LSSC 2003: Large-Scale Scientific Computing pp 204-213 | Cite as
Multivariate Rational Interpolation of Scattered Data
Abstract
Rational data fitting has proved extremely useful in a number of scientific applications. We refer among others to its use in some network problems [6,7,15,16], to the modelling of electro-magnetic components [20,13], to model reduction of linear shift-invariant systems [2,3,8] and so on.
When computing a rational interpolant in one variable, all existing techniques deliver the same rational function, because all rational functions that satisfy the interpolation conditions reduce to the same unique irreducible form. When switching from one to many variables, the situation is entirely different. Not only does one have a large choice of multivariate rational functions, but moreover, different algorithms yield different rational interpolants and apply to different situations.
The rational interpolation of function values that are given at a set of points lying on a multidimensional grid, has extensively been dealt with in [11,10,5]. The case where the interpolation data are scattered in the multivariate space, is far less discussed and is the subject of this paper. We present a fast solver for the linear block Cauchy-Vandermonde system that translates the interpolation conditions, and combine it with an interval arithmetic verification step.
Keywords
Interval Arithmetic Interpolation Condition Rational Interpolation Rational Interpolant Irreducible FormPreview
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References
- 1.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 (1999)Google Scholar
- 2.Bultheel, A.: Algorithms to compute the reflection coefficients of digital filters. In: Collatz, L., et al. (eds.) Numer. Math. Approx. Th., vol. 7, pp. 33–50 (1984)Google Scholar
- 3.Bultheel, A., Van Barel, M.: Padé techniques for model reduction in linear system theory: a survey. J. Comput. Appl. Math. 14, 401–438 (1986)CrossRefMathSciNetGoogle Scholar
- 4.Chaffy, C.: Interpolation polynomiale et rationnelle d’une fonction de plusieurs variables complexes. Thèse, Institut National Polytechnique de Grenoble (1984)Google Scholar
- 5.Cuyt, A.: A recursive computation scheme for multivariate rational interpolants. SIAM J. Numer. Anal. 24, 228–239 (1987)MATHCrossRefMathSciNetGoogle Scholar
- 6.Cuyt, A., Lenin, R.B.: Computing packet loss probabilities in multiplexer models using adaptive rational interpolation with optimal pole placement. IEEE Transactions on Computers (submitted for publication)Google Scholar
- 7.Cuyt, A., Lenin, R.B.: Multivariate rational approximants for multiclass closed queuing networks. IEEE Transactions on Computers 50, 1279–1288 (2001)CrossRefGoogle Scholar
- 8.Cuyt, A., Ogawa, S., Verdonk, B.: Model reduction of multidimensional linear shift-invariant recursive systems using Padé techniques. Multidimensional systems and Signal processing 3, 309–321 (1992)MATHCrossRefMathSciNetGoogle Scholar
- 9.Cuyt, A., Verdonk, B.: Different techniques for the construction of multivariate rational interpolants. In: Cuyt, A. (ed.) Nonlinear numerical methods and rational approximation (Wilrijk, 1987), pp. 167–190. Reidel, Dordrecht (1988)Google Scholar
- 10.Cuyt, A.: A multivariate qd-like algorithm. BIT 28, 98–112 (1988)MATHCrossRefMathSciNetGoogle Scholar
- 11.Cuyt, A., Verdonk, B.: General order Newton-Padé approximants for multivariate functions. Numer. Math. 43, 293–307 (1984)MATHCrossRefMathSciNetGoogle Scholar
- 12.Cuyt, A., Verdonk, B.: A review of branched continued fraction theory for the construction of multivariate rational approximants. Appl. Numer. Math. 4, 263–271 (1988)MATHCrossRefMathSciNetGoogle Scholar
- 13.De Geest, J., Dhaene, T., Faché, N., De Zutter, D.: Adaptive CAD-model building algorithm for general planar microwave structures. IEEE Transactions on microwave theory and techniques 47, 1801–1809 (1999)CrossRefGoogle Scholar
- 14.Gohberg, I., Kailath, T., Olshevsky, V.: Fast Gaussian elimination with partial pivoting for matrices with displacement structure. Math. Comp. 64, 1557–1576 (1995)MATHCrossRefMathSciNetGoogle Scholar
- 15.Gong, W.B., Nananukul, S.: Rational interpolation for rare event probabilities. In: Stochastic Networks: Lecture Notes in Statistics, pp. 139–168 (1996)Google Scholar
- 16.Gong, W.B., Yang, H.: Rational approximants for some performance analysis problems. IEEE Trans. Comput. 44, 1394–1404 (1995)MATHCrossRefGoogle Scholar
- 17.Hammer, R., Hocks, M., Kulisch, U., Ratz, D.: Numerical Toolbox for verified computing I. Springer, Heidelberg (1993)MATHGoogle Scholar
- 18.Kailath, T., Kung, S.Y., Morf, M.: Displacement ranks of matrices and linear equations. J. Math. Anal. Appl. 68, 395–407 (1979)MATHCrossRefMathSciNetGoogle Scholar
- 19.Kuchminskaya, K.I.: On approximation of functions by two-dimensional continued fractions. In: Gilewicz, J., Pindor, M., Siemaszko, W. (eds.) Rational approximation and applications in mathematics and physics (Łańcut, 1985). Lecture Notes in Mathematics, vol. 1237, pp. 207–216. Springer, Berlin (1987)CrossRefGoogle Scholar
- 20.Lehmensiek, R., Meyer, P.: An efficient adaptive frequency sampling algorithm for model-based parameter estimation as applied to aggressive space mapping. Microwave and Optical Technology Letters 24, 71–78 (2000)CrossRefGoogle Scholar
- 21.O’Donohoe, M.: Applications of continued fractions in one and more variables. PhD thesis, Brunel University (1974)Google Scholar
- 22.Siemaszko, W.: Thiele-type branched continued fractions for two-variable functions. J. Comput. Appl. Math. 9, 137–153 (1983)MATHCrossRefMathSciNetGoogle Scholar