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 SIAM J. Matrix Anal. Appl.34(3) 1341–1360
1: – IntegerInput
On entry: , the order of the matrix .
2: – doubleInput/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 matrix th power, .
3: – IntegerInput
On entry: the stride separating matrix row elements in the array a.
4: – doubleInput
On entry: the required power of .
5: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error 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 had an illegal value.
On entry, . Constraint: .
On entry, and . Constraint: .
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.
has eigenvalues on the negative real line. The principal th power is not defined. f01fqc can be used to find a complex, non-principal th power.
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.
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.
8Parallelism and Performance
f01eqc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
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 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.
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 f01jec should be used.