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).

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

Parameters

Compulsory Input Parameters

1: $fun$ – string

Indicates which matrix function will be computed.

$fun ='exp'$: The matrix exponential, $e^{A}$ , will be computed.
$fun ='sin'$: The matrix sine, $\sin (A)$ , will be computed.
$fun ='cos'$: The matrix cosine, $\cos (A)$ , will be computed.
$fun ='sinh'$: The hyperbolic matrix sine, $\sinh (A)$ , will be computed.
$fun ='cosh'$: The hyperbolic matrix cosine, $\cosh (A)$ , will be computed.

Constraint:

fun ='exp'

'sin'

'cos'

'sinh'

'cosh'

2: $a (lda :)$ – double array

The first dimension of the array a must be at least

n

The second dimension of the array a must be at least

n

The

n

n

matrix

A

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the first dimension of the array a.
$n$ , the order of the matrix $A$ .

Constraint: $n \geq 0$ .

Output Parameters

1: $a (lda :)$ – double array: The first dimension of the array a will be $n$ .
The second dimension of the array a will be $n$ .

The $n$ by $n$ matrix, $f (A)$ .
2: $imnorm$ – double scalar: If $A$ has complex eigenvalues, nag_matop_real_gen_matrix_fun_std (f01ek) 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.
If $A$ has real eigenvalues, nag_matop_real_gen_matrix_fun_std (f01ek) uses real arithmetic and $imnorm = 0$ .
3: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$: A Taylor series failed to converge.

$ifail = 2$: An unexpected internal error occurred when evaluating the function at a point. Please contact NAG.

$ifail = 3$: 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.

$ifail = 4$: 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.

$ifail = 5$: An unexpected internal error occurred. Please contact NAG.

$ifail = 6$: 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.

$ifail = - 1$: On entry, $fun =_$ was an illegal value.

$ifail = - 2$: Input argument number $_$ is invalid.

$ifail = - 4$: On entry, argument lda is invalid.
Constraint: $lda \geq n$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

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).

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_matop_real_gen_matrix_exp (f01ec) be used. nag_matop_real_gen_matrix_exp (f01ec) uses an algorithm which is, in general, more accurate than the Schur–Parlett algorithm used by nag_matop_real_gen_matrix_fun_std (f01ek).

If estimates of the condition number of the matrix function are required then nag_matop_real_gen_matrix_cond_std (f01ja) should be used.

nag_matop_complex_gen_matrix_fun_std (f01fk) can be used to find the matrix exponential, sin, cos, sinh or cosh of a complex matrix.

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}) .

Open in the MATLAB editor: f01ek_example

function f01ek_example


fprintf('f01ek example results\n\n');

a =  [2,  0,  1,  0
      0,  2, -2,  1
      0,  2,  3,  1
      1,  4,  0,  0];

% Compute cos(a)
fun = 'cos';
[cosa, imnorm, ifail] = f01ek(fun, a);

disp('f(A) = cos(A)');
disp(cosa);

fprintf('Imnorm = %6.2f\n',imnorm);

f01ek example results

f(A) = cos(A)
   -0.2998    1.5003   -0.7849    0.4677
   -0.2385   -3.2657    0.5812   -1.1460
    0.4677    0.3008   -4.0853   -0.2200
   -0.2107   -2.8199   -1.2964   -0.8325

Imnorm =   0.00

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox