nag_matop_real_gen_matrix_cond_usd (f01jcc) (PDF version)

f01 Chapter Contents

f01 Chapter Introduction

NAG Library Manual

NAG Library Function Document

nag_matop_real_gen_matrix_cond_usd (f01jcc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

nag_matop_real_gen_matrix_cond_usd (f01jcc) 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, using analytical derivatives of

f

you have supplied.

2 Specification

#include <nag.h>

#include <nagf01.h>

void

nag_matop_real_gen_matrix_cond_usd (Integer n, double 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)

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}

. nag_matop_real_gen_matrix_cond_usd (f01jcc) 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 function 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. nag_matop_real_gen_matrix_cond_usd (f01jcc) 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

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

5 Arguments

1: n – IntegerInput

On entry:

n

, the order of the matrix

A

.

Constraint:

n \geq 0

.

2: a[ $\dim$ ] – doubleInput/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

by

n

matrix

A

.

On exit: the

n

by

n

matrix,

f (A)

.

3: pda – IntegerInput

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

Constraint:

pda \geq n

.

4: 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)

may not be defined. If iflag is returned as nonzero then nag_matop_real_gen_matrix_cond_usd (f01jcc) will terminate the computation, with

fail . code =

3: nz – IntegerInput

On entry:

n_{z}

, the number of function or derivative values required.

4: z[nz] – 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[nz] – 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}

. If

z_{i}

lies on the real line, then so must

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

.

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_real_gen_matrix_cond_usd (f01jcc) you may allocate memory and initialize these pointers with various quantities for use by f when called from nag_matop_real_gen_matrix_cond_usd (f01jcc) (see Section 3.2.1.1 in the Essential Introduction).

5: comm – Nag_Comm *Communication Structure

The NAG communication argument (see Section 3.2.1.1 in the Essential Introduction).

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 =

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 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Allocation of memory failed.
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.
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 nag_matop_real_gen_matrix_fun_usd (f01emc) with the matrix $A$ and the function $f$ .
NE_USER_STOP: iflag has been set nonzero by the user-supplied function.

7 Accuracy

nag_matop_real_gen_matrix_cond_usd (f01jcc) uses the norm estimation routine nag_linsys_real_gen_norm_rcomm (f04ydc) 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 nag_linsys_real_gen_norm_rcomm (f04ydc).

8 Parallelism and Performance

nag_matop_real_gen_matrix_cond_usd (f01jcc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_matop_real_gen_matrix_cond_usd (f01jcc) 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.

In these implementations, this may make calls to the user supplied functions from within an OpenMP parallel region. Thus OpenMP directives within the user functions should be avoided, unless you are using the same OpenMP runtime library (which normally means using the same compiler) as that used to build your NAG Library implementation, as listed in the Installers' Note. 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.

Please 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 nag_matop_real_gen_matrix_fun_usd (f01emc). 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 function is nag_matop_complex_gen_matrix_cond_usd (f01kcc).

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 (f01jcce.c)

10.2 Program Data

Program Data (f01jcce.d)

10.3 Program Results

Program Results (f01jcce.r)

nag_matop_real_gen_matrix_cond_usd (f01jcc) (PDF version)

f01 Chapter Contents

f01 Chapter Introduction

NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014