NAG Library Manual, Mark 28.6

Interfaces: FL CL CPP AD PY MB

NAG FL Interface Introduction

F01 (Matop) Chapter Contents

F01 (Matop) Chapter Introduction

f01jc: FL CL CPP AD PY MB

NAG FL Interface
f01jcf (real_gen_matrix_cond_usd)

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NAG Library Manual, Mark 28.6

Interfaces: FL CL CPP AD PY MB

NAG FL Interface Introduction

F01 (Matop) Chapter Contents

F01 (Matop) Chapter Introduction

f01jc: FL CL CPP AD PY MB

▸▿ Contents

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

8 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

© The Numerical Algorithms Group Ltd. 2022

1 Purpose

f01jcf computes an estimate of the absolute condition number of a matrix function

f

at a real

n \times n

matrix

A

in the

1

-norm, using analytical derivatives of

f

you have supplied.

2 Specification

Fortran Interface

Subroutine f01jcf (

n, a, lda, f, iuser, ruser, iflag, conda, norma, normfa, ifail)

Integer, Intent (In)	::	n, lda
Integer, Intent (Inout)	::	iuser(*), ifail
Integer, Intent (Out)	::	iflag
Real (Kind=nag_wp), Intent (Inout)	::	a(lda,), ruser()
Real (Kind=nag_wp), Intent (Out)	::	conda, norma, normfa
External	::	f

C Header Interface

#include <nag.h>

void

f01jcf_ (const Integer *n, double a[], const Integer *lda,
void (NAG_CALL *f)(const Integer *m, Integer *iflag, const Integer *nz, const Complex z[], Complex fz[], Integer iuser[], double ruser[]),
Integer iuser[], double ruser[], Integer *iflag, double *conda, double *norma, double *normfa, Integer *ifail)

The routine may be called by the names f01jcf or nagf_matop_real_gen_matrix_cond_usd.

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)

, which is defined by

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

where

L (X, E)

is 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}

. f01jcf 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 (generally complex) points

z_{i}

, for

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

. For any

z

on the real line,

f (z)

must also be real. f01jcf 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 Arguments

1: $n$ – Integer Input

On entry:

n

, the order of the matrix

A

.

Constraint:

n \geq 0

.

2: $a (lda, *)$ – Real (Kind=nag_wp) array Input/Output

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

n

.

On entry: the

n \times n

matrix

A

.

On exit: the

n \times n

matrix,

f (A)

.

3: $lda$ – Integer Input

On entry: the first dimension of the array a as declared in the (sub)program from which f01jcf is called.

Constraint:

lda \geq 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:

Fortran Interface

Subroutine f (

m, iflag, nz, z, fz, iuser, ruser)

Integer, Intent (In)	::	m, nz
Integer, Intent (Inout)	::	iflag, iuser(*)
Real (Kind=nag_wp), Intent (Inout)	::	ruser(*)
Complex (Kind=nag_wp), Intent (In)	::	z(nz)
Complex (Kind=nag_wp), Intent (Out)	::	fz(nz)

C Header Interface

void	f (const Integer m, Integer iflag, const Integer *nz, const Complex z[], Complex fz[], Integer iuser[], double ruser[])

1: $m$ – Integer Input: 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: $iflag$ – Integer Input/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 f01jcf will terminate the computation, with $ifail = 3$ .
3: $nz$ – Integer Input: On entry: $n_{z}$ , the number of function or derivative values required.
4: $z (nz)$ – Complex (Kind=nag_wp) array Input: 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) array Output: On exit: the $n_{z}$ function or derivative values. $fz (i)$ should return the value $f^{(m)} (z_{i})$ , for $i = 1, 2, \dots, n_{z}$ . If $z_{i}$ lies on the real line, then so must $f^{(m)} (z_{i})$ .
6: $iuser (*)$ – Integer array User Workspace
7: $ruser (*)$ – Real (Kind=nag_wp) array User Workspace: f is called with the arguments iuser and ruser as supplied to f01jcf. You should use the arrays iuser and ruser to supply information to f.

f must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which f01jcf is called. Arguments denoted as Input must not be changed by this procedure.

Note: f should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by f01jcf. If your code inadvertently does return any NaNs or infinities, f01jcf is likely to produce unexpected results.

5: $iuser (*)$ – Integer array User Workspace

6: $ruser (*)$ – Real (Kind=nag_wp) array User Workspace

iuser and ruser are not used by f01jcf, but are passed directly to f and may be used to pass information to this routine.

7: $iflag$ – Integer Output

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: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

,

−1

or

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

−1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

−1

or

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. 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 f01emf with the matrix $A$ and the function $f$ .

$ifail = 3$: Termination requested in f.

$ifail = - 1$: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .

$ifail = - 3$: On entry, argument lda is invalid.
Constraint: $lda \geq n$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

f01jcf 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 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

f01jcf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library. In these implementations, this routine may make calls to the user-supplied functions from within an OpenMP parallel region. Thus OpenMP directives within the user functions can only be used if you are compiling the user-supplied function and linking the executable in accordance with the instructions in the Users' Note for your implementation. The user workspace arrays iuser and ruser are classified as OpenMP shared memory and use of iuser and ruser has to take account of this in order to preserve thread safety whenever information is written back to either of these arrays. If at all possible, it is recommended that these arrays are only used to supply read-only data to the user functions when a multithreaded implementation is being used.

f01jcf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The matrix function is computed using the underlying matrix function routine f01emf. 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.

If only

f (A)

is required, without an estimate of the condition number, then it is far more efficient to use the underlying matrix function routine directly.

The complex analogue of this routine is f01kcf.

10 Example

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

e^{2 A}

where

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

10.1 Program Text

Program Text (f01jcfe.f90)

10.2 Program Data

Program Data (f01jcfe.d)

10.3 Program Results

Program Results (f01jcfe.r)

NAG Library Manual, Mark 28.6

Interfaces: FL CL CPP AD PY MB

NAG FL Interface Introduction

F01 (Matop) Chapter Contents

F01 (Matop) Chapter Introduction

f01jc: FL CL CPP AD PY MB

© The Numerical Algorithms Group Ltd. 2022