Computes the p'th power A^p of the matrix A for arbitrary real -1<p<1 and A with no nonpositive real eigenvalues, by the Schur-Pade algorithm. It also computes the Frechet derivative of A^p in any direction E and estimates the condition number for computing the matrix power.
This submission contains two functions: powerm_pade_fre.m uses complex arithmetic;
powerm_pade_fre_real.m uses real arithmetic which is intended for the case where both A and E are real.
The codes can be called in the following ways (same for powerm_pade_fre_real.m):
X = POWERM_PADE_FRE(A,P)
[X,~,COND] = POWERM_PADE_FRE(A,P)
[X,F] = POWERM_PADE_FRE(A,P,E)
[X,F,COND] = POWERM_PADE_FRE(A,P,E)
[X,F,COND,NSQ,M] = POWERM_PADE_FRE(A,P,E)
where X is A^p, F is the Frechet derivative at A in the direction E, COND is the condition number estimate, NSQ is the number of matrix square roots computed and M is the degree of the Pade approximant used in the algorithm.
Function TEST_GALLERY.M runs a simple test of the codes. Matrix Function Toolbox (MFT) must be installed. Obtain it from http://www.maths.manchester.ac.uk/~higham/mftoolbox
More details can be found in:
N. J. Higham and L. Lin,
An Improved Schur--Pade Algorithm for Fractional Powers of a Matrix and their Frechet Derivatives
MIMS Eprint 2013.1, January 2013, revised May 2013.
Lijing Lin (2022). Fractional Matrix Powers with Frechet Derivatives and Condition Number Estimate (https://www.mathworks.com/matlabcentral/fileexchange/41621-fractional-matrix-powers-with-frechet-derivatives-and-condition-number-estimate), MATLAB Central File Exchange. Retrieved .
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