NAG CL Interface
f01kfc (complex_​gen_​matrix_​frcht_​pow)

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1 Purpose

f01kfc computes the Fréchet derivative L(A,E) of the pth power (where p is real) of the complex n×n matrix A applied to the complex n×n matrix E. The principal matrix power Ap is also returned.

2 Specification

#include <nag.h>
void  f01kfc (Integer n, Complex a[], Integer pda, Complex e[], Integer pde, double p, NagError *fail)
The function may be called by the names: f01kfc or nag_matop_complex_gen_matrix_frcht_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}). If A is nonsingular but has negative real eigenvalues, the principal logarithm is not defined, but a non-principal pth power can be defined by using a non-principal logarithm.
The Fréchet derivative of the matrix pth power of A is the unique linear mapping EL(A,E) such that for any matrix E
(A+E)p - Ap - L(A,E) = o(E) .  
The derivative describes the first-order effect of perturbations in A on the matrix power Ap.
f01kfc uses the algorithms of Higham and Lin (2011) and Higham and Lin (2013) to compute Ap and L(A,E). 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, a Padé approximant and the scaling and squaring method. The Padé approximant is differentiated in order to obtain the Fréchet derivative of Aq and L(A,E) is then computed using a combination of the chain rule and the product rule for Fréchet derivatives.

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: the n×n principal matrix pth power, Ap. Alternatively if fail.code= NE_NEGATIVE_EIGVAL, a non-principal pth power is returned.
3: pda Integer Input
On entry: the stride separating matrix row elements in the array a.
Constraint: pdan.
4: e[dim] Complex Input/Output
Note: the dimension, dim, of the array e must be at least pde×n.
The (i,j)th element of the matrix E is stored in e[(j-1)×pde+i-1].
On entry: the n×n matrix E.
On exit: the Fréchet derivative L(A,E).
5: pde Integer Input
On entry: the stride separating matrix row elements in the array e.
Constraint: pden.
6: p double Input
On entry: the required power of A.
7: 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

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, n=value.
Constraint: n0.
NE_INT_2
On entry, pda=value and n=value.
Constraint: pdan.
On entry, pde=value and n=value.
Constraint: pden.
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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NEGATIVE_EIGVAL
A has eigenvalues on the negative real line. The principal pth power is not defined in this case, so a non-principal power was returned.
NE_NO_LICENCE
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.
NE_SINGULAR
A is singular so the pth power cannot be computed.
NW_SOME_PRECISION_LOSS
Ap has been computed using an IEEE double precision Padé approximant, although the arithmetic precision is higher than IEEE double precision.

7 Accuracy

For a normal matrix A (for which AHA=AAH), the Schur decomposition is diagonal and the computation of the fractional part of the matrix power 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. See Higham and Lin (2011) and Higham and Lin (2013) for details and further discussion.
If the condition number of the matrix power is required then f01kec should be used.

8 Parallelism and Performance

f01kfc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01kfc 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 complex allocatable memory required by the algorithm is approximately 6×n2.
The cost of the algorithm is O(n3) floating-point operations; see Higham and Lin (2011) and Higham and Lin (2013).
If the matrix pth power alone is required, without the Fréchet derivative, then f01fqc should be used. If the condition number of the matrix power is required then f01kec should be used. The real analogue of this function is f01jfc.

10 Example

This example finds Ap and the Fréchet derivative of the matrix power L(A,E), where p=0.2,
A = ( 2i+ 3i+0 2i+0 1+3i 2+i 1i+0 1i+0 2+0i 0+i 2+2i 0+2i 0+4i 3i+ 0+0i 3i+0 1i+0 )   and   E = ( 0+i 3i+0 2i+0 1+3i 0+i 1i+0 3+3i 0+0i 0+i 2+2i 0+2i 0i+0 2i+ 0+0i 1i+0 1i+0 ) .  

10.1 Program Text

Program Text (f01kfce.c)

10.2 Program Data

Program Data (f01kfce.d)

10.3 Program Results

Program Results (f01kfce.r)