NAG Library Routine Document

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

C Header Interface

#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.

3

Description

f07tgf (dtrcon) estimates the condition number of a real 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: $n$ – IntegerInput

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

5: $a (lda *)$ – Real (Kind=nag_wp) arrayInput

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: $lda$ – IntegerInput

On entry: the first dimension of the array a as declared in the (sub)program from which f07tgf (dtrcon) 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 if the estimate underflows. If rcond is less than machine precision,

A

is singular to working precision.

8: $work (3 \times n)$ – Real (Kind=nag_wp) arrayWorkspace

9: $iwork (n)$ – Integer arrayWorkspace

10: $info$ – IntegerOutput

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

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.

9

Further Comments

A call to f07tgf (dtrcon) involves solving a number of systems of linear equations of the form

A x = b

A^{T} x = b

; the number is usually

4

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).

10

Example

This example estimates the condition number in the

1

-norm of the matrix

A

, where

A = (\begin{matrix} 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 \end{matrix}) .

The true condition number in the

1

-norm is

116.41

NAG Library Routine Document

f07tgf (dtrcon)

▸▿ 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

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