NAG CL Interface
f01fkc (complex_​gen_​matrix_​fun_​std)

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1 Purpose

f01fkc computes the matrix exponential, sine, cosine, sinh or cosh, of a complex n×n matrix A using the Schur–Parlett algorithm.

2 Specification

#include <nag.h>
void  f01fkc (Nag_OrderType order, Nag_MatFunType fun, Integer n, Complex a[], Integer pda, NagError *fail)
The function may be called by the names: f01fkc or nag_matop_complex_gen_matrix_fun_std.

3 Description

f(A), where f 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_OrderType Input
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 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2: fun Nag_MatFunType Input
On entry: indicates which matrix function will be computed.
The matrix exponential, eA, will be computed.
The matrix sine, sin(A), will be computed.
The matrix cosine, cos(A), will be computed.
The hyperbolic matrix sine, sinh(A), will be computed.
The hyperbolic matrix cosine, cosh(A), will be computed.
Constraint: fun=Nag_Exp, Nag_Sin, Nag_Cos, Nag_Sinh or Nag_Cosh.
3: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
4: a[dim] Complex Input/Output
Note: the dimension, dim, of the array a must be at least pda×n.
The (i,j)th element of the matrix A is stored in
  • a[(j-1)×pda+i-1] when order=Nag_ColMajor;
  • a[(i-1)×pda+j-1] when order=Nag_RowMajor.
On entry: the n×n matrix A.
On exit: the n×n matrix, f(A).
5: pda Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: pdan.
6: 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

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument value had an illegal value.
A Taylor series failed to converge.
On entry, n=value.
Constraint: n0.
On entry, pda=value and n=value.
Constraint: pdan.
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 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.
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.
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 AHA=AAH), 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

Background information to multithreading can be found in the Multithreading documentation.
f01fkc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01fkc 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, and the Complex allocatable memory required is approximately 9n2.
The cost of the Schur–Parlett algorithm depends on the spectrum of A, but is roughly between 28n3 and n4/3 floating-point operations; see Algorithm 9.6 of Higham (2008).
If the matrix exponential is required then it is recommended that f01fcc be used. f01fcc uses an algorithm which is, in general, more accurate than the Schur–Parlett algorithm used by f01fkc.
If estimates of the condition number of the matrix function are required then f01kac should be used.
f01ekc can be used to find the matrix exponential, sin, cos, sinh or cosh of a real matrix A.

10 Example

This example finds the matrix sinh of the matrix
A = ( 1.0+1.0i 0.0+0.0i 1.0+3.0i 0.0+0.0i 0.0+0.0i 2.0+0.0i 0.0+0.0i 1.0+2.0i 3.0+1.0i 0.0+4.0i 1.0+1.0i 0.0+0.0i 1.0+1.0i 0.0+2.0i 0.0+0.0i 1.0+0.0i ) .  

10.1 Program Text

Program Text (f01fkce.c)

10.2 Program Data

Program Data (f01fkce.d)

10.3 Program Results

Program Results (f01fkce.r)