void	nag_matop_real_gen_matrix_cond_std (Nag_MatFunType fun, Integer n, double a[], Integer pda, double conda, double norma, double normfa, NagError fail)

3

Description

The absolute condition number of

f

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)

n^{2} \times n^{2}

. nag_matop_real_gen_matrix_cond_std (f01jac) 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: $fun$ – Nag_MatFunTypeInput

On entry: indicates which matrix function will be used.

$fun = Nag_Exp$: The matrix exponential, $e^{A}$ , will be used.
$fun = Nag_Sin$: The matrix sine, $\sin (A)$ , will be used.
$fun = Nag_Cos$: The matrix cosine, $\cos (A)$ , will be used.
$fun = Nag_Sinh$: The hyperbolic matrix sine, $\sinh (A)$ , will be used.
$fun = Nag_Cosh$: The hyperbolic matrix cosine, $\cosh (A)$ , will be used.
$fun = Nag_Loga$: The matrix logarithm, $\log (A)$ , will be used.

Constraint:

fun = Nag_Exp

Nag_Sin

Nag_Cos

Nag_Sinh

Nag_Cosh

Nag_Loga

2: $n$ – IntegerInput

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

3: $a [\dim]$ – 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

n

matrix

A

On exit: the

n

n

matrix,

f (A)

4: $pda$ – IntegerInput

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

Constraint:

pda \geq n

5: $conda$ – double *Output

On exit: an estimate of the absolute condition number of

f

A

6: $norma$ – double *Output

On exit: the

1

-norm of

A

7: $normfa$ – double *Output

On exit: the

1

-norm of

f (A)

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_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 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_real_gen_matrix_exp (f01ecc), nag_matop_real_gen_matrix_log (f01ejc) or nag_matop_real_gen_matrix_fun_std (f01ekc) with the matrix $A$ .
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.

7

Accuracy

nag_matop_real_gen_matrix_cond_std (f01jac) uses the norm estimation function 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_std (f01jac) 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.

nag_matop_real_gen_matrix_cond_std (f01jac) 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

The matrix function is computed using one of three underlying matrix function routines:

if $fun = Nag_Exp$ , nag_real_gen_matrix_exp (f01ecc) is used;
if $fun = Nag_Loga$ , nag_matop_real_gen_matrix_log (f01ejc) is used;
else, nag_matop_real_gen_matrix_fun_std (f01ekc) is used.

Approximately

6 n^{2}

of real allocatable memory is required by the routine, in addition to the memory used by these underlying matrix function routines.

If only

f (A)

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

nag_matop_complex_gen_matrix_cond_std (f01kac) can be used to find the condition number of the exponential, logarithm, sine, cosine, sinh or cosh matrix functions at a complex matrix.

10

Example

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

A = (\begin{array}{r} 2 & 1 & 3 & 1 \\ 3 & - 1 & 0 & 2 \\ 1 & 0 & 3 & 1 \\ 1 & 2 & 0 & 3 \end{array}) .

NAG Library Function Document

nag_matop_real_gen_matrix_cond_std (f01jac)

▸▿ 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

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