F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF07VHF (DTBRFS)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 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 ${A}^{\mathrm{T}}X=B$.

## 2  Specification

 SUBROUTINE F07VHF ( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
 INTEGER N, KD, NRHS, LDAB, LDB, LDX, IWORK(N), INFO REAL (KIND=nag_wp) AB(LDAB,*), B(LDB,*), X(LDX,*), FERR(NRHS), BERR(NRHS), WORK(3*N) CHARACTER(1) UPLO, TRANS, 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 ${A}^{\mathrm{T}}X=B$. The routine handles each right-hand side vector (stored as a column of the matrix $B$) independently, so we describe the function of 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 $\beta$. 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.

## 4  References

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

## 5  Parameters

1:     $\mathrm{UPLO}$ – CHARACTER(1)Input
On entry: specifies whether $A$ is upper or lower triangular.
${\mathbf{UPLO}}=\text{'U'}$
$A$ is upper triangular.
${\mathbf{UPLO}}=\text{'L'}$
$A$ is lower triangular.
Constraint: ${\mathbf{UPLO}}=\text{'U'}$ or $\text{'L'}$.
2:     $\mathrm{TRANS}$ – CHARACTER(1)Input
On entry: indicates the form of the equations.
${\mathbf{TRANS}}=\text{'N'}$
The equations are of the form $AX=B$.
${\mathbf{TRANS}}=\text{'T'}$ or $\text{'C'}$
The equations are of the form ${A}^{\mathrm{T}}X=B$.
Constraint: ${\mathbf{TRANS}}=\text{'N'}$, $\text{'T'}$ or $\text{'C'}$.
3:     $\mathrm{DIAG}$ – CHARACTER(1)Input
On entry: indicates whether $A$ is a nonunit or unit triangular matrix.
${\mathbf{DIAG}}=\text{'N'}$
$A$ is a nonunit triangular matrix.
${\mathbf{DIAG}}=\text{'U'}$
$A$ is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be $1$.
Constraint: ${\mathbf{DIAG}}=\text{'N'}$ or $\text{'U'}$.
4:     $\mathrm{N}$ – INTEGERInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
5:     $\mathrm{KD}$ – INTEGERInput
On entry: ${k}_{d}$, the number of superdiagonals of the matrix $A$ if ${\mathbf{UPLO}}=\text{'U'}$, or the number of subdiagonals if ${\mathbf{UPLO}}=\text{'L'}$.
Constraint: ${\mathbf{KD}}\ge 0$.
6:     $\mathrm{NRHS}$ – INTEGERInput
On entry: $r$, the number of right-hand sides.
Constraint: ${\mathbf{NRHS}}\ge 0$.
7:     $\mathrm{AB}\left({\mathbf{LDAB}},*\right)$ – REAL (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 $n$ by $n$ triangular band matrix $A$.
The matrix is stored in rows $1$ to ${k}_{d}+1$, more precisely,
• if ${\mathbf{UPLO}}=\text{'U'}$, the elements of the upper triangle of $A$ within the band must be stored with element ${A}_{ij}$ in ${\mathbf{AB}}\left({k}_{d}+1+i-j,j\right)\text{​ for ​}\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j-{k}_{d}\right)\le i\le j$;
• if ${\mathbf{UPLO}}=\text{'L'}$, the elements of the lower triangle of $A$ within the band must be stored with element ${A}_{ij}$ in ${\mathbf{AB}}\left(1+i-j,j\right)\text{​ for ​}j\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+{k}_{d}\right)\text{.}$
If ${\mathbf{DIAG}}=\text{'U'}$, the diagonal elements of $A$ are assumed to be $1$, and are not referenced.
8:     $\mathrm{LDAB}$ – INTEGERInput
On entry: the first dimension of the array AB as declared in the (sub)program from which F07VHF (DTBRFS) is called.
Constraint: ${\mathbf{LDAB}}\ge {\mathbf{KD}}+1$.
9:     $\mathrm{B}\left({\mathbf{LDB}},*\right)$ – REAL (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 $B$.
10:   $\mathrm{LDB}$ – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F07VHF (DTBRFS) is called.
Constraint: ${\mathbf{LDB}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
11:   $\mathrm{X}\left({\mathbf{LDX}},*\right)$ – REAL (KIND=nag_wp) arrayInput
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 F07VEF (DTBTRS).
12:   $\mathrm{LDX}$ – INTEGERInput
On entry: the first dimension of the array X as declared in the (sub)program from which F07VHF (DTBRFS) is called.
Constraint: ${\mathbf{LDX}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
13:   $\mathrm{FERR}\left({\mathbf{NRHS}}\right)$ – 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 $\mathit{j}=1,2,\dots ,r$.
14:   $\mathrm{BERR}\left({\mathbf{NRHS}}\right)$ – 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 $\mathit{j}=1,2,\dots ,r$.
15:   $\mathrm{WORK}\left(3×{\mathbf{N}}\right)$ – REAL (KIND=nag_wp) arrayWorkspace
16:   $\mathrm{IWORK}\left({\mathbf{N}}\right)$ – INTEGER arrayWorkspace
17:   $\mathrm{INFO}$ – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error 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.

## 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.

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 ${A}^{\mathrm{T}}x=b$; the number is usually $4$ or $5$ and never more than $11$. Each solution involves approximately $2nk$ floating-point operations (assuming $n\gg k$).
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)