NAG Library Routine Document

f07vhf (dtbrfs)

1
Purpose

f07vhf (dtbrfs) returns error bounds for the solution of a real triangular band system of linear equations with multiple right-hand sides, AX=B or ATX=B.

2
Specification

Fortran Interface
Subroutine f07vhf ( uplo, trans, diag, n, kd, nrhs, ab, ldab, b, ldb, x, ldx, ferr, berr, work, iwork, info)
Integer, Intent (In):: n, kd, nrhs, ldab, ldb, ldx
Integer, Intent (Out):: iwork(n), info
Real (Kind=nag_wp), Intent (In):: ab(ldab,*), b(ldb,*), x(ldx,*)
Real (Kind=nag_wp), Intent (Out):: ferr(nrhs), berr(nrhs), work(3*n)
Character (1), Intent (In):: uplo, trans, diag
C Header Interface
#include <nagmk26.h>
void  f07vhf_ (const char *uplo, const char *trans, const char *diag, const Integer *n, const Integer *kd, const Integer *nrhs, const double ab[], const Integer *ldab, const double b[], const Integer *ldb, const double x[], const Integer *ldx, double ferr[], double berr[], double work[], Integer iwork[], Integer *info, const Charlen length_uplo, const Charlen length_trans, const Charlen length_diag)
The routine may be called by its LAPACK name dtbrfs.

3
Description

f07vhf (dtbrfs) returns the backward errors and estimated bounds on the forward errors for the solution of a real triangular 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 f07vhf (dtbrfs) 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:
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:     uplo – Character(1)Input
On entry: specifies whether A is upper or lower triangular.
uplo='U'
A is upper triangular.
uplo='L'
A is lower triangular.
Constraint: uplo='U' or 'L'.
2:     trans – Character(1)Input
On entry: indicates the form of the equations.
trans='N'
The equations are of the form AX=B.
trans='T' or 'C'
The equations are of the form ATX=B.
Constraint: trans='N', 'T' or 'C'.
3:     diag – Character(1)Input
On entry: indicates whether A is a nonunit or unit triangular matrix.
diag='N'
A is a nonunit triangular matrix.
diag='U'
A is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be 1.
Constraint: diag='N' or 'U'.
4:     n – IntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
5:     kd – IntegerInput
On entry: kd, the number of superdiagonals of the matrix A if uplo='U', or the number of subdiagonals if uplo='L'.
Constraint: kd0.
6:     nrhs – IntegerInput
On entry: r, the number of right-hand sides.
Constraint: nrhs0.
7:     abldab* – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array ab must be at least max1,n.
On entry: the n by n triangular band matrix A.
The matrix is stored in rows 1 to kd+1, more precisely,
  • if uplo='U', the elements of the upper triangle of A within the band must be stored with element Aij in abkd+1+i-jj​ for ​max1,j-kdij;
  • if uplo='L', the elements of the lower triangle of A within the band must be stored with element Aij in ab1+i-jj​ for ​jiminn,j+kd.
If diag='U', the diagonal elements of A are assumed to be 1, and are not referenced.
8:     ldab – IntegerInput
On entry: the first dimension of the array ab as declared in the (sub)program from which f07vhf (dtbrfs) is called.
Constraint: ldabkd+1.
9:     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.
10:   ldb – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f07vhf (dtbrfs) is called.
Constraint: ldbmax1,n.
11:   xldx* – Real (Kind=nag_wp) arrayInput
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 f07vef (dtbtrs).
12:   ldx – IntegerInput
On entry: the first dimension of the array x as declared in the (sub)program from which f07vhf (dtbrfs) is called.
Constraint: ldxmax1,n.
13:   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.
14:   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.
15:   work3×n – Real (Kind=nag_wp) arrayWorkspace
16:   iworkn – Integer arrayWorkspace
17:   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

f07vhf (dtbrfs) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07vhf (dtbrfs) 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

A call to f07vhf (dtbrfs), for each right-hand side, 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 2nk floating-point operations (assuming nk).
The complex analogue of this routine is f07vvf (ztbrfs).

10
Example

This example solves the system of equations AX=B and to compute forward and backward error bounds, where
A= -4.16 0.00 0.00 0.00 -2.25 4.78 0.00 0.00 0.00 5.86 6.32 0.00 0.00 0.00 -4.82 0.16   and   B= -16.64 -4.16 -13.78 -16.59 13.10 -4.94 -14.14 -9.96 .  

10.1
Program Text

Program Text (f07vhfe.f90)

10.2
Program Data

Program Data (f07vhfe.d)

10.3
Program Results

Program Results (f07vhfe.r)