Abstract
When constructing multivariate Padé approximants, highly structured linear systems arise in almost all existing definitions [10]. Until now little or no attention has been paid to fast algorithms for the computation of multivariate Padé approximants, with the exception of [17]. In this paper we show that a suitable arrangement of the unknowns and equations, for the multivariate definitions of Padé approximant under consideration, leads to a Toeplitz-block linear system with coefficient matrix of low displacement rank. Moreover, the matrix is very sparse, especially in higher dimensions. In Section 2 we discuss this for the so-called equation lattice definition and in Section 3 for the homogeneous definition of the multivariate Padé approximant. We do not discuss definitions based on multivariate generalizations of continued fractions [12, 25], or approaches that require some symbolic computations [6, 18]. In Section 4 we present an explicit formula for the factorization of the matrix that results from applying the displacement operator to the Toeplitz-block coefficient matrix. We then generalize the well-known fast Gaussian elimination procedure with partial pivoting developed in [14, 19], to deal with a rectangular block structure where the number and size of the blocks vary. We do not aim for a superfast solver because of the higher risk for instability. Instead we show how the developed technique can be combined with an easy interval arithmetic verification step. Numerical results illustrate the technique in Section 5.
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References
Brent, R.P.: Stability of fast algorithms for structured linear systems. In: Kailath, T., Sayed, A.H. (eds.) Fast Reliable Algorithms for Matrices with Structure, pp. 103–116. SIAM, Philadelphia, Pennsylvania (1999)
Bultheel, A.: Epsilon and qd algorithms for matrix- and 2D-Padé approximation. Technical Report TW57, Department of Computer Science, K.U. Leuven (1982)
Bultheel, A.: Low displacement-rank problems in 2-D Padé approximation. In: Outils et modèles mathématiques pour l’automatique, l’analyse de systèmes et le traitement du signal, vol. 2, pp. 563–576. Editions du CNRS, Paris (1982)
A. Bultheel. Recursive computation of triangular 2D-Padé approximants. Technical report, Department of Computer Science, K.U. Leuven (1982)
Bunch, J.R.: Stability of methods for solving Toeplitz systems of equations. SIAM J. Sci. Statist. Comput. 6(2), 349–364 (1985)
Chaffy, C.: \(\text{({P}ad\'e)}\sb y\) of \(\text{({P}ad\'e)}\sb x\) approximants of \({F}(x,y)\). In: Cuyt, A. (ed.) Nonlinear Numerical Methods and Rational Approximation (Wilrijk, 1987), pp. 155–166. Reidel, Dordrecht (1988)
Chisholm, J.S.R.: Rational approximants defined from double power series. Math. Comput. 27, 841–848 (1973)
Critchfield, C.L., Gammel, J.L.: Rational approximants for inverse functions of two variables. Rocky Mt. J. Math. 4, 339–349 (1974)
Cuyt, A.: Padé Approximants for Operators: Theory and Applications, vol. 1065 of Lecture Notes in Mathematics. Springer, Berlin Heidelberg New York (1984)
Cuyt, A.: How well can the concept of Padé approximant be generalized to the multivariate case? J. Comput. Appl. Math. 105(1–2), 25–50 (1999)
Cuyt, A., Verdonk, B.: General order Newton–Padé approximants for multivariate functions. Numer. Math. 43(2), 293–307 (1984)
Cuyt, A., Verdonk, B.: A review of branched continued fraction theory for the construction of multivariate rational approximants. Appl. Numer. Math. 4(2–4), 263–271 (1988)
Cuyt, A., Verdonk, B.: The need for knowledge and reliability in numeric computation: Case study of multivariate Padé approximation. Acta Appl. Math. 33, 273–302 (1993)
Gohberg, I., Kailath, T., Olshevsky, V.: Fast Gaussian elimination with partial pivoting for matrices with displacement structure. Math. Comput. 64(212), 1557–1576 (1995)
Gohberg, I., Olshevsky, V.: Complexity of multiplication with vectors for structured matrices. Linear Algebra Appl. 202, 163–192 (1994)
Gončar, A.A.: A local condition for the single-valuedness of analytic functions of several variables. Math. USSR Sb. 22(2), 305–322 (1974)
Graves-Morris, P.R., Hughes Jones, R., Makinson, G.J.: The calculation of some rational approximants in two variables. J. Inst. Math. Appl. 13, 311–320 (1974)
Guillaume, P.: Nested multivariate Padé approximants. J. Comput. Appl. Math. 82(1–2), 149–158 (1997)
Heinig, G.: Inversion of generalized Cauchy matrices and other classes of structured matrices. In: Linear algebra for signal processing (Minneapolis, MN, 1992), vol. 69 of IMA Vol. Math. Appl., pp. 63–81. Springer, Berlin Heidelberg New York(1995)
Hughes Jones, R.: General rational approximants in \({N}\)-variables. J. Approx. Theory 16(3), 201–233 (1976)
Hughes Jones, R., Makinson, G.J.: The generation of Chisholm rational polynomial approximants to power series in two variables. J. Inst. Math. Appl. 13, 299–310 (1974)
Kailath, T., Kung, S.Y., Morf, M.: Displacement ranks of matrices and linear equations. J. Math. Anal. Appl. 68(2), 395–407 (1979)
Karan, B.M., Srivastava, M.C.: A new multidimensional rational approximant. J. Indian Inst. Sci. 67(9–10), 351–360 (1987)
Karlsson, J., Wallin, H.: Rational approximation by an interpolation procedure in several variables. In: Saff, E.B., Varga, R.S. (eds.) Padé and Rational Approximation: Theory and Applications (Proc. Internat. Sympos., Univ. South Florida, Tampa, Fla., 1976), pp. 83–100. Academic Press, Berlin Heidelberg New York (1977)
Kuchminskaya, K.I.: Corresponding and associated branching continued fractions for a double power series. Dokl. Akad. Nauk Ukr. SSR, Ser. A 7, 613–617, 669 (1978)
Levin, D.: General order Padé-type rational approximants defined from double power series. J. Inst. Math. Appl. 18(1), 1–8 (1976)
Levy, B., Morf, M., Kung, S.-Y.: Some algorithms for the recursive input-output modeling of 2-d systems. Technical Report LIDS-P-962, MIT, Cambridge (1979)
Lutterodt, C.H.: A two-dimensional analogue of Padé approximant theory. J. Phys. A 7, 1027–1037 (1974)
Lutterodt, C.H.: Addendum to: “A two-dimensional analogue of Padé approximant theory” (J. Phys. A 7 (1974), 1027–1037). J. Phys. A 8, 427–428 (1975)
Lutterodt, C.H.: Rational approximants to holomorphic functions in \(n\)-dimensions. J. Math. Anal. Appl. 53(1), 89–98 (1976)
Paraskevopoulos, P.N.: Padé-type order reduction of two-dimensional systems. IEEE Trans. Circuits Syst. 27, 413–416 (1980)
Rump, S.M.: Kleine Fehlerschranken bei Matrixproblemen. PhD thesis, Universität Karlsruhe (1980)
Rump, S.M.: INTLAB – INTerval LABoratory. In: Csendes, T. (ed.) Developments in Reliable Computing, pp. 77–104. Kluwer, Dordrecht (1999)
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Research partly funded by FWO-Vlaanderen.
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Becuwe, S., Cuyt, A. On the fast solution of Toeplitz-block linear systems arising in multivariate approximation theory. Numer Algor 43, 1–24 (2006). https://doi.org/10.1007/s11075-006-9032-8
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DOI: https://doi.org/10.1007/s11075-006-9032-8