NAG Library Routine Document

F07TUF (ZTRCON)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

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(2N)
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:

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 Parameters

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, $*$ ) – COMPLEX (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 F07TUF (ZTRCON) 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) arrayWorkspace

9: RWORK(N) – REAL (KIND=nag_wp) arrayWorkspace

10: INFO – INTEGEROutput

On exit:

INFO = 0

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

6 Error Indicators and Warnings

Errors or warnings detected by the routine:

$INFO < 0$: If $INFO = - i$ , 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

ρ

, and in practice is nearly always less than

10 ρ

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

8 Further Comments

A call to F07TUF (ZTRCON) 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 (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 = (\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

NAG Library Routine DocumentF07TUF (ZTRCON)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

F07TUF (ZTRCON)