F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF07TUF (ZTRCON)

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

F07TUF (ZTRCON) estimates the condition number of a complex triangular matrix.

## 2  Specification

 SUBROUTINE F07TUF ( NORM, UPLO, DIAG, N, A, LDA, RCOND, WORK, RWORK, INFO)
 INTEGER N, LDA, INFO REAL (KIND=nag_wp) RCOND, RWORK(N) COMPLEX (KIND=nag_wp) A(LDA,*), WORK(2*N) CHARACTER(1) NORM, UPLO, DIAG
The routine may be called by its LAPACK name ztrcon.

## 3  Description

F07TUF (ZTRCON) estimates the condition number of a complex 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 ${‖{A}^{-1}‖}_{\infty }$.

## 4  References

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

## 5  Parameters

1:     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:     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:     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:     A(LDA,$*$) – COMPLEX (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:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F07TUF (ZTRCON) is called.
Constraint: ${\mathbf{LDA}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
7:     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 the estimate underflows. If RCOND is less than machine precision, $A$ is singular to working precision.
8:     WORK($2×{\mathbf{N}}$) – COMPLEX (KIND=nag_wp) arrayWorkspace
9:     RWORK(N) – REAL (KIND=nag_wp) arrayWorkspace
10:   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 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.

A call to F07TUF (ZTRCON) 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 $4{n}^{2}$ real floating point operations but takes considerably longer than a call to F07TSF (ZTRTRS) with one right-hand side, because extra care is taken to avoid overflow when $A$ is approximately singular.
The real analogue of this routine is F07TGF (DTRCON).

## 9  Example

This example estimates the condition number in the $1$-norm of the matrix $A$, where
 $A= 4.78+4.56i 0.00+0.00i 0.00+0.00i 0.00+0.00i 2.00-0.30i -4.11+1.25i 0.00+0.00i 0.00+0.00i 2.89-1.34i 2.36-4.25i 4.15+0.80i 0.00+0.00i -1.89+1.15i 0.04-3.69i -0.02+0.46i 0.33-0.26i .$
The true condition number in the $1$-norm is $70.27$.

### 9.1  Program Text

Program Text (f07tufe.f90)

### 9.2  Program Data

Program Data (f07tufe.d)

### 9.3  Program Results

Program Results (f07tufe.r)