NAG Library Manual, Mark 29.3

Interfaces: FL CL CPP AD PY MB

NAG CL Interface Introduction

F01 (Matop) Chapter Contents

F01 (Matop) Chapter Introduction

f01kc: FL CL CPP AD PY MB

NAG CL Interface
f01kcc (complex_gen_matrix_cond_usd)

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NAG Library Manual, Mark 29.3

Interfaces: FL CL CPP AD PY MB

NAG CL Interface Introduction

F01 (Matop) Chapter Contents

F01 (Matop) Chapter Introduction

f01kc: 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. 2023

CL Name Style:
Short (a00aac)
Long (impl_details)
Full (nag_info_impl_details)

1 Purpose

f01kcc computes an estimate of the absolute condition number of a matrix function

f

of a complex

n \times n

matrix

A

in the

1

-norm, using analytical derivatives of

f

you have supplied.

2 Specification

#include <nag.h>

void

f01kcc (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, double *conda, double *norma, double *normfa, NagError *fail)

The function may be called by the names: f01kcc or nag_matop_complex_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}

. f01kcc 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 Arguments

1: $n$ – Integer Input

On entry:

n

, the order of the matrix

A

.

Constraint:

n \geq 0

.

2: $a [\dim]$ – Complex Input/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]

.

On entry: the

n \times n

matrix

A

.

On exit: the

n \times n

matrix,

f (A)

.

3: $pda$ – Integer Input

On entry: the stride separating matrix row elements in the array a.

Constraint:

pda \geq n

.

4: $f$ – function, supplied by the user External 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: $m$ – 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 f01kcc will terminate the computation, with

fail . code =

NE_INT, NE_INT_2 or NE_USER_STOP.

3: $nz$ – Integer Input

On entry:

n_{z}

, the number of function or derivative values required.

4: $z [\dim]$ – const Complex 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: $fz [\dim]$ – Complex Output

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 f01kcc you may allocate memory and initialize these pointers with various quantities for use by f when called from f01kcc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).

Note: f should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by f01kcc. If your code inadvertently does return any NaNs or infinities, f01kcc is likely to produce unexpected results.

5: $comm$ – Nag_Comm *

The NAG communication argument (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).

6: $iflag$ – 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 =

NE_INT, NE_INT_2 or NE_USER_STOP.

7: $conda$ – double * Output

On exit: an estimate of the absolute condition number of

f

at

A

.

8: $norma$ – double * Output

On exit: the

1

-norm of

A

.

9: $normfa$ – double * Output

On exit: the

1

-norm of

f (A)

.

10: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
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 7.5 in the Introduction to the NAG Library CL Interface for further information.

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

An internal error occurred when evaluating the matrix function $f (A)$ . You can investigate further by calling f01fmc with the matrix $A$ and the function $f$ .
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_USER_STOP: Termination requested in f.

7 Accuracy

f01kcc uses the norm estimation function f04zdc 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 f04zdc.

8 Parallelism and Performance

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

f01kcc 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.

f01kcc 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

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 f01fmc.

f01kcc returns the matrix function

f (A)

. This is computed using f01fmc. If only

f (A)

is required, without an estimate of the condition number, then it is far more efficient to use f01fmc directly.

The real analogue of this function is f01jcc.

10 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}) .

10.1 Program Text

Program Text (f01kcce.c)

10.2 Program Data

Program Data (f01kcce.d)

10.3 Program Results

Program Results (f01kcce.r)

NAG Library Manual, Mark 29.3

Interfaces: FL CL CPP AD PY MB

NAG CL Interface Introduction

F01 (Matop) Chapter Contents

F01 (Matop) Chapter Introduction

f01kc: FL CL CPP AD PY MB

© The Numerical Algorithms Group Ltd. 2023