# NAG Library Routine Document

## 1Purpose

f07tgf (dtrcon) estimates the condition number of a real triangular matrix.

## 2Specification

Fortran Interface
 Subroutine f07tgf ( norm, uplo, diag, n, a, lda, work, info)
 Integer, Intent (In) :: n, lda Integer, Intent (Out) :: iwork(n), info Real (Kind=nag_wp), Intent (In) :: a(lda,*) Real (Kind=nag_wp), Intent (Out) :: rcond, work(3*n) Character (1), Intent (In) :: norm, uplo, diag
#include <nagmk26.h>
 void f07tgf_ (const char *norm, const char *uplo, const char *diag, const Integer *n, const double a[], const Integer *lda, double *rcond, double work[], Integer iwork[], Integer *info, const Charlen length_norm, const Charlen length_uplo, const Charlen length_diag)
The routine may be called by its LAPACK name dtrcon.

## 3Description

f07tgf (dtrcon) estimates the condition number of a real triangular matrix $A$, in either the $1$-norm or the $\infty$-norm:
 $κ1 A = A1 A-11 or κ∞ A = A∞ A-1∞ .$
Note that ${\kappa }_{\infty }\left(A\right)={\kappa }_{1}\left({A}^{\mathrm{T}}\right)$.
Because the condition number is infinite if $A$ is singular, the routine actually returns an estimate of the reciprocal of the condition number.
The routine computes ${‖A‖}_{1}$ or ${‖A‖}_{\infty }$ exactly, and uses Higham's implementation of Hager's method (see Higham (1988)) to estimate ${‖{A}^{-1}‖}_{1}$ or ${‖{A}^{-1}‖}_{\infty }$.

## 4References

Higham N J (1988) FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation ACM Trans. Math. Software 14 381–396

## 5Arguments

1:     $\mathbf{norm}$ – Character(1)Input
On entry: indicates whether ${\kappa }_{1}\left(A\right)$ or ${\kappa }_{\infty }\left(A\right)$ is estimated.
${\mathbf{norm}}=\text{'1'}$ or $\text{'O'}$
${\kappa }_{1}\left(A\right)$ is estimated.
${\mathbf{norm}}=\text{'I'}$
${\kappa }_{\infty }\left(A\right)$ is estimated.
Constraint: ${\mathbf{norm}}=\text{'1'}$, $\text{'O'}$ or $\text{'I'}$.
2:     $\mathbf{uplo}$ – 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 $\text{'L'}$.
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:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – 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.
6:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f07tgf (dtrcon) is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
7:     $\mathbf{rcond}$ – Real (Kind=nag_wp)Output
On exit: an estimate of the reciprocal of the condition number of $A$. rcond is set to zero if exact singularity is detected or if the estimate underflows. If rcond is less than machine precision, $A$ is singular to working precision.
8:     $\mathbf{work}\left(3×{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
9:     $\mathbf{iwork}\left({\mathbf{n}}\right)$ – Integer arrayWorkspace
10:   $\mathbf{info}$ – 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 computed estimate rcond is never less than the true value $\rho$, and in practice is nearly always less than $10\rho$, although examples can be constructed where rcond is much larger.

## 8Parallelism and Performance

f07tgf (dtrcon) 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 f07tgf (dtrcon) 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 but takes considerably longer than a call to f07tef (dtrtrs) with one right-hand side, because extra care is taken to avoid overflow when $A$ is approximately singular.
The complex analogue of this routine is f07tuf (ztrcon).

## 10Example

This example estimates the condition number in the $1$-norm of the matrix $A$, where
 $A= 4.30 0.00 0.00 0.00 -3.96 -4.87 0.00 0.00 0.40 0.31 -8.02 0.00 -0.27 0.07 -5.95 0.12 .$
The true condition number in the $1$-norm is $116.41$.

### 10.1Program Text

Program Text (f07tgfe.f90)

### 10.2Program Data

Program Data (f07tgfe.d)

### 10.3Program Results

Program Results (f07tgfe.r)