored in
abku+1+i-jj  for ​max1,j-kuiminn,j+kl. 
See Section 9 in f07baf (dgbsv) for further details.
7:     ldab – IntegerInput
On entry: the first dimension of the array ab as declared in the (sub)program from which f07bhf (dgbrfs) is called.
Constraint: ldabkl+ku+1.
8:     afbldafb* – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array afb must be at least max1,n.
On entry: the LU factorization of A, as returned by f07bdf (dgbtrf).
9:     ldafb – IntegerInput
On entry: the first dimension of the array afb as declared in the (sub)program from which f07bhf (dgbrfs) is called.
Constraint: ldafb2×kl+ku+1.
10:   ipiv* – Integer arrayInput
Note: the dimension of the array ipiv must be at least max1,n.
On entry: the pivot indices, as returned by f07bdf (dgbtrf).
11:   bldb* – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array b must be at least max1,nrhs.
On entry: the n by r right-hand side matrix B.
12:   ldb – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f07bhf (dgbrfs) is called.
Constraint: ldbmax1,n.
13:   xldx* – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array x must be at least max1,nrhs.
On entry: the n by r solution matrix X, as returned by f07bef (dgbtrs).
On exit: the improved solution matrix X.
14:   ldx – IntegerInput
On entry: the first dimension of the array x as declared in the (sub)program from which f07bhf (dgbrfs) is called.
Constraint: ldxmax1,n.
15:   ferrnrhs – Real (Kind=nag_wp) arrayOutput
On exit: ferrj contains an estimated error bound for the jth solution vector, that is, the jth column of X, for j=1,2,,r.
16:   berrnrhs – Real (Kind=nag_wp) arrayOutput
On exit: berrj 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.
17:   work3×n – Real (Kind=nag_wp) arrayWorkspace
18:   iworkn – Integer arrayWorkspace
19:   info – IntegerOutput
On exit: info=0 unless the routine detects an error (see Section 6).

6
Error Indicators and Warnings

info<0
If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.

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

f07bhf (dgbrfs) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07bhf (dgbrfs) 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 routine. Please also 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 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 routine is f07bvf (zgbrfs).

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= -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 f07bdf (dgbtrf).

10.1
Program Text

Program Text (f07bhfe.f90)

10.2
Program Data

Program Data (f07bhfe.d)

10.3
Program Results

Program Results (f07bhfe.r)