NAG FL Interfacef07bhf (dgbrfs)

1Purpose

f07bhf returns error bounds for the solution of a real band system of linear equations with multiple right-hand sides, $AX=B$ or ${A}^{\mathrm{T}}X=B$. It improves the solution by iterative refinement, in order to reduce the backward error as much as possible.

2Specification

Fortran Interface
 Subroutine f07bhf ( n, kl, ku, nrhs, ab, ldab, afb, ipiv, b, ldb, x, ldx, ferr, berr, work, 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
#include <nag.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 the names f07bhf, nagf_lapacklin_dgbrfs or its LAPACK name dgbrfs.

3Description

f07bhf 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 ${A}^{\mathrm{T}}X=B$. The routine handles each right-hand side vector (stored as a column of the matrix $B$) independently, so we describe the function of f07bhf 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 $\beta$. 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
 $A+δAx=b+δb δaij≤βaij and δbi≤βbi .$
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 $\stackrel{^}{x}$ is the true solution.
For details of the method, see the F07 Chapter Introduction.

4References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5Arguments

1: $\mathbf{trans}$Character(1) Input
On entry: indicates the form of the linear equations for which $X$ is the computed solution.
${\mathbf{trans}}=\text{'N'}$
The linear equations are of the form $AX=B$.
${\mathbf{trans}}=\text{'T'}$ or $\text{'C'}$
The linear equations are of the form ${A}^{\mathrm{T}}X=B$.
Constraint: ${\mathbf{trans}}=\text{'N'}$, $\text{'T'}$ or $\text{'C'}$.
2: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3: $\mathbf{kl}$Integer Input
On entry: ${k}_{l}$, the number of subdiagonals within the band of the matrix $A$.
Constraint: ${\mathbf{kl}}\ge 0$.
4: $\mathbf{ku}$Integer Input
On entry: ${k}_{u}$, the number of superdiagonals within the band of the matrix $A$.
Constraint: ${\mathbf{ku}}\ge 0$.
5: $\mathbf{nrhs}$Integer Input
On entry: $r$, the number of right-hand sides.
Constraint: ${\mathbf{nrhs}}\ge 0$.
6: $\mathbf{ab}\left({\mathbf{ldab}},*\right)$Real (Kind=nag_wp) array Input
Note: the second dimension of the array ab must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the original $n$ by $n$ band matrix $A$ as supplied to f07bdf.
The matrix is stored in rows $1$ to ${k}_{l}+{k}_{u}+1$, more precisely, the element ${A}_{ij}$ must be stored in
 $abku+1+i-jj for ​max1,j-ku≤i≤minn,j+kl.$
See Section 9 in f07baf for further details.
7: $\mathbf{ldab}$Integer Input
On entry: the first dimension of the array ab as declared in the (sub)program from which f07bhf is called.
Constraint: ${\mathbf{ldab}}\ge {\mathbf{kl}}+{\mathbf{ku}}+1$.
8: $\mathbf{afb}\left({\mathbf{ldafb}},*\right)$Real (Kind=nag_wp) array Input
Note: the second dimension of the array afb must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $LU$ factorization of $A$, as returned by f07bdf.
9: $\mathbf{ldafb}$Integer Input
On entry: the first dimension of the array afb as declared in the (sub)program from which f07bhf is called.
Constraint: ${\mathbf{ldafb}}\ge 2×{\mathbf{kl}}+{\mathbf{ku}}+1$.
10: $\mathbf{ipiv}\left(*\right)$Integer array Input
Note: the dimension of the array ipiv must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the pivot indices, as returned by f07bdf.
11: $\mathbf{b}\left({\mathbf{ldb}},*\right)$Real (Kind=nag_wp) array Input
Note: the second dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
On entry: the $n$ by $r$ right-hand side matrix $B$.
12: $\mathbf{ldb}$Integer Input
On entry: the first dimension of the array b as declared in the (sub)program from which f07bhf is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
13: $\mathbf{x}\left({\mathbf{ldx}},*\right)$Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array x must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
On entry: the $n$ by $r$ solution matrix $X$, as returned by f07bef.
On exit: the improved solution matrix $X$.
14: $\mathbf{ldx}$Integer Input
On entry: the first dimension of the array x as declared in the (sub)program from which f07bhf is called.
Constraint: ${\mathbf{ldx}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
15: $\mathbf{ferr}\left({\mathbf{nrhs}}\right)$Real (Kind=nag_wp) array Output
On exit: ${\mathbf{ferr}}\left(\mathit{j}\right)$ contains an estimated error bound for the $\mathit{j}$th solution vector, that is, the $\mathit{j}$th column of $X$, for $\mathit{j}=1,2,\dots ,r$.
16: $\mathbf{berr}\left({\mathbf{nrhs}}\right)$Real (Kind=nag_wp) array Output
On exit: ${\mathbf{berr}}\left(\mathit{j}\right)$ contains the component-wise backward error bound $\beta$ for the $\mathit{j}$th solution vector, that is, the $\mathit{j}$th column of $X$, for $\mathit{j}=1,2,\dots ,r$.
17: $\mathbf{work}\left(3×{\mathbf{n}}\right)$Real (Kind=nag_wp) array Workspace
18: $\mathbf{iwork}\left({\mathbf{n}}\right)$Integer array Workspace
19: $\mathbf{info}$Integer Output
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

6Error Indicators and Warnings

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

7Accuracy

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.

8Parallelism and Performance

f07bhf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07bhf 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.

For each right-hand side, computation of the backward error involves a minimum of $4n\left({k}_{l}+{k}_{u}\right)$ floating-point operations. Each step of iterative refinement involves an additional $2n\left(4{k}_{l}+3{k}_{u}\right)$ operations. This assumes $n\gg {k}_{l}$ and $n\gg {k}_{u}$. 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 ${A}^{\mathrm{T}}x=b$; the number is usually $4$ or $5$ and never more than $11$. Each solution involves approximately $2n\left(2{k}_{l}+{k}_{u}\right)$ operations.
The complex analogue of this routine is f07bvf.

10Example

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.

10.1Program Text

Program Text (f07bhfe.f90)

10.2Program Data

Program Data (f07bhfe.d)

10.3Program Results

Program Results (f07bhfe.r)