NAG CL Interface
f01fqc (complex_​gen_​matrix_​pow)

Settings help

CL Name Style:

1 Purpose

f01fqc computes an abitrary real power Ap of a complex n×n matrix A.

2 Specification

#include <nag.h>
void  f01fqc (Integer n, Complex a[], Integer pda, double p, NagError *fail)
The function may be called by the names: f01fqc or nag_matop_complex_gen_matrix_pow.

3 Description

For a matrix A with no eigenvalues on the closed negative real line, Ap (p) can be defined as
Ap= exp(plog(A))  
where log(A) is the principal logarithm of A (the unique logarithm whose spectrum lies in the strip {z:-π<Im(z)<π}).
Ap is computed using the Schur–Padé algorithm described in Higham and Lin (2011) and Higham and Lin (2013).
The real number p is expressed as p=q+r where q(−1,1) and r. Then Ap=AqAr. The integer power Ar is found using a combination of binary powering and, if necessary, matrix inversion. The fractional power Aq is computed using a Schur decomposition and a Padé approximant.

4 References

Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA
Higham N J and Lin L (2011) A Schur–Padé algorithm for fractional powers of a matrix SIAM J. Matrix Anal. Appl. 32(3) 1056–1078
Higham N J and Lin L (2013) An improved Schur–Padé algorithm for fractional powers of a matrix and their Fréchet derivatives SIAM J. Matrix Anal. Appl. 34(3) 1341–1360

5 Arguments

1: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
2: 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].
On entry: the n×n matrix A.
On exit: if fail.code= NE_NOERROR, the n×n matrix pth power, Ap. Alternatively, if fail.code= NE_NEGATIVE_EIGVAL, contains an n×n non-principal power of A.
3: pda Integer Input
On entry: the stride separating matrix row elements in the array a.
Constraint: pdan.
4: p double Input
On entry: the required power of A.
5: 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.
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.
A has eigenvalues on the negative real line. The principal pth power is not defined so a non-principal power is returned.
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.
A is singular so the pth power cannot be computed.
Ap has been computed using an IEEE double precision Padé approximant, although the arithmetic precision is higher than IEEE double precision.

7 Accuracy

For positive integer p, the algorithm reduces to a sequence of matrix multiplications. For negative integer p, the algorithm consists of a combination of matrix inversion and matrix multiplications.
For a normal matrix A (for which AHA=AAH) and non-integer p, the Schur decomposition is diagonal and the algorithm reduces to evaluating powers of the eigenvalues of A and then constructing Ap using the Schur vectors. This should give a very accurate result. In general however, no error bounds are available for the algorithm.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f01fqc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01fqc 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 cost of the algorithm is O(n3). The exact cost depends on the matrix A but if p(−1,1) then the cost is independent of p. O(4×n2) complex allocatable memory is required by the function.
If estimates of the condition number of Ap are required then f01kec should be used.

10 Example

This example finds Ap where p=0.2 and
A = ( 2i+ 3i+0 2i+0 1+3i 2+i 1i+0 1i+0 2+2i 2+i 2+2i 2i 2+4i 3i+ 2+2i 3i+0 1i+0 ) .  

10.1 Program Text

Program Text (f01fqce.c)

10.2 Program Data

Program Data (f01fqce.d)

10.3 Program Results

Program Results (f01fqce.r)