NAG CL Interface
f01kec (complex_​gen_​matrix_​cond_​pow)

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1 Purpose

f01kec computes an estimate of the relative condition number κAp of the pth power (where p is real) of a complex n×n matrix A, in the 1-norm. The principal matrix power Ap is also returned.

2 Specification

#include <nag.h>
void  f01kec (Integer n, Complex a[], Integer pda, double p, double *condpa, NagError *fail)
The function may be called by the names: f01kec or nag_matop_complex_gen_matrix_cond_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}).
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.
The relative condition number of the matrix pth power can be defined by
κAp = L(A) A Ap ,  
where L(A) is the norm of the Fréchet derivative of the matrix power at A.
f01kec uses the algorithms of Higham and Lin (2011) and Higham and Lin (2013) to compute κAp and Ap. 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.
To obtain the estimate of κAp, f01kec first estimates L(A) by computing an estimate γ of a quantity K[n-1L(A)1,nL(A)1], such that γK. This requires multiple Fréchet derivatives to be computed. Fréchet derivatives of Aq are obtained by differentiating the Padé approximant. Fréchet derivatives of Ap are then computed using a combination of the chain rule and the product rule for Fréchet derivatives.
If A is nonsingular but has negative real eigenvalues f01kec will return a non-principal matrix pth power and its condition number.

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, unless fail.code= NE_NEGATIVE_EIGVAL, in which case 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: p double Input
On entry: the required power of A.
5: condpa double * Output
On exit: if fail.code= NE_NOERROR or NW_SOME_PRECISION_LOSS, an estimate of the relative condition number of the matrix pth power, κAp. Alternatively, if fail.code= NE_RCOND, the absolute condition number of the matrix pth power.
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.
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 in this case, so a non-principal power was 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.
The relative condition number is infinite. The absolute condition number was returned instead.
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

f01kec uses the norm estimation function f04zdc to produce an estimate γ of a quantity K[n−1L(A)1,nL(A)1], such that γK. For further details on the accuracy of norm estimation, see the documentation for f04zdc.
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.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f01kec is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01kec 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 amount of complex allocatable memory required by the algorithm is typically of the order 10×n2.
The cost of the algorithm is O(n3) floating-point operations; see Higham and Lin (2013).
If the matrix pth power alone is required, without an estimate of the condition number, then f01fqc should be used. If the Fréchet derivative of the matrix power is required then f01kfc should be used. The real analogue of this function is f01jec.

10 Example

This example estimates the relative condition number of the matrix power Ap, where p=0.4 and
A = ( 1+2i 3 2 1+3i 1+i 1 1 2+i 1 2 1 2i 3 i 2+i 1 ) .  

10.1 Program Text

Program Text (f01kece.c)

10.2 Program Data

Program Data (f01kece.d)

10.3 Program Results

Program Results (f01kece.r)