f01kfc computes the Fréchet derivative of the th power (where is real) of the complex matrix applied to the complex matrix . The principal matrix power is also returned.
The function may be called by the names: f01kfc or nag_matop_complex_gen_matrix_frcht_pow.
3Description
For a matrix with no eigenvalues on the closed negative real line, () can be defined as
where is the principal logarithm of (the unique logarithm whose spectrum lies in the strip ). If is nonsingular but has negative real eigenvalues, the principal logarithm is not defined, but a non-principal th power can be defined by using a non-principal logarithm.
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 .
f01kfc uses the algorithms of Higham and Lin (2011) and Higham and Lin (2013) to compute and . 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 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 and is then computed using a combination of the chain rule and the product rule for Fréchet derivatives.
4References
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
5Arguments
1: – IntegerInput
On entry: , the order of the matrix .
Constraint:
.
2: – ComplexInput/Output
Note: the dimension, dim, of the array a
must be at least
.
The th element of the matrix is stored in .
On entry: the matrix .
On exit: the principal matrix th power, . Alternatively if NE_NEGATIVE_EIGVAL, a non-principal th power is returned.
3: – IntegerInput
On entry: the stride separating matrix row elements in the array a.
Constraint:
.
4: – ComplexInput/Output
Note: the dimension, dim, of the array e
must be at least
.
The th element of the matrix is stored in .
On entry: the matrix .
On exit: the Fréchet derivative .
5: – IntegerInput
On entry: the stride separating matrix row elements in the array e.
Constraint:
.
6: – doubleInput
On entry: the required power of .
7: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error 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 had an illegal value.
NE_INT
On entry, . Constraint: .
NE_INT_2
On entry, and . Constraint: .
On entry, and . Constraint: .
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
has eigenvalues on the negative real line. The principal th 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
is singular so the th power cannot be computed.
NW_SOME_PRECISION_LOSS
has been computed using an IEEE double precision Padé approximant, although the arithmetic precision is higher than IEEE double precision.
7Accuracy
For a normal matrix (for which ), 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.
If the condition number of the matrix power is required then f01kec should be used.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
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.
9Further Comments
The complex allocatable memory required by the algorithm is approximately .
If the matrix th 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.
10Example
This example finds and the Fréchet derivative of the matrix power , where ,