# NAG Library Routine Document

## 1Purpose

f07thf (dtrrfs) returns error bounds for the solution of a real triangular system of linear equations with multiple right-hand sides, $AX=B$ or ${A}^{\mathrm{T}}X=B$.

## 2Specification

Fortran Interface
 Subroutine f07thf ( uplo, diag, n, nrhs, a, lda, b, ldb, x, ldx, ferr, berr, work, info)
 Integer, Intent (In) :: n, nrhs, lda, ldb, ldx Integer, Intent (Out) :: iwork(n), info Real (Kind=nag_wp), Intent (In) :: a(lda,*), b(ldb,*), x(ldx,*) Real (Kind=nag_wp), Intent (Out) :: ferr(nrhs), berr(nrhs), work(3*n) Character (1), Intent (In) :: uplo, trans, diag
#include nagmk26.h
 void f07thf_ (const char *uplo, const char *trans, const char *diag, const Integer *n, const Integer *nrhs, const double a[], const Integer *lda, const double b[], const Integer *ldb, const double x[], const Integer *ldx, double ferr[], double berr[], double work[], Integer iwork[], Integer *info, const Charlen length_uplo, const Charlen length_trans, const Charlen length_diag)
The routine may be called by its LAPACK name dtrrfs.

## 3Description

f07thf (dtrrfs) returns the backward errors and estimated bounds on the forward errors for the solution of a real triangular 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 f07thf (dtrrfs) 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:     $\mathbf{uplo}$ – 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:     $\mathbf{trans}$ – 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:     $\mathbf{diag}$ – 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:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
5:     $\mathbf{nrhs}$ – IntegerInput
On entry: $r$, the number of right-hand sides.
Constraint: ${\mathbf{nrhs}}\ge 0$.
6:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $n$ by $n$ triangular matrix $A$.
• If ${\mathbf{uplo}}=\text{'U'}$, $A$ is upper triangular and the elements of the array below the diagonal are not referenced.
• If ${\mathbf{uplo}}=\text{'L'}$, $A$ is lower triangular and the elements of the array above the diagonal are not referenced.
• If ${\mathbf{diag}}=\text{'U'}$, the diagonal elements of $A$ are assumed to be $1$, and are not referenced.
7:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f07thf (dtrrfs) is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
8:     $\mathbf{b}\left({\mathbf{ldb}},*\right)$ – 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 $B$.
9:     $\mathbf{ldb}$ – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f07thf (dtrrfs) is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
10:   $\mathbf{x}\left({\mathbf{ldx}},*\right)$ – 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 f07tef (dtrtrs).
11:   $\mathbf{ldx}$ – IntegerInput
On entry: the first dimension of the array x as declared in the (sub)program from which f07thf (dtrrfs) is called.
Constraint: ${\mathbf{ldx}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
12:   $\mathbf{ferr}\left({\mathbf{nrhs}}\right)$ – 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 $\mathit{j}=1,2,\dots ,r$.
13:   $\mathbf{berr}\left({\mathbf{nrhs}}\right)$ – 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 $\mathit{j}=1,2,\dots ,r$.
14:   $\mathbf{work}\left(3×{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
15:   $\mathbf{iwork}\left({\mathbf{n}}\right)$ – Integer arrayWorkspace
16:   $\mathbf{info}$ – 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

f07thf (dtrrfs) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07thf (dtrrfs) 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 f07thf (dtrrfs), 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 ${n}^{2}$ floating-point operations.
The complex analogue of this routine is f07tvf (ztrrfs).

## 10Example

This example solves the system of equations $AX=B$ and to compute forward and backward error bounds, where
 $A= 4.30 0.00 0.00 0.00 -3.96 -4.87 0.00 0.00 0.40 0.31 -8.02 0.00 -0.27 0.07 -5.95 0.12 and B= -12.90 -21.50 16.75 14.93 -17.55 6.33 -11.04 8.09 .$

### 10.1Program Text

Program Text (f07thfe.f90)

### 10.2Program Data

Program Data (f07thfe.d)

### 10.3Program Results

Program Results (f07thfe.r)

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