NAG Library Function Document
nag_matop_complex_gen_matrix_cond_pow (f01kec) computes an estimate of the relative condition number of the th power (where is real) of a complex by matrix , in the -norm. The principal matrix power is also returned.
||nag_matop_complex_gen_matrix_cond_pow (Integer n,
For a matrix
with no eigenvalues on the closed negative real line,
) can be defined as
is the principal logarithm of
(the unique logarithm whose spectrum lies in the strip
The Fréchet derivative of the matrix
th power of
is the unique linear mapping
such that for any matrix
The derivative describes the first-order effect of perturbations in on the matrix power .
The relative condition number of the matrix
th power can be defined by
is the norm of the Fréchet derivative of the matrix power at
nag_matop_complex_gen_matrix_cond_pow (f01kec) uses the algorithms of Higham and Lin (2011)
and Higham and Lin (2013)
. The real number
is expressed as
. The integer power
is found using a combination of binary powering and, if necessary, matrix inversion. The fractional power
is computed using a Schur decomposition, a Padé approximant and the scaling and squaring method.
To obtain the estimate of , nag_matop_complex_gen_matrix_cond_pow (f01kec) first estimates by computing an estimate of a quantity , such that . This requires multiple Fréchet derivatives to be computed. Fréchet derivatives of are obtained by differentiating the Padé approximant. Fréchet derivatives of are then computed using a combination of the chain rule and the product rule for Fréchet derivatives.
If is nonsingular but has negative real eigenvalues nag_matop_complex_gen_matrix_cond_pow (f01kec) will return a non-principal matrix th power and its condition number.
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 MIMS Eprint 2013.1
Manchester Institute for Mathematical Sciences, School of Mathematics, University of Manchester http://eprints.ma.man.ac.uk/
n – IntegerInput
On entry: , the order of the matrix .
a – ComplexInput/Output
the dimension, dim
, of the array a
must be at least
The th element of the matrix is stored in .
On entry: the by matrix .
, unless NE_NEGATIVE_EIGVAL
, in which case a non-principal
th power is returned.
pda – IntegerInput
: the stride separating matrix row elements in the array a
p – doubleInput
On entry: the required power of .
condpa – double *Output
NE_NOERROR or NW_SOME_PRECISION_LOSS
, an estimate of the relative condition number of the matrix
. Alternatively, if NE_RCOND
, the absolute condition number of the matrix
fail – NagError *Input/Output
The NAG error argument (see Section 3.6
in the Essential Introduction).
6 Error Indicators and Warnings
Dynamic memory allocation failed.
On entry, argument had an illegal value.
On entry, .
On entry, and .
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
has eigenvalues on the negative real line.
The principal th power is not defined in this case, so a non-principal power was returned.
The relative condition number is infinite. The absolute condition number was returned instead.
is singular so the th power cannot be computed.
has been computed using an IEEE double precision Padé approximant, although the arithmetic precision is higher than IEEE double precision.
nag_matop_complex_gen_matrix_cond_pow (f01kec) uses the norm estimation function nag_linsys_complex_gen_norm_rcomm (f04zdc)
to produce an estimate
of a quantity
, such that
. For further details on the accuracy of norm estimation, see the documentation for nag_linsys_complex_gen_norm_rcomm (f04zdc)
For a normal matrix
), the Schur decomposition is diagonal and the computation of the fractional part of the matrix power reduces to evaluating powers of the eigenvalues of
and then constructing
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
nag_matop_complex_gen_matrix_cond_pow (f01kec) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_matop_complex_gen_matrix_cond_pow (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 Users' Note
for your implementation for any additional implementation-specific information.
The amount of complex allocatable memory required by the algorithm is typically of the order .
The cost of the algorithm is
floating-point operations; see Higham and Lin (2013)
If the matrix
th power alone is required, without an estimate of the condition number, then nag_matop_complex_gen_matrix_pow (f01fqc)
should be used. If the Fréchet derivative of the matrix power is required then nag_matop_complex_gen_matrix_frcht_pow (f01kfc)
should be used. The real analogue of this function is nag_matop_real_gen_matrix_cond_pow (f01jec)
This example estimates the relative condition number of the matrix power
10.1 Program Text
Program Text (f01kece.c)
10.2 Program Data
Program Data (f01kece.d)
10.3 Program Results
Program Results (f01kece.r)