Integer, Intent (In)	::	n, lda
Integer, Intent (Out)	::	info
Real (Kind=nag_wp), Intent (Out)	::	rcond, rwork(n)
Complex (Kind=nag_wp), Intent (In)	::	a(lda,*)
Complex (Kind=nag_wp), Intent (Out)	::	work(2*n)
Character (1), Intent (In)	::	norm, uplo, diag

C Header Interface

#include <nag.h>

void	f07tuf_ (const char norm, const char uplo, const char diag, const Integer n, const Complex a[], const Integer lda, double rcond, Complex work[], double rwork[], Integer *info, const Charlen length_norm, const Charlen length_uplo, const Charlen length_diag)

The routine may be called by the names f07tuf, nagf_lapacklin_ztrcon or its LAPACK name ztrcon.

3 Description

f07tuf estimates the condition number of a complex triangular matrix

A

, in either the

1

-norm or the

\infty

-norm:

κ_{1} (A) = {‖A‖}_{1} {‖A^{- 1}‖}_{1} or κ_{\infty} (A) = {‖A‖}_{\infty} {‖A^{- 1}‖}_{\infty} .

Note that

κ_{\infty} (A) = κ_{1} (A^{T})

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}

{‖A‖}_{\infty}

exactly, and uses Higham's implementation of Hager's method (see Higham (1988)) to estimate

{‖A^{- 1}‖}_{1}

{‖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 Arguments

1: $norm$ – Character(1) Input

On entry: indicates whether

κ_{1} (A)

κ_{\infty} (A)

is estimated.

$norm ='1'$ or $'O'$: $κ_{1} (A)$ is estimated.
$norm ='I'$: $κ_{\infty} (A)$ is estimated.

Constraint:

norm ='1'

'O'

'I'

2: $uplo$ – Character(1) Input

On entry: specifies whether

A

is upper or lower triangular.

$uplo ='U'$: $A$ is upper triangular.
$uplo ='L'$: $A$ is lower triangular.

Constraint:

uplo ='U'

'L'

3: $diag$ – Character(1) Input

On entry: indicates whether

A

is a nonunit or unit triangular matrix.

$diag ='N'$: $A$ is a nonunit triangular matrix.
$diag ='U'$: $A$ is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be $1$ .

Constraint:

diag ='N'

'U'

4: $n$ – Integer Input

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

5: $a (lda *)$ – Complex (Kind=nag_wp) array Input

Note: the second dimension of the array a must be at least

\max (1, n)

On entry: the

n

n

triangular matrix

A

If $uplo ='U'$ , $A$ is upper triangular and the elements of the array below the diagonal are not referenced.
If $uplo ='L'$ , $A$ is lower triangular and the elements of the array above the diagonal are not referenced.
If $diag ='U'$ , the diagonal elements of $A$ are assumed to be $1$ , and are not referenced.

6: $lda$ – Integer Input

On entry: the first dimension of the array a as declared in the (sub)program from which f07tuf is called.

Constraint:

lda \geq \max (1, n)

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: $work (2 \times n)$ – Complex (Kind=nag_wp) array Workspace

9: $rwork (n)$ – Real (Kind=nag_wp) array Workspace

10: $info$ – Integer Output

On exit:

info = 0

unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

$info < 0$: If $info = - i$ , argument $i$ 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

ρ

, and in practice is nearly always less than

10 ρ

, although examples can be constructed where rcond is much larger.

8 Parallelism and Performance

f07tuf 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.

9 Further Comments

A call to f07tuf involves solving a number of systems of linear equations of the form

A x = b

A^{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 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.

10 Example

This example estimates the condition number in the

1

-norm of the matrix

A

, where

A = (\begin{array}{r} 4.78 + 4.56 i & 0.00 + 0.00 i & 0.00 + 0.00 i & 0.00 + 0.00 i \\ 2.00 - 0.30 i & - 4.11 + 1.25 i & 0.00 + 0.00 i & 0.00 + 0.00 i \\ 2.89 - 1.34 i & 2.36 - 4.25 i & 4.15 + 0.80 i & 0.00 + 0.00 i \\ - 1.89 + 1.15 i & 0.04 - 3.69 i & - 0.02 + 0.46 i & 0.33 - 0.26 i \end{array}) .

The true condition number in the

1

-norm is

70.27

Interfaces: FL CL AD

NAG FL Interface Introduction

F07 (Lapacklin) Chapter Contents

F07 (Lapacklin) Chapter Introduction

f07tu: FL CL AD

NAG FL Interfacef07tuf (ztrcon)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
f07tuf (ztrcon)