×

A new algorithm to approximate bivariate matrix function via Newton-Thiele type formula. (English) Zbl 1266.65015

Summary: A new method for computing the approximation of bivariate matrix function is introduced. It uses the construction of bivariate Newton-Thiele type matrix rational interpolants on a rectangular grid. The rational interpolant is of the form motivated by Tan and Fang (2000), which is combined by Newton interpolant and branched continued fractions, with scalar denominator. The matrix quotients are based on the generalized inverse for a matrix which is introduced by C. Gu the author of this paper, and it is effective in continued fraction interpolation. The algorithm and some other important conclusions such as divisibility and characterization are given. In the end, two examples are also given to show the effectiveness of the algorithm. The numerical results of the second example show that the algorithm of this paper is better than the method of Thieletype matrix-valued rational interpolant in Gu (1997).

MSC:

65D05 Numerical interpolation
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] G. A. Baker and P. R. Graves-Morris, Padé Approximation, Part II, Addison-Wesley Publishing Company, Reading, Mass, USA, 1981.
[2] I. Kh. Kuchminska, O. M. Sus, and S. M. Vozna, “Approximation properties of two-dimensional continued fractions,” Ukrainian Mathematical Journal, vol. 55, pp. 36-54, 2003. · Zbl 1035.40003
[3] J. Tan and Y. Fang, “Newton-Thiele’s rational interpolants,” Numerical Algorithms, vol. 24, no. 1-2, pp. 141-157, 2000. · Zbl 0965.41007
[4] P. R. Graves-Morris, “Vector valued rational interpolants I,” Numerische Mathematik, vol. 42, no. 3, pp. 331-348, 1983. · Zbl 0525.41014
[5] G. Chuan-qing and C. Zhibing, “Matrix valued rational interpolants and its error formula,” Mathematica Numerica Sinica, vol. 17, pp. 73-77, 1995. · Zbl 0860.41018
[6] G. Chuan-qing, “Matrix Padé type approximat and directional matrix Padé approximat in the inner product space,” Journal of Computational and Applied Mathematics, vol. 164-165, pp. 365-385, 2004. · Zbl 1050.41012
[7] C. Gu, “Thiele-type and Lagrange-type generalized inverse rational interpolation for rectangular complex matrices,” Linear Algebra and Its Applications, vol. 295, no. 1-3, pp. 7-30, 1999. · Zbl 0932.93020
[8] C. Gu, “A practical two-dimensional Thiele-type matrix Padé approximation,” IEEE Transactions on Automatic Control, vol. 48, no. 12, pp. 2259-2263, 2003. · Zbl 1364.93135
[9] C. Gu, “Bivariate Thiele-type matrix-valued rational interpolants,” Journal of Computational and Applied Mathematics, vol. 80, no. 1, pp. 71-82, 1997. · Zbl 0877.65005
[10] N. K. Bose and S. Basu, “Two-dimensional matrix Padé approximants: existence, nonuniqueness and recursive computation,” IEEE Transactions on Automatic Control, vol. 25, no. 3, pp. 509-514, 1980. · Zbl 0433.93017
[11] A. Cuyt and B. Verdonk, “Multivariate reciprocal differences for branched Thiele continued fraction expansions,” Journal of Computational and Applied Mathematics, vol. 21, no. 2, pp. 145-160, 1988. · Zbl 0638.65014
[12] W. Siemaszko, “Thiele-tyoe branched continued fractions for two-variable functions,” Journal of Computational and Applied Mathematics, vol. 9, no. 2, pp. 137-153, 1983. · Zbl 0515.41015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.