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:     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:     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:     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:     N – INTEGERInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
5:     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:     NRHS – INTEGERInput
On entry: $r$, the number of right-hand sides.
Constraint: ${\mathbf{NRHS}}\ge 0$.
7:     AB(LDAB,$*$) – 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:     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:     B(LDB,$*$) – 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:   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:   X(LDX,$*$) – 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:   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:   FERR(NRHS) – 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:   BERR(NRHS) – 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:   WORK($3×{\mathbf{N}}$) – REAL (KIND=nag_wp) arrayWorkspace
16:   IWORK(N) – INTEGER arrayWorkspace
17:   INFO – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error Indicators and Warnings

Errors or warnings detected by the routine:
${\mathbf{INFO}}<0$
If ${\mathbf{INFO}}=-i$, the $i$th parameter 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.

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).

## 9  Example

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

### 9.1  Program Text

Program Text (f07vhfe.f90)

### 9.2  Program Data

Program Data (f07vhfe.d)

### 9.3  Program Results

Program Results (f07vhfe.r)