NAG Library Routine Document

F01JAF

+− 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

F01JAF computes an estimate of the absolute condition number of a matrix function

f

at a real

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 F01JAF (

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

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

. F01JAF 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, $*$ ) – REAL (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 F01JAF 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 evaluating the matrix function $f (A)$ . Please contact NAG.

$IFAIL = 2$: An internal error occurred when estimating the norm of the Fréchet derivative of $f$ at $A$ . Please contact NAG.

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

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

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

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

7 Accuracy

F01JAF uses the norm estimation routine F04YDF 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 F04YDF.

8 Further Comments

Approximately

6 n^{2}

of real allocatable memory is required by the routine, in addition to the memory used by the underlying matrix function routines F01ECF, F01EJF or F01EKF.

F01JAF returns the matrix function

f (A)

. This is computed using F01ECF if

FUN ='EXP'

, F01EJF if

FUN ='LOG'

and F01EKF otherwise. If only

f (A)

is required, without an estimate of the condition number, then it is far more efficient to use F01ECF, F01EJF or F01EKF directly.

F01KAF can be used to find the condition number of the exponential, logarithm, sine, cosine, sinh or cosh at a complex matrix.

9 Example

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

A = (\begin{array}{r} 2 & 1 & 3 & 1 \\ 3 & - 1 & 0 & 2 \\ 1 & 0 & 3 & 1 \\ 1 & 2 & 0 & 3 \end{array}) .

NAG Library Routine DocumentF01JAF

+− 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

F01JAF