Computing matrix functions arising in engineering models with orthogonal matrix polynomials

Trigonometric matrix functions play a fundamental role in the solution of second order differential equations. Hermite series truncation together with Paterson-Stockmeyer method and the double angle formula technique allow efficient computation of the matrix cosine. A careful error bound analysis of the Hermite approximation is given and a theoretical estimate for the optimal value of its parameters is obtained. Based on the ideas above, an efficient and highly-accurate Hermite algorithm is presented. A MATLAB implementation of this algorithm has also been developed and made available online. This implementation has been compared to other efficient state-ofthe-art implementations on a large class of matrices for different dimensions, obtaining higher accuracy and lower computational costs in the majority of cases.

[1]  Ya Yan Lu,et al.  Computing a Matrix Function for Exponential Integrators , 2003 .

[2]  S. Serbin,et al.  An Algorithm for Computing the Matrix Cosine , 1980 .

[3]  R. Company,et al.  Hermite matrix polynomials and second order matrix differential equations , 1996, Approximation Theory and its Applications.

[4]  David Westreich,et al.  A practical method for computing the exponential of a matrix and its integral , 1990 .

[5]  Nicholas J. Higham,et al.  A Schur-Parlett Algorithm for Computing Matrix Functions , 2003, SIAM J. Matrix Anal. Appl..

[6]  Emilio Defez,et al.  Efficient orthogonal matrix polynomial based method for computing matrix exponential , 2011, Appl. Math. Comput..

[7]  Shing-Tung Yau,et al.  Reducing the Symmetric Matrix Eigenvalue Problem to Matrix Multiplications , 1993, SIAM J. Sci. Comput..

[8]  Emilio Defez,et al.  Some applications of the Hermite matrix polynomials series expansions , 1998 .

[9]  M. M. Tung,et al.  Improvement on the bound of Hermite matrix polynomials , 2011 .

[10]  Nicholas J. Higham,et al.  The Scaling and Squaring Method for the Matrix Exponential Revisited , 2005, SIAM J. Matrix Anal. Appl..

[11]  Nicholas J. Higham,et al.  Functions of matrices - theory and computation , 2008 .

[12]  S. Serbin Rational approximations of trigonometric matrices with application to second-order systems of differential equations , 1979 .

[13]  Nicholas J. Higham,et al.  Computing the Matrix Cosine , 2004, Numerical Algorithms.

[14]  L. Dieci,et al.  Padé approximation for the exponential of a block triangular matrix , 2000 .

[15]  R. Ward Numerical Computation of the Matrix Exponential with Accuracy Estimate , 1977 .

[16]  Emilio Defez,et al.  Computing matrix functions solving coupled differential models , 2009, Math. Comput. Model..

[17]  N. Higham The Test Matrix Toolbox for MATLAB , 1993 .

[18]  Larry J. Stockmeyer,et al.  On the Number of Nonscalar Multiplications Necessary to Evaluate Polynomials , 1973, SIAM J. Comput..

[19]  Timothy F. Havel,et al.  Derivatives of the Matrix Exponential and Their Computation , 1995 .

[20]  C. Loan,et al.  Nineteen Dubious Ways to Compute the Exponential of a Matrix , 1978 .

[21]  Nicholas J. Higham,et al.  Efficient algorithms for the matrix cosine and sine , 2005, Numerical Algorithms.

[22]  Jorge J. Moré,et al.  Digital Object Identifier (DOI) 10.1007/s101070100263 , 2001 .

[23]  Cleve B. Moler,et al.  Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later , 1978, SIAM Rev..