A superfast algorithm for multi-dimensional Padé systems

For a vector ofk+1 matrix power series, a superfast algorithm is given for the computation of multi-dimensional Padé systems. The algorithm provides a method for obtaining matrix Padé, matrix Hermite Padé and matrix simultaneous Padé approximants. When the matrix power series is normal or perfect, the algorithm is shown to calculate multi-dimensional matrix Padé systems of type (n0,...,nk) inO(‖n‖ · log2‖n‖) block-matrix operations, where ‖n‖=n0+...+nk. Whenk=1 and the power series is scalar, this is the same complexity as that of other superfast algorithms for computing Padé systems. Whenk>1, the fastest methods presently compute these matrix Padé approximants with a complexity ofO(‖n‖2). The algorithm succeeds also in the non-normal and non-perfect case, but with a possibility of an increase in the cost complexity.

[1]  H. Jager,et al.  A Multidimensional Generalization of the Padé Table. II , 1964 .

[2]  George Labahn,et al.  Inversion components of block Hankel-like matrices , 1992 .

[3]  K. Mahler.  Zur Approximation der Exponentialfunktion und des Logarithmus. Teil I. , 1932 .

[4]  Yasuo Sugiyama,et al.  An algorithm for solving discrete-time Wiener-Hopf equations based upon Euclid's algorithm , 1986, IEEE Trans. Inf. Theory.

[5]  David Y. Y. Yun,et al.  Fast Solution of Toeplitz Systems of Equations and Computation of Padé Approximants , 1980, J. Algorithms.

[6]  M. Tismenetsky,et al.  Generalized Bezoutian and the inversion problem for block matrices, I. General scheme , 1986 .

[7]  J. Dora,et al.  Approximants de pade-hermite. 1ère partie: theorie , 1984 .

[8]  P. R. Graves-Morris,et al.  Definition and uniqueness of integral approximants , 1990 .

[9]  M. de Bruin,et al.  The interruption phenomenon for generalized continued fractions , 1978, Bulletin of the Australian Mathematical Society.

[10]  George Labahn,et al.  The Inverses of Block Hankel and Block Toeplitz Matrices , 1990, SIAM J. Comput..

[11]  Adhemar Bultheel,et al.  The computation of non-perfect Padé-Hermite approximants , 2005, Numerical Algorithms.

[12]  Marcel G. de-Bruin.  Some aspects of simultaneous rational approximation , 1990 .

[13]  Stanley Cabay,et al.  Algebraic Computations of Scaled Padé Fractions , 1986, SIAM J. Comput..

[14]  W. Gragg,et al.  The Padé Table and Its Relation to Certain Algorithms of Numerical Analysis , 1972 .

[15]  George Labahn,et al.  On the theory and computation of nonperfect Pade´-Hermite approximants , 1992 .

[16]  Bernhard Beckermann,et al.  The structure of the singular solution table of the M-Pade´ approximation problem , 1990 .

[17]  J. Dora,et al.  Approximants de Padé-Hermite 2ème partie: programmation , 1984 .

[18]  George Labahn,et al.  Matrix Padé Fractions and Their Computation , 1989, SIAM J. Comput..

[19]  J. Bunch,et al.  Direct Methods for Solving Symmetric Indefinite Systems of Linear Equations , 1971 .

[20]  P. R. Graves-Morris.  Vector-valued Rational Interpolants II , 1984 .

[21]  H. Padé.  Sur la représentation approchée d'une fonction par des fractions rationnelles , 1892 .

[22]  Zhishun A. Liu,et al.  A Look Ahead Lanczos Algorithm for Unsymmetric Matrices , 1985 .

[23]  Adhemar Bultheel,et al.  Minimal vector Pade´ approximation , 1989 .

[24]  John Coates.  On the algebraic approximation of functions. III , 1966 .

[25]  Bernhard Beckermann,et al.  A reliable method for computing M-Pade´ approximants on arbitrary staircases , 1992 .

[26]  Stefan Paszkowski,et al.  Recurrence relations in Padé-Hermite approximation , 1987 .