Matrix Padé-Type Method for Computing the Matrix Exponential

ABSTRACT

Matrix Padé approximation is a widely used method for computing matrix functions. In this paper, we apply matrix Padé-type approximation instead of typical Padé approximation to computing the matrix exponential. In our approach the scaling and squaring method is also used to make the approximant more accurate. We present two algorithms for computing and for computing with many espectively. Numerical experiments comparing the proposed method with other existing methods which are MATLAB’s functions expm and funm show that our approach is also very effective and reliable for computing the matrix exponential . Moreover, there are two main advantages of our approach. One is that there is no inverse of a matrix required in this method. The other is that this method is more convenient when computing for a fixed matrix A with many t ≥ 0.

Matrix Padé approximation is a widely used method for computing matrix functions. In this paper, we apply matrix Padé-type approximation instead of typical Padé approximation to computing the matrix exponential. In our approach the scaling and squaring method is also used to make the approximant more accurate. We present two algorithms for computing and for computing with many espectively. Numerical experiments comparing the proposed method with other existing methods which are MATLAB’s functions expm and funm show that our approach is also very effective and reliable for computing the matrix exponential . Moreover, there are two main advantages of our approach. One is that there is no inverse of a matrix required in this method. The other is that this method is more convenient when computing for a fixed matrix A with many t ≥ 0.

Cite this paper

nullC. Li, X. Zhu and C. Gu, "Matrix Padé-Type Method for Computing the Matrix Exponential,"*Applied Mathematics*, Vol. 2 No. 2, 2011, pp. 247-253. doi: 10.4236/am.2011.22028.

nullC. Li, X. Zhu and C. Gu, "Matrix Padé-Type Method for Computing the Matrix Exponential,"

References

[1] C. B. Moler and C. F. Van Loan, “Nineteen Dubious Ways to Compute the Exponential of a Matrix,” SIAM Review, Vol. 20, No. 4, 1978, pp. 801-836. doi:10.1137/1020098

[2] C. B. Moler and C. F. Van Loan, “Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later,” SIAM Review, Vol. 45, No. 1, 2003, pp. 3-49. doi:10.1137/S00361445024180

[3] N. J. Higham, “The Scaling and Squaring Method for the Matrix Exponential Revisited,” SIAM Journal on Matrix Analysis and Application, Vol. 26, No. 4, 2005, pp. 1179-1193. doi:10.1137/04061101X

[4] N. J. Higham, “The Scaling and Squaring Method for the Matrix Exponential Revisited,” SIAM Review, Vol. 51, No. 4, 2009, pp. 747-764. doi:10.1137/090768539

[5] C. Gu, “Matrix Padé-Type Approximant and Directional Matrix Padé Approximant in the Inner Product Space,” Journal of Computational and Applied Mathematics, Vol. 164-165, No. 1, 2004, pp. 365-385. doi:10.1016/S0377-0427(03)00487-4

[6] C. Brezinski, “Rational Approximation to Formal Power Series,” Journal of Approximation Theory, Vol. 25, No. 4, 1979, pp. 295-317. doi:10.1016/0021-9045(79)90019-4

[7] C. Brezinski, “Padé-Type Approximation and General Orthogonal Polynomials,” Birkh?auser-Verlag, Basel, 1980.

[8] A. Draux, “Approximants de Type Padé et de Table,” Little: Publication A, University of Lille, Lille, No. 96, 1983.

[9] C. Gu, “Generalized Inverse Matrix Padé Approximation on the Basis of Scalar Products,” Linear Algebra and Its Applications, Vol. 322, No. 1-3, 2001, pp. 141-167. doi: 10.1016/S0024-3795(00)00230-5

[10] C. Gu, “A Practical Two-Dimensional Thiele-Type Matrix Padé Approximation,” IEEE Transactions on Automatic Control, Vol. 48, No. 12, 2003, pp. 2259-2263. doi: 10.1109/TAC.2003.820163

[11] N. J. Higham, “Functions of Matrices: Theory and Computation,” SIAM Publisher, Philadelphia, 2008. doi:10.11 37/1.9780898717778

