F01JBF (PDF version)

F01 Chapter Contents

F01 Chapter Introduction

NAG Library Manual

NAG Library Routine Document

F01JBF

+− 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

F01JBF computes an estimate of the absolute condition number of a matrix function

f

at a real

n

by

n

matrix

A

in the

1

-norm. Numerical differentiation is used to evaluate the derivatives of

f

when they are required.

2 Specification

SUBROUTINE F01JBF (

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

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

. F01JBF 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

is supplied via subroutine F which evaluates

f (z_{i})

at a number of points

z_{i}

.

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, $*$ ) – REAL (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 F01JBF is called.

Constraint:

LDA \geq \max (1, N)

.

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

The subroutine F evaluates

f (z_{i})

at a number of points

z_{i}

.

The specification of F is:

SUBROUTINE F (

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

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

1: 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 F01JBF will terminate the computation, with $IFAIL = 3$ .
2: NZ – INTEGERInput: On entry: $n_{z}$ , the number of function values required.
3: 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.
4: FZ(NZ) – COMPLEX (KIND=nag_wp) arrayOutput: On exit: the $n_{z}$ function values. $FZ (i)$ should return the value $f (z_{i})$ , for $i = 1, 2, \dots, n_{z}$ . If $z_{i}$ lies on the real line, then so must $f (z_{i})$ .
5: IUSER( $*$ ) – INTEGER arrayUser Workspace
6: RUSER( $*$ ) – REAL (KIND=nag_wp) arrayUser Workspace: F is called with the parameters IUSER and RUSER as supplied to F01JBF. 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 F01JBF 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 F01JBF, 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 F01ELF with the matrix $A$ and the function $f$ .

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

$IFAIL = - 1$: On entry, $N < 0$ .
Input argument number $⟨value⟩$ is invalid.

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

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

7 Accuracy

F01JBF 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 routine F01ELF.

F01JBF returns the matrix function

f (A)

. This is computed using F01ELF. If only

f (A)

is required, without an estimate of the condition number, then it is far more efficient to use F01ELF directly.

The complex analogue of this routine is F01KBF.

9 Example

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

\cos 2 A

where

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

9.1 Program Text

Program Text (f01jbfe.f90)

9.2 Program Data

Program Data (f01jbfe.d)

9.3 Program Results

Program Results (f01jbfe.r)

F01JBF (PDF version)

F01 Chapter Contents

F01 Chapter Introduction

NAG Library Manual

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