NAG Library Function Document
nag_matop_real_gen_matrix_pow (f01eqc) computes the principal real power , for arbitrary , of a real by matrix .
||nag_matop_real_gen_matrix_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
is computed using the real version of the Schur–Padé algorithm described in Higham and Lin (2011)
and Higham and Lin (2013)
The real number is expressed as where and . Then . The integer power is found using a combination of binary powering and, if necessary, matrix inversion. The fractional power is computed, entirely in real arithmetic, using a real Schur decomposition and a Padé approximant.
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 – doubleInput/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 .
On exit: the by matrix th power, .
pda – IntegerInput
: the stride separating matrix row elements in the array a
p – doubleInput
On entry: the required power of .
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.
th power is not defined. nag_matop_complex_gen_matrix_pow (f01fqc)
can be used to find a complex, non-principal
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.
For positive integer , the algorithm reduces to a sequence of matrix multiplications. For negative integer , the algorithm consists of a combination of matrix inversion and matrix multiplications.
For a normal matrix (for which ) and non-integer , the Schur decomposition is diagonal and the algorithm 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.
8 Parallelism and Performance
nag_matop_real_gen_matrix_pow (f01eqc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_matop_real_gen_matrix_pow (f01eqc) 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 cost of the algorithm is . The exact cost depends on the matrix but if then the cost is independent of .
of real allocatable memory is required by the function.
If estimates of the condition number of
are required then nag_matop_real_gen_matrix_cond_pow (f01jec)
should be used.
This example finds
10.1 Program Text
Program Text (f01eqce.c)
10.2 Program Data
Program Data (f01eqce.d)
10.3 Program Results
Program Results (f01eqce.r)