NAG Library Routine Document

F01KAF

+− 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

F01KAF computes an estimate of the absolute condition number of a matrix function

f

at a complex

n

n

matrix

A

in the

1

-norm, where

f

is either the exponential, logarithm, sine, cosine, hyperbolic sine (sinh) or hyperbolic cosine (cosh). The evaluation of the matrix function,

f (A)

, is also returned.

2 Specification

SUBROUTINE F01KAF (

FUN, N, A, LDA, CONDA, NORMA, NORMFA, IFAIL)

INTEGER	N, LDA, IFAIL
REAL (KIND=nag_wp)	CONDA, NORMA, NORMFA
COMPLEX (KIND=nag_wp)	A(LDA,*)
CHARACTER(*)	FUN

3 Description

The absolute condition number of

f

A

{cond}_{abs} (f, A)

is given by the norm of the Fréchet derivative of

f

L (A, E)

, which is defined by

‖L (X)‖ := \max_{E \neq 0} \frac{‖L (X, E)‖}{‖E‖} .

The Fréchet derivative in the direction

E

L (X, E)

is linear in

E

and can therefore be written as

vec (L (X, E)) = K (X) vec (E),

where the

vec

operator stacks the columns of a matrix into one vector, so that

K (X)

n^{2} \times n^{2}

. F01KAF computes an estimate

γ

such that

γ \leq {‖K (X)‖}_{1}

, where

{‖K (X)‖}_{1} \in [n^{- 1} {‖L (X)‖}_{1}, n {‖L (X)‖}_{1}]

. The relative condition number can then be computed via

{cond}_{rel} (f, A) = \frac{{cond}_{abs} (f, A) {‖A‖}_{1}}{{‖f (A)‖}_{1}} .

The algorithm used to find

γ

is detailed in Section 3.4 of Higham (2008).

4 References

Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA

5 Parameters

1: FUN – CHARACTER(*)Input

On entry: indicates which matrix function will be used.

$FUN ='EXP'$: The matrix exponential, $e^{A}$ , will be used.
$FUN ='SIN'$: The matrix sine, $\sin (A)$ , will be used.
$FUN ='COS'$: The matrix cosine, $\cos (A)$ , will be used.
$FUN ='SINH'$: The hyperbolic matrix sine, $\sinh (A)$ , will be used.
$FUN ='COSH'$: The hyperbolic matrix cosine, $\cosh (A)$ , will be used.
$FUN ='LOG'$: The matrix logarithm, $\log (A)$ , will be used.

Constraint:

FUN ='EXP'

'SIN'

'COS'

'SINH'

'COSH'

'LOG'

2: N – INTEGERInput

On entry:

n

, the order of the matrix

A

Constraint:

N \geq 0

3: A(LDA, $*$ ) – COMPLEX (KIND=nag_wp) arrayInput/Output

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

N

On entry: the

n

n

matrix

A

On exit: the

n

n

matrix,

f (A)

4: LDA – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F01KAF is called.

Constraint:

LDA \geq \max (1, N)

5: CONDA – REAL (KIND=nag_wp)Output

On exit: an estimate of the absolute condition number of

f

A

6: NORMA – REAL (KIND=nag_wp)Output

On exit: the

1

-norm of

A

7: NORMFA – REAL (KIND=nag_wp)Output

On exit: the

1

-norm of

f (A)

8: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$: An internal error occurred when estimating the norm of the Fréchet derivative of $f$ at $A$ . Please contact NAG.

$IFAIL = 2$: An internal error occurred when evaluating the matrix function $f (A)$ . You can investigate further by calling F01FCF, F01FJF or F01FKF with the matrix $A$ .

$IFAIL = - 1$: On entry, $FUN \neq'EXP'$ , $'SIN'$ , $'COS'$ , $'SINH'$ , $'COSH'$ or $'LOG'$ .
Input argument number $⟨value⟩$ is invalid.

$IFAIL = - 2$: On entry, $N < 0$ .
Input argument number $⟨value⟩$ is invalid.

$IFAIL = - 4$: On entry, parameter LDA is invalid.
Constraint: $LDA \geq N$ .

$IFAIL = - 999$: Allocation of memory failed.

7 Accuracy

F01KAF uses the norm estimation routine F04ZDF to estimate a quantity

γ

, where

γ \leq {‖K (X)‖}_{1}

and

{‖K (X)‖}_{1} \in [n^{- 1} {‖L (X)‖}_{1}, n {‖L (X)‖}_{1}]

. For further details on the accuracy of norm estimation, see the documentation for F04ZDF.

8 Further Comments

Approximately

6 n^{2}

of complex allocatable memory is required by the routine, in addition to the memory used by the underlying matrix function routines F01FCF, F01FJF or F01FKF.

F01KAF returns the matrix function

f (A)

. This is computed using F01FCF if

FUN ='EXP'

, F01FJF if

FUN ='LOG'

and F01FKF otherwise. If only

f (A)

is required, without an estimate of the condition number, then it is far more efficient to use F01FCF, F01FJF or F01FKF directly.

F01JAF can be used to find the condition number of the exponential, logarithm, sine, cosine, sinh or cosh at a real matrix.

9 Example

This example estimates the absolute and relative condition numbers of the matrix sinh function for

A = (\begin{array}{r} 0.0 + 1.0 i & - 1.0 + 0.0 i & 1.0 + 0.0 i & 2.0 + 0.0 i \\ 2.0 + 1.0 i & 0.0 - 1.0 i & 0.0 + 0.0 i & 1.0 + 0.0 i \\ 0.0 + 1.0 i & 0.0 + 0.0 i & 1.0 + 1.0 i & 0.0 + 2.0 i \\ 1.0 + 0.0 i & 2.0 + 0.0 i & - 2.0 + 3.0 i & 0.0 + 1.0 i \end{array}) .

NAG Library Routine DocumentF01KAF

+− 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

F01KAF