# Matrix Padé fractions

## Abstract

For matrix power series with coefficients over a field, the notion of a matrix power series remainder sequence and its corresponding cofactor sequence are introduced and developed. An algorithm for constructing these sequences is presented.

It is shown that the cofactor sequence yields directly a sequence of Padé fractions for a matrix power series represented as a quotient *B*(*z*)^{−1}*A*(*z*). When *B*(*z*)^{−1}*A*(*z*) is normal, the complexity of the algorithm for computing a Padé fraction of type (m,n) is O(*p*^{3}(*m+n*)^{2}), where p is the order of the matrices A(z) and B(z).

For power series which are abnormal, for a given (m,n), Padé fractions may not exist. However, it is shown that a generalized notion of Padé fraction, the Padé form, introduced in this paper does always exist and can be computed by the algorithm. In the abnormal case, the algorithm can reach a complexity of O(*p*^{3}(*m+n*)^{3}), depending on the nature of the abnormalities. In the special case of a scalar power series, however, the algorithm complexity is O((*m+n*)^{2}), even in the abnormal case.

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## References

- 1.N.K. Bose and S. Basu, “Theory and Recursive Computation of 1-D Matrix Padé Approximants,”
*IEEE Trans. on Circuits and Systems*,**4**, pp. 323–325 (1980).CrossRefGoogle Scholar - 2.R. Brent, F.G. Gustavson, and D.Y.Y. Yun, “Fast Solution of Toeplitz Systems of Equations and Computation of Padé Approximants,”
*J. of Algorithms*,**1**pp. 259–295 (1980).CrossRefGoogle Scholar - 3.A. Bultheel, “Recursive Algorithms for the Matrix Padé Table,”
*Math. of Computation*,**35**pp. 875–892 (1980).Google Scholar - 4.A. Bultheel, “Recursive Algorithms for Nonnormal Padé Tables,”
*SIAM J. Appl. Math*,**39**pp. 106–118 (1980).CrossRefGoogle Scholar - 5.S. Cabay and D.K. Choi, “Algebraic Computations of Scaled Padé Fractions,”
*SIAM J. of Computation*,**15**pp. 243–270 (1986).Google Scholar - 6.S. Cabay and P. Kossowski, “Power Series Remainder Sequences and Padé Fractions over Integral Domains,”
*J. of Symbolic Computation*, (to appear).Google Scholar - 7.W.B. Gragg, “The Padé Table and its Relation to Certain Algorithms of Numerical Analysis,”
*SIAM Rev.*,**14**pp. 1–61 (1972).Google Scholar - 8.G. Labahn,
*Matrix Padé Approximants*, M.Sc. Thesis, Dep't of Computing Science, University of Alberta, Edmonton, Canada (1986).Google Scholar - 9.J. Rissanen, “Recursive Evaluation of Padé Approximants for Matrix Sequences,”
*IBM J. Res. Develop.*, pp. 401–406 (1972).Google Scholar - 10.J. Rissanen, “Algorithms for Triangular Decomposition of Block Hankel and Toeplitz Matrices with Application to Factoring Positive Matrix Polynomials,”
*Math. Comp.*,**27**pp. 147–154 (1973).Google Scholar - 11.J. Rissanen, “Solution of Linear Equations with Hankel and Toeplitz Matrices,”
*Numer. Math*,**22**pp. 361–366 (1974).Google Scholar - 12.Y. Starkand, “Explicit Formulas for Matrix-valued Padé Approximants,”
*J. of Comp. and Appl. Math*,**5**pp. 63–65 (1979).Google Scholar