F01KCF (PDF version)

F01 Chapter Contents

F01 Chapter Introduction

NAG Library Manual

NAG Library Routine Document

F01KCF

+− 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

F01KCF computes an estimate of the absolute condition number of a matrix function

f

at a complex

n

by

n

matrix

A

in the

1

-norm, using analytical derivatives of

f

you have supplied.

2 Specification

SUBROUTINE F01KCF (

N, A, LDA, F, IUSER, RUSER, IFLAG, CONDA, NORMA, NORMFA, IFAIL)

INTEGER	N, LDA, IUSER(*), IFLAG, IFAIL
REAL (KIND=nag_wp)	RUSER(*), CONDA, NORMA, NORMFA
COMPLEX (KIND=nag_wp)	A(LDA,*)
EXTERNAL	F

3 Description

The absolute condition number of

f

at

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)

is

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

. F01KCF 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).

The function

f

, and the derivatives of

f

, are returned by subroutine F which, given an integer

m

, evaluates

f^{(m)} (z_{i})

at a number of points

z_{i}

, for

i = 1, 2, \dots, n_{z}

, on the complex plane. F01KCF is therefore appropriate for routines that can be evaluated on the complex plane and whose derivatives, of arbitrary order, can also be evaluated on the complex plane.

4 References

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

5 Parameters

1: N – INTEGERInput

On entry:

n

, the order of the matrix

A

.

Constraint:

N \geq 0

.

2: 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

by

n

matrix

A

.

On exit: the

n

by

n

matrix,

f (A)

.

3: LDA – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F01KCF is called.

Constraint:

LDA \geq \max (1, N)

.

4: F – SUBROUTINE, supplied by the user.External Procedure

Given an integer

m

, the subroutine F evaluates

f^{(m)} (z_{i})

at a number of points

z_{i}

.

The specification of F is:

SUBROUTINE F (

M, IFLAG, NZ, Z, FZ, IUSER, RUSER)

INTEGER	M, IFLAG, NZ, IUSER(*)
REAL (KIND=nag_wp)	RUSER(*)
COMPLEX (KIND=nag_wp)	Z(NZ), FZ(NZ)

1: M – INTEGERInput: On entry: the order, $m$ , of the derivative required.
If $M = 0$ , $f (z_{i})$ should be returned. For $M > 0$ , $f^{(m)} (z_{i})$ should be returned.
2: IFLAG – INTEGERInput/Output: On entry: IFLAG will be zero.

On exit: IFLAG should either be unchanged from its entry value of zero, or may be set nonzero to indicate that there is a problem in evaluating the function $f (z)$ ; for instance $f (z)$ may not be defined. If IFLAG is returned as nonzero then F01KCF will terminate the computation, with $IFAIL = 3$ .
3: NZ – INTEGERInput: On entry: $n_{z}$ , the number of function or derivative values required.
4: Z(NZ) – COMPLEX (KIND=nag_wp) arrayInput: On entry: the $n_{z}$ points $z_{1}, z_{2}, \dots, z_{n_{z}}$ at which the function $f$ is to be evaluated.
5: FZ(NZ) – COMPLEX (KIND=nag_wp) arrayOutput: On exit: the $n_{z}$ function or derivative values. $FZ (i)$ should return the value $f^{(m)} (z_{i})$ , for $i = 1, 2, \dots, n_{z}$ .
6: IUSER( $*$ ) – INTEGER arrayUser Workspace
7: RUSER( $*$ ) – REAL (KIND=nag_wp) arrayUser Workspace: F is called with the parameters IUSER and RUSER as supplied to F01KCF. You are free to use the arrays IUSER and RUSER to supply information to F as an alternative to using COMMON global variables.

F must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which F01KCF is called. Parameters denoted as Input must not be changed by this procedure.

5: IUSER( $*$ ) – INTEGER arrayUser Workspace

6: RUSER( $*$ ) – REAL (KIND=nag_wp) arrayUser Workspace

IUSER and RUSER are not used by F01KCF, but are passed directly to F and may be used to pass information to this routine as an alternative to using COMMON global variables.

7: IFLAG – INTEGEROutput

On exit:

IFLAG = 0

, unless IFLAG has been set nonzero inside F, in which case IFLAG will be the value set and IFAIL will be set to

IFAIL = 3

.

8: CONDA – REAL (KIND=nag_wp)Output

On exit: an estimate of the absolute condition number of

f

at

A

.

9: NORMA – REAL (KIND=nag_wp)Output

On exit: the

1

-norm of

A

.

10: NORMFA – REAL (KIND=nag_wp)Output

On exit: the

1

-norm of

f (A)

.

11: 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

or

- 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 F01FMF with the matrix $A$ and the function $f$ .

$IFAIL = 3$: IFLAG has been set nonzero by the user-supplied subroutine.

$IFAIL = - 1$: On entry, $N < 0$ .

$IFAIL = - 3$: On entry, parameter LDA is invalid.
Constraint: $LDA \geq N$ .

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

7 Accuracy

F01KCF 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 routine F01FMF.

F01KCF returns the matrix function

f (A)

. This is computed using F01FMF. If only

f (A)

is required, without an estimate of the condition number, then it is far more efficient to use F01FMF directly.

The real analogue of this routine is F01JCF.

9 Example

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

e^{3 A}

where

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

9.1 Program Text

Program Text (f01kcfe.f90)

9.2 Program Data

Program Data (f01kcfe.d)

9.3 Program Results

Program Results (f01kcfe.r)

F01KCF (PDF version)

F01 Chapter Contents

F01 Chapter Introduction

NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012