NAG C Library Manual, Mark 26

NAG Library Manual, Mark 26

NAG AD Library Manual, Mark 26

f01 Chapter Contents

f01 Chapter Introduction

NAG C Library Function Document

nag_matop_complex_gen_matrix_fun_usd (f01fmc)

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NAG C Library Manual, Mark 26

NAG Library Manual, Mark 26

NAG AD Library Manual, Mark 26

f01 Chapter Contents

f01 Chapter Introduction

▸▿ 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. 2018

1

Purpose

nag_matop_complex_gen_matrix_fun_usd (f01fmc) computes the matrix function,

f (A)

, of a complex

n

by

n

matrix

A

, using analytical derivatives of

f

you have supplied.

2

Specification

#include <nag.h>

#include <nagf01.h>

void

nag_matop_complex_gen_matrix_fun_usd (Nag_OrderType order, Integer n, Complex a[], Integer pda,

void	(f)(Integer m, Integer iflag, Integer nz, const Complex z[], Complex fz[], Nag_Comm *comm),

Nag_Comm *comm, Integer *iflag, NagError *fail)

3

Description

f (A)

is computed using the Schur–Parlett algorithm described in Higham (2008) and Davies and Higham (2003).

The scalar function

f

, and the derivatives of

f

, are returned by the function f which, given an integer

m

, should evaluate

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

at a number of points

z_{i}

, for

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

, on the complex plane. nag_matop_complex_gen_matrix_fun_usd (f01fmc) is therefore appropriate for functions that can be evaluated on the complex plane and whose derivatives, of arbitrary order, can also be evaluated on the complex plane.

4

References

Davies P I and Higham N J (2003) A Schur–Parlett algorithm for computing matrix functions. SIAM J. Matrix Anal. Appl. 25(2) 464–485

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

5

Arguments

1: $order$ – Nag_OrderTypeInput

On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by

order = Nag_RowMajor

. See Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

or

Nag_ColMajor

.

2: $n$ – IntegerInput

On entry:

n

, the order of the matrix

A

.

Constraint:

n \geq 0

.

3: $a [\dim]$ – ComplexInput/Output

Note: the dimension, dim, of the array a must be at least

pda \times n

.

The

(i, j)

th element of the matrix

A

is stored in

$a [(j - 1) \times pda + i - 1]$ when $order = Nag_ColMajor$ ;
$a [(i - 1) \times pda + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the

n

by

n

matrix

A

.

On exit: the

n

by

n

matrix,

f (A)

.

4: $pda$ – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array a.

Constraint:

pda \geq n

.

5: $f$ – function, supplied by the userExternal Function

Given an integer

m

, the function f evaluates

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

at a number of points

z_{i}

.

The specification of f is:

void	f (Integer m, Integer iflag, Integer nz, const Complex z[], Complex fz[], Nag_Comm comm)

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: $iflag$ – 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_{i})

may not be defined for a particular

z_{i}

. If iflag is returned as nonzero then nag_matop_complex_gen_matrix_fun_usd (f01fmc) will terminate the computation, with

fail . code =

3: $nz$ – IntegerInput

On entry:

n_{z}

, the number of function or derivative values required.

4: $z [\dim]$ – const ComplexInput

On entry: the

n_{z}

points

z_{1}, z_{2}, \dots, z_{n_{z}}

at which the function

f

is to be evaluated.

5: $fz [\dim]$ – ComplexOutput

On exit: the

n_{z}

function or derivative values.

fz [i - 1]

should return the value

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

, for

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

.

6: $comm$ – Nag_Comm *

Pointer to structure of type Nag_Comm; the following members are relevant to f.

user – double *
iuser – Integer *
p – Pointer: The type Pointer will be void *. Before calling nag_matop_complex_gen_matrix_fun_usd (f01fmc) you may allocate memory and initialize these pointers with various quantities for use by f when called from nag_matop_complex_gen_matrix_fun_usd (f01fmc) (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).

Note: f should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by nag_matop_complex_gen_matrix_fun_usd (f01fmc). If your code inadvertently does return any NaNs or infinities, nag_matop_complex_gen_matrix_fun_usd (f01fmc) is likely to produce unexpected results.

6: $comm$ – Nag_Comm *

The NAG communication argument (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).

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 fail will be set to

fail . code =

8: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_CONVERGENCE: A Taylor series failed to converge.
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .
NE_INT_2: On entry, $pda = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pda \geq n$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.

An unexpected internal error occurred when ordering the eigenvalues of $A$ . Please contact NAG.

The function was unable to compute the Schur decomposition of $A$ .
Note: this failure should not occur and suggests that the function has been called incorrectly.

There was an error whilst reordering the Schur form of $A$ .
Note: this failure should not occur and suggests that the function has been called incorrectly.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_USER_STOP: iflag has been set nonzero by the user.

7

Accuracy

For a normal matrix

A

(for which

A^{H} A = A A^{H}

), the Schur decomposition is diagonal and the algorithm reduces to evaluating

f

at the eigenvalues of

A

and then constructing

f (A)

using the Schur vectors. This should give a very accurate result. In general, however, no error bounds are available for the algorithm. See Section 9.4 of Higham (2008) for further discussion of the Schur–Parlett algorithm.

8

Parallelism and Performance

nag_matop_complex_gen_matrix_fun_usd (f01fmc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library. In these implementations, this function may make calls to the user-supplied functions from within an OpenMP parallel region. Thus OpenMP pragmas 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. You must also ensure that you use the NAG communication argument comm in a thread safe manner, which is best achieved by only using it to supply read-only data to the user functions.

nag_matop_complex_gen_matrix_fun_usd (f01fmc) 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9

Further Comments

Up to

6 n^{2}

of Complex allocatable memory is required.

The cost of the Schur–Parlett algorithm depends on the spectrum of

A

, but is roughly between

28 n^{3}

and

n^{4} / 3

floating-point operations. There is an additional cost in evaluating

f

and its derivatives. If the derivatives of

f

are not known analytically, then nag_matop_complex_gen_matrix_fun_num (f01flc) can be used to evaluate

f (A)

using numerical differentiation. If

A

is complex Hermitian then it is recommended that nag_matop_complex_herm_matrix_fun (f01ffc) be used as it is more efficient and, in general, more accurate than nag_matop_complex_gen_matrix_fun_usd (f01fmc).

Note that

f

must be analytic in the region of the complex plane containing the spectrum of

A

.

For further information on matrix functions, see Higham (2008).

If estimates of the condition number of the matrix function are required then nag_matop_complex_gen_matrix_cond_usd (f01kcc) should be used.

nag_matop_real_gen_matrix_fun_usd (f01emc) can be used to find the matrix function

f (A)

for a real matrix

A

.

10

Example

This example finds the

e^{3 A}

where

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

10.1

Program Text

Program Text (f01fmce.c)

10.2

Program Data

Program Data (f01fmce.d)

10.3

Program Results

Program Results (f01fmce.r)

NAG C Library Manual, Mark 26

NAG Library Manual, Mark 26

NAG AD Library Manual, Mark 26

f01 Chapter Contents

f01 Chapter Introduction

© The Numerical Algorithms Group Ltd. 2018