Recursiveness in matrix rational interpolation problems

Abstract We consider the problem of computing solutions to a variety of matrix rational interpolation problems. These include the partial realization problem for matrix power series and Newton-Pade, Hermite-Pade, simultaneous Pade, M-Pade and multipoint Pade approximation problems along with their matrix generalizations. A general recurrence relation is given for solving these problems. Unlike other previous recursive methods, our recurrence works along arbitrary computational paths. When restricted to specific paths, the recurrence relation generalizes previous work of Antoulas, Cabay and Labahn, Beckermann, Van Barel and Bultheel and Gutknecht along with others. Our results rely on the concept of extended M-Pade approximation introduced in this paper. This is a natural generalization of the two-point Pade approximation problem extended to multiple interpolation points (including infinity) and matrix Laurent and Newton series. By using module-theoretic techniques we determine complete parameterizations of all solutions to this problem. Our recurrence relation then efficiently computes these parameterizations. This recursion requires no conditions on the input data. We also discuss the concept of duality which was shown to be of particular interest for a stable computation of those approximants. Finally, we show the invariance of our approximation problem under linear transformations of the extended complex plane.

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

[2]  Adhemar Bultheel,et al.  A new approach to the rational interpolation problem: the vector case , 1990 .

[3]  Martin H. Gutknecht,et al.  Stable row recurrences for the Padé table and generically superfast lookahead solvers for non-Hermitian Toeplitz systems , 1993 .

[4]  M. Gutknecht.  The multipoint Padé table and general recurrences for rational interpolation , 1993 .

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

[6]  M. Barel,et al.  Duality in vector Pade´-Hermite approximation problems , 1996 .

[7]  W. Dahmen,et al.  Approximation theory VII , 1994 .

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

[9]  George Labahn,et al.  Computation of Numerical Padé-Hermite and Simultaneous Padé Systems II: A Weakly Stable Algorithm , 1996, SIAM J. Matrix Anal. Appl..

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

[11]  B. Beckermann,et al.  A Uniform Approach for the Fast Computation of Matrix-Type Padé Approximants , 1994, SIAM J. Matrix Anal. Appl..

[12]  Adhemar Bultheel,et al.  A lookahead algorithm for the solution of block toeplitz systems , 1997 .

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

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

[15]  A. Antoulas.  On recursiveness and related topics in linear systems , 1986 .

[16]  S. Cabay,et al.  A weakly stable algorithm for Pade´ approximants and the inversion of Hankel matrices , 1993 .

[17]  Jeannette Van Iseghem,et al.  Algebraic Aspects of Matrix Orthogonality for Vector Polynomials , 1997 .

[18]  J. McCabe,et al.  Continued Fractions which Correspond to Power Series Expansions at Two Points , 1976 .

[19]  B. Beckermann,et al.  The Diagonal of the Padé Table and the Approximation of the Weyl Function of Second-Order Difference Operators , 1997 .

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

[21]  G. Claessens,et al.  On the Newton-Padé approximation problem , 1978 .

[22]  Stefan Paszkowski.  Hermite-Pade´ approximation (basic notions and theorems) , 1990 .

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

[24]  J. V. Iseghem.  Vector orthogonal relations. Vector QD-algorithm , 1987 .

[25]  M. Barel,et al.  A look-ahead method for computing vector Padé-Hermite approximants , 1995 .

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

[27]  M. Barel,et al.  A new formal approach to the rational interpolation problem , 1992 .