NAG FL Interface
f07bvf (zgbrfs)

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1 Purpose

f07bvf returns error bounds for the solution of a complex band system of linear equations with multiple right-hand sides, AX=B, ATX=B or AHX=B. It improves the solution by iterative refinement, in order to reduce the backward error as much as possible.

2 Specification

Fortran Interface
Subroutine f07bvf ( trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info)
Integer, Intent (In) :: n, kl, ku, nrhs, ldab, ldafb, ipiv(*), ldb, ldx
Integer, Intent (Out) :: info
Real (Kind=nag_wp), Intent (Out) :: ferr(nrhs), berr(nrhs), rwork(n)
Complex (Kind=nag_wp), Intent (In) :: ab(ldab,*), afb(ldafb,*), b(ldb,*)
Complex (Kind=nag_wp), Intent (Inout) :: x(ldx,*)
Complex (Kind=nag_wp), Intent (Out) :: work(2*n)
Character (1), Intent (In) :: trans
C Header Interface
#include <nag.h>
void  f07bvf_ (const char *trans, const Integer *n, const Integer *kl, const Integer *ku, const Integer *nrhs, const Complex ab[], const Integer *ldab, const Complex afb[], const Integer *ldafb, const Integer ipiv[], const Complex b[], const Integer *ldb, Complex x[], const Integer *ldx, double ferr[], double berr[], Complex work[], double rwork[], Integer *info, const Charlen length_trans)
The routine may be called by the names f07bvf, nagf_lapacklin_zgbrfs or its LAPACK name zgbrfs.

3 Description

f07bvf returns the backward errors and estimated bounds on the forward errors for the solution of a complex band system of linear equations with multiple right-hand sides AX=B, ATX=B or AHX=B. The routine handles each right-hand side vector (stored as a column of the matrix B) independently, so we describe the function of f07bvf 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+δA)x=b+δb |δaij|β|aij|   and   |δbi|β|bi| .  
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 as follows:
trans='N'
The linear equations are of the form AX=B.
trans='T'
The linear equations are of the form ATX=B.
trans='C'
The linear equations are of the form AHX=B.
Constraint: trans='N', 'T' or 'C'.
2: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
3: kl Integer Input
On entry: kl, the number of subdiagonals within the band of the matrix A.
Constraint: kl0.
4: ku Integer Input
On entry: ku, the number of superdiagonals within the band of the matrix A.
Constraint: ku0.
5: nrhs Integer Input
On entry: r, the number of right-hand sides.
Constraint: nrhs0.
6: ab(ldab,*) Complex (Kind=nag_wp) array Input
Note: the second dimension of the array ab must be at least max(1,n).
On entry: the original n×n band matrix A as supplied to f07brf.
The matrix is stored in rows 1 to kl+ku+1, more precisely, the element Aij must be stored in
ab(ku+1+i-j,j)  for ​max(1,j-ku)imin(n,j+kl).  
See Section 9 in f07bnf for further details.
7: ldab Integer Input
On entry: the first dimension of the array ab as declared in the (sub)program from which f07bvf is called.
Constraint: ldabkl+ku+1.
8: afb(ldafb,*) Complex (Kind=nag_wp) array Input
Note: the second dimension of the array afb must be at least max(1,n).
On entry: the LU factorization of A, as returned by f07brf.
9: ldafb Integer Input
On entry: the first dimension of the array afb as declared in the (sub)program from which f07bvf is called.
Constraint: ldafb2×kl+ku+1.
10: ipiv(*) Integer array Input
Note: the dimension of the array ipiv must be at least max(1,n).
On entry: the pivot indices, as returned by f07brf.
11: b(ldb,*) Complex (Kind=nag_wp) array Input
Note: the second dimension of the array b must be at least max(1,nrhs).
On entry: the n×r right-hand side matrix B.
12: ldb Integer Input
On entry: the first dimension of the array b as declared in the (sub)program from which f07bvf is called.
Constraint: ldbmax(1,n).
13: x(ldx,*) Complex (Kind=nag_wp) array Input/Output
Note: the second dimension of the array x must be at least max(1,nrhs).
On entry: the n×r solution matrix X, as returned by f07bsf.
On exit: the improved solution matrix X.
14: ldx Integer Input
On entry: the first dimension of the array x as declared in the (sub)program from which f07bvf is called.
Constraint: ldxmax(1,n).
15: ferr(nrhs) Real (Kind=nag_wp) array Output
On exit: ferr(j) contains an estimated error bound for the jth solution vector, that is, the jth column of X, for j=1,2,,r.
16: berr(nrhs) Real (Kind=nag_wp) array Output
On exit: berr(j) 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: work(2×n) Complex (Kind=nag_wp) array Workspace
18: rwork(n) Real (Kind=nag_wp) array Workspace
19: info Integer Output
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

Background information to multithreading can be found in the Multithreading documentation.
f07bvf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07bvf 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 16n(kl+ku) real floating-point operations. Each step of iterative refinement involves an additional 8n(4kl+3ku) real 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 AHx=b; the number is usually 5 and never more than 11. Each solution involves approximately 8n(2kl+ku) real operations.
The real analogue of this routine is f07bhf.

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= ( -1.65+2.26i -2.05-0.85i 0.97-2.84i 0.00+0.00i 0.00+6.30i -1.48-1.75i -3.99+4.01i 0.59-0.48i 0.00+0.00i -0.77+2.83i -1.06+1.94i 3.33-1.04i 0.00+0.00i 0.00+0.00i 4.48-1.09i -0.46-1.72i )  
and
B= ( -1.06+21.50i 12.85+02.84i -22.72-53.90i -70.22+21.57i 28.24-38.60i -20.73-01.23i -34.56+16.73i 26.01+31.97i ) .  
Here A is nonsymmetric and is treated as a band matrix, which must first be factorized by f07brf.

10.1 Program Text

Program Text (f07bvfe.f90)

10.2 Program Data

Program Data (f07bvfe.d)

10.3 Program Results

Program Results (f07bvfe.r)