# NAG Library Routine Document

## 1Purpose

f07bvf (zgbrfs) returns error bounds for the solution of a complex band system of linear equations with multiple right-hand sides, $AX=B$, ${A}^{\mathrm{T}}X=B$ or ${A}^{\mathrm{H}}X=B$. It improves the solution by iterative refinement, in order to reduce the backward error as much as possible.

## 2Specification

Fortran Interface
 Subroutine f07bvf ( 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) :: 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
#include nagmk26.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 its LAPACK name zgbrfs.

## 3Description

f07bvf (zgbrfs) 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 $AX=B$, ${A}^{\mathrm{T}}X=B$ or ${A}^{\mathrm{H}}X=B$. The routine handles each right-hand side vector (stored as a column of the matrix $B$) independently, so we describe the function of f07bvf (zgbrfs) 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:
 $maxixi-x^i/maxixi$
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 as follows:
${\mathbf{trans}}=\text{'N'}$
The linear equations are of the form $AX=B$.
${\mathbf{trans}}=\text{'T'}$
The linear equations are of the form ${A}^{\mathrm{T}}X=B$.
${\mathbf{trans}}=\text{'C'}$
The linear equations are of the form ${A}^{\mathrm{H}}X=B$.
Constraint: ${\mathbf{trans}}=\text{'N'}$, $\text{'T'}$ or $\text{'C'}$.
2:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     $\mathbf{kl}$ – IntegerInput
On entry: ${k}_{l}$, the number of subdiagonals within the band of the matrix $A$.
Constraint: ${\mathbf{kl}}\ge 0$.
4:     $\mathbf{ku}$ – IntegerInput
On entry: ${k}_{u}$, the number of superdiagonals within the band of the matrix $A$.
Constraint: ${\mathbf{ku}}\ge 0$.
5:     $\mathbf{nrhs}$ – IntegerInput
On entry: $r$, the number of right-hand sides.
Constraint: ${\mathbf{nrhs}}\ge 0$.
6:     $\mathbf{ab}\left({\mathbf{ldab}},*\right)$ – Complex (Kind=nag_wp) arrayInput
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 f07brf (zgbtrf).
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 f07bnf (zgbsv) for further details.
7:     $\mathbf{ldab}$ – IntegerInput
On entry: the first dimension of the array ab as declared in the (sub)program from which f07bvf (zgbrfs) is called.
Constraint: ${\mathbf{ldab}}\ge {\mathbf{kl}}+{\mathbf{ku}}+1$.
8:     $\mathbf{afb}\left({\mathbf{ldafb}},*\right)$ – Complex (Kind=nag_wp) arrayInput
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 f07brf (zgbtrf).
9:     $\mathbf{ldafb}$ – IntegerInput
On entry: the first dimension of the array afb as declared in the (sub)program from which f07bvf (zgbrfs) is called.
Constraint: ${\mathbf{ldafb}}\ge 2×{\mathbf{kl}}+{\mathbf{ku}}+1$.
10:   $\mathbf{ipiv}\left(*\right)$ – Integer arrayInput
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 f07brf (zgbtrf).
11:   $\mathbf{b}\left({\mathbf{ldb}},*\right)$ – Complex (Kind=nag_wp) arrayInput
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:   $\mathbf{ldb}$ – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f07bvf (zgbrfs) is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
13:   $\mathbf{x}\left({\mathbf{ldx}},*\right)$ – Complex (Kind=nag_wp) arrayInput/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 f07bsf (zgbtrs).
On exit: the improved solution matrix $X$.
14:   $\mathbf{ldx}$ – IntegerInput
On entry: the first dimension of the array x as declared in the (sub)program from which f07bvf (zgbrfs) 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) arrayOutput
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) arrayOutput
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:   $\mathbf{work}\left(2×{\mathbf{n}}\right)$ – Complex (Kind=nag_wp) arrayWorkspace
18:   $\mathbf{rwork}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
19:   $\mathbf{info}$ – IntegerOutput
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

f07bvf (zgbrfs) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07bvf (zgbrfs) 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 $16n\left({k}_{l}+{k}_{u}\right)$ real floating-point operations. Each step of iterative refinement involves an additional $8n\left(4{k}_{l}+3{k}_{u}\right)$ real 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 ${A}^{\mathrm{H}}x=b$; the number is usually $5$ and never more than $11$. Each solution involves approximately $8n\left(2{k}_{l}+{k}_{u}\right)$ real operations.
The real analogue of this routine is f07bhf (dgbrfs).

## 10Example

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 (zgbtrf).

### 10.1Program Text

Program Text (f07bvfe.f90)

### 10.2Program Data

Program Data (f07bvfe.d)

### 10.3Program Results

Program Results (f07bvfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017