[12] A. Sidi, “Rational Approximations from Power Series of Vector-Valued Meromorphic Functions,” Journal of Approximation Theory, Vol. 77, No. 1, 1994, pp. 89-111. doi:10.1006/jath.1994.1036

[13] R. Mathias, “Approximation of Matrix-Valued Functions,” SIAM Journal on Matrix Analysis and Application, Vol. 14, No. 4, 1993, pp. 1061-1063. doi:10.1137/0614070

[14] J. D. Lawson, “Generalized Runge-Kutta Processes for Stable Systems with Large Lipschitz Constants,” SIAM Journal on Numerical Analysis, Vol. 4, No. 3, 1967, pp. 372-380. doi:10.1137/0704033

[15] A. H. Al-Mohy and N. J. Higham, “A New Scaling and Squaring Algorithm for the Matrix,” SIAM Journal on Matrix Analysis and Application, Vol. 31, No. 3, 2009, pp. 970-989. doi:10.1137/09074721X

[1] C. B. Moler and C. F. Van Loan, “Nineteen Dubious Ways to Compute the Exponential of a Matrix,” SIAM Review, Vol. 20, No. 4, 1978, pp. 801-836. doi:10.1137/1020098

[2] C. B. Moler and C. F. Van Loan, “Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later,” SIAM Review, Vol. 45, No. 1, 2003, pp. 3-49. doi:10.1137/S00361445024180

[3] N. J. Higham, “The Scaling and Squaring Method for the Matrix Exponential Revisited,” SIAM Journal on Matrix Analysis and Application, Vol. 26, No. 4, 2005, pp. 1179-1193. doi:10.1137/04061101X

[4] N. J. Higham, “The Scaling and Squaring Method for the Matrix Exponential Revisited,” SIAM Review, Vol. 51, No. 4, 2009, pp. 747-764. doi:10.1137/090768539

[5] C. Gu, “Matrix Padé-Type Approximant and Directional Matrix Padé Approximant in the Inner Product Space,” Journal of Computational and Applied Mathematics, Vol. 164-165, No. 1, 2004, pp. 365-385. doi:10.1016/S0377-0427(03)00487-4

[6] C. Brezinski, “Rational Approximation to Formal Power Series,” Journal of Approximation Theory, Vol. 25, No. 4, 1979, pp. 295-317. doi:10.1016/0021-9045(79)90019-4

[7] C. Brezinski, “Padé-Type Approximation and General Orthogonal Polynomials,” Birkh?auser-Verlag, Basel, 1980.

[8] A. Draux, “Approximants de Type Padé et de Table,” Little: Publication A, University of Lille, Lille, No. 96, 1983.

[9] C. Gu, “Generalized Inverse Matrix Padé Approximation on the Basis of Scalar Products,” Linear Algebra and Its Applications, Vol. 322, No. 1-3, 2001, pp. 141-167. doi: 10.1016/S0024-3795(00)00230-5

[10] C. Gu, “A Practical Two-Dimensional Thiele-Type Matrix Padé Approximation,” IEEE Transactions on Automatic Control, Vol. 48, No. 12, 2003, pp. 2259-2263. doi: 10.1109/TAC.2003.820163

[11] N. J. Higham, “Functions of Matrices: Theory and Computation,” SIAM Publisher, Philadelphia, 2008. doi:10.11 37/1.9780898717778

[12] A. Sidi, “Rational Approximations from Power Series of Vector-Valued Meromorphic Functions,” Journal of Approximation Theory, Vol. 77, No. 1, 1994, pp. 89-111. doi:10.1006/jath.1994.1036

[13] R. Mathias, “Approximation of Matrix-Valued Functions,” SIAM Journal on Matrix Analysis and Application, Vol. 14, No. 4, 1993, pp. 1061-1063. doi:10.1137/0614070

[14] J. D. Lawson, “Generalized Runge-Kutta Processes for Stable Systems with Large Lipschitz Constants,” SIAM Journal on Numerical Analysis, Vol. 4, No. 3, 1967, pp. 372-380. doi:10.1137/0704033

[15] A. H. Al-Mohy and N. J. Higham, “A New Scaling and Squaring Algorithm for the Matrix,” SIAM Journal on Matrix Analysis and Application, Vol. 31, No. 3, 2009, pp. 970-989. doi:10.1137/09074721X