is either the exponential, sine, cosine, sinh or cosh, is computed using the Schur–Parlett algorithm described in Higham (2008) and Davies and Higham (2003).

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

Nag_ColMajor

2: $fun$ – Nag_MatFunTypeInput

On entry: indicates which matrix function will be computed.

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

Constraint:

fun = Nag_Exp

Nag_Sin

Nag_Cos

Nag_Sinh

Nag_Cosh

3: $n$ – IntegerInput

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

4: $a [\dim]$ – 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

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

On entry: the

n

n

matrix

A

On exit: the

n

n

matrix,

f (A)

5: $pda$ – IntegerInput

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

Constraint:

pda \geq n

6: $imnorm$ – double *Output

On exit: if

A

has complex eigenvalues, nag_matop_real_gen_matrix_fun_std (f01ekc) will use complex arithmetic to compute the matrix function. The imaginary part is discarded at the end of the computation, because it will theoretically vanish. imnorm contains the

1

-norm of the imaginary part, which should be used to check that the routine has given a reliable answer.

A

has real eigenvalues, nag_matop_real_gen_matrix_fun_std (f01ekc) uses real arithmetic and

imnorm = 0

7: $fail$ – 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 evaluating the function at a point. Please contact NAG.

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_SINGULAR: The linear equations to be solved are nearly singular and the Padé approximant used to compute the exponential may have no correct figures.
Note: this failure should not occur and suggests that the function has been called incorrectly.

7

Accuracy

For a normal matrix

A

(for which

A^{T} A = A A^{T}

), 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.

For further discussion of the Schur–Parlett algorithm see Section 9.4 of Higham (2008).

8

Parallelism and Performance

nag_matop_real_gen_matrix_fun_std (f01ekc) 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_fun_std (f01ekc) 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 Integer allocatable memory required is

n

. If

A

has real eigenvalues then up to

9 n^{2}

of double allocatable memory may be required. If

A

has complex eigenvalues then up to

9 n^{2}

of Complex allocatable memory may be 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; see Algorithm 9.6 of Higham (2008).

If the matrix exponential is required then it is recommended that nag_real_gen_matrix_exp (f01ecc) be used. nag_real_gen_matrix_exp (f01ecc) uses an algorithm which is, in general, more accurate than the Schur–Parlett algorithm used by nag_matop_real_gen_matrix_fun_std (f01ekc).

If estimates of the condition number of the matrix function are required then nag_matop_real_gen_matrix_cond_std (f01jac) should be used.

nag_matop_complex_gen_matrix_fun_std (f01fkc) can be used to find the matrix exponential, sin, cos, sinh or cosh of a complex matrix.

10

Example

This example finds the matrix cosine of the matrix

A = (\begin{array}{r} 2 & 0 & 1 & 0 \\ 0 & 2 & - 2 & 1 \\ 0 & 2 & 3 & 1 \\ 1 & 4 & 0 & 0 \end{array}) .

NAG Library Function Document

nag_matop_real_gen_matrix_fun_std (f01ekc)

▸▿ 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