×

zbMATH — the first resource for mathematics

The need for knowledge and reliability in numeric computation: Case study of multivariate Padé approximation. (English) Zbl 0799.65012
The authors review results of their theoretical and numerical investigations on the multivariate Padé approximation. They show how the knowledge about the problem environment can guide a number of choices needed before the computation. Secondly, they show how the defect correction algorithm adapted for interval arithmetic allows to improve numerical results. The computation of bivariate Padé approximation is illustrated in the case of the beta function.

MSC:
65D15 Algorithms for approximation of functions
65D20 Computation of special functions and constants, construction of tables
65G30 Interval and finite arithmetic
41A21 Padé approximation
33B15 Gamma, beta and polygamma functions
41A63 Multidimensional problems (should also be assigned at least one other classification number from Section 41-XX)
Software:
SENAC
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] A. Beardon, The convergence of Padé approximants,J. Math. Anal. Appl. 21 (1968), pp. 344-346. · Zbl 0182.40501 · doi:10.1016/0022-247X(68)90216-3
[2] C. Brezinski, and M. Redivo-Zaglia, Breakdowns in the computation of orthogonal polynomials, In A. Cuyt [17] (to appear). · Zbl 0812.65008
[3] Kevin A. Broughan, SENAC: A high-level interface for the NAG library,ACM Transactions on Mathematical Software 17 No. 4 (1991), pp. 462-480. · Zbl 0900.65423 · doi:10.1145/210232.210236
[4] C. W. Cryer, The ESPRIT project FOCUS, In P. W. Gaffney and E. N. Houstis [21], pp. 371-380. · Zbl 0793.68152
[5] A. Cuyt, The epsilon-algorithm and multivariate Padé approximants,Numer. Math. 40 (1982), pp. 39-46. · Zbl 0499.65005 · doi:10.1007/BF01459073
[6] A. Cuyt, A comparison of some multivariate Padé approximants,SIAM J. Math. Anal. 14 (1983), pp. 195-202. · Zbl 0508.41016 · doi:10.1137/0514016
[7] A. Cuyt, The QD-algorithm and multivariate Padé approximants,Numer. Math 42 (1983), pp. 259-269. · Zbl 0536.65011 · doi:10.1007/BF01389572
[8] A. Cuyt,Padé Approximants for Operators: Theory and Applications, Lecture Notes in Mathematics1065, Springer-Verlag, Berlin, 1984. · Zbl 0538.41024
[9] A. Cuyt, A Montessus de Ballore theorem for multivariate Padé approximants,J. Approx. Theory 43 (1985), pp. 43-52. · Zbl 0556.41012 · doi:10.1016/0021-9045(85)90147-9
[10] A. Cuyt, A review of multivariate Padé approximation theory,J. Comput. Appl. Math. 12/13 (1985), pp. 221-232. · Zbl 0572.41010 · doi:10.1016/0377-0427(85)90019-6
[11] A. Cuyt, General order multivariate rational Hermite interpolants, Monograph, 1986, University of Antwerp (UIA).
[12] A. Cuyt, Multivariate Padé approximants revisited,BIT 26 (1986), pp. 71-79. · Zbl 0622.65011 · doi:10.1007/BF01939363
[13] A. Cuyt, A recursive computation scheme for multivariate rational interpolants,SIAM J. Num. Anal. 24 (1987), pp. 228-238. · Zbl 0616.65012 · doi:10.1137/0724019
[14] A. Cuyt, A multivariate qd-like algorithm,BIT 28 (1988), pp. 98-112. · Zbl 0637.65013 · doi:10.1007/BF01934698
[15] A. Cuyt, A multivariate convergence theorem of the ?de Montessus de Ballore? type,J. Comp. Appl. Math. 32 (1990), pp. 47-57. · Zbl 0715.41023 · doi:10.1016/0377-0427(90)90415-V
[16] A. Cuyt, Extension of ?A multivariate convergence theorem of the de Montessus de Bailore type? to multipoles,J. Comp. Appl. Math. 41 (1992), pp. 323-330. · Zbl 0756.41023 · doi:10.1016/0377-0427(92)90139-O
[17] A. Cuyt (ed.),Nonlinear Numerical Methods and Rational Approximation II, Dordrecht, Kluwer Academic Publishers, 1994 (to appear).
[18] A. Cuyt, Where do the columns of the multivariate qd-algorithm go? A proof constructed with the aid of Mathematica,Numer. Math. (1994) (submitted).
[19] A. Cuyt, and B. Verdonk, General order Newton-Padé approximants for multivariate functions,Numer. Math. 43 (1984), pp. 293-307. · Zbl 0513.41008 · doi:10.1007/BF01390129
[20] R. de Montessus de Ballore, Sur les fractions continues algébriques,Rend. Circ. Mat. Palermo 19 (1905), pp. 1-73. · JFM 36.0295.04
[21] P. W. Gaffney, and E. N. Houstis (eds.),Programming Environments for High-Level Scientific Problem Solving, Amsterdam, Elsevier Science Publishers, 1992.
[22] P. Henrici,Applied and Computational Complex Analysis, Vols. 1 & 2, John Wiley, New York, 1974. · Zbl 0313.30001
[23] R. Klatte, U. Kulish, M. Neaga, D. Ratz, and Ch. Ullrich,Pascal-XSC: Language Reference with Examples, Springer-Verlag, Berlin, 1992. · Zbl 0875.68228
[24] U. Kulish, and W. L. Miranker (eds.),A New Approach to Scientific Computation, Academic Press, Orlando, 1983.
[25] D. Levin, General-order Padé-type approximants defined from double power series,J. Inst. Maths. Appl 18 (1976), pp. 1-8. · Zbl 0352.41015 · doi:10.1093/imamat/18.1.1
[26] J. Nuttall, The convergence of Padé approximants of meromorphic functions,J. Math. Anal Appl 31 (1970), pp. 147-153. · Zbl 0204.08602 · doi:10.1016/0022-247X(70)90126-5
[27] O. Perron,Die Lehre von den Kettenbruchen II, Teubner, Stuttgart, 1977.
[28] S. Rump, Solving algebraic probems with high accuracy, In U. Kulish, and W. L. Miranker [24], pp. 51-120.
[29] G. Schumacher, and J. Wolff von Gudenberg, Highly accurate numerical algorithms, In Ch. Ullrich, and J. Wolff von Gudenberg [30], pp. 1-58. · Zbl 0692.65026
[30] Ch. Ullrich, and J. Wolff von Gudenberg (eds.),Accurate Numerical Algorithms, A Collection of Research Papers, Research Reports ESPRIT, Vol. 1, Springer-Verlag, Berlin, 1989. · Zbl 0672.00009
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.