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Some extensions of the Lanczos-Ortiz theory of canonical polynomials in the Tau Method

Author: M. E. Froes Bunchaft
Journal: Math. Comp. 66 (1997), 609-621
MSC (1991): Primary 65L05, 65L10, 65D99
MathSciNet review: 1397440
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Abstract: Lanczos and Ortiz placed the canonical polynomials (c.p.'s) in a central position in the Tau Method. In addition, Ortiz devised a recursive process for determining c.p.'s consisting of a generating formula and a complementary algorithm coupled to the formula. In this paper a) We extend the theory so as to include in the formalism also the ordinary linear differential operators with polynomial coefficients $D$ with negative height

\begin{equation*}h=\underset {{n\in N}}{\max } \{m_{n}-n\}<0, \end{equation*}

where $m_{n}$ denotes the degree of $Dx^{n}$. b) We establish a basic classification of the c.p.'s $Q_{m}(x)$ and their orders $m\in M$, as primary or derived, depending, respectively, on whether $\exists n\in \mathbf {N}\colon m_{n}=m$ or such $n$ does not exist; and we state a classification of the indices $n\in \mathbf {N}$, as generic $(m_{n}=n+h)$, singular $(m_{n}<n+h)$, and indefinite $(Dx^{n}\equiv 0)$. Then a formula which gives the set of primary orders is proved. c) In the rather frequent case in which all c.p.'s are primary, we establish, for differential operators $D$ with any height $h$, a recurrency formula which generates bases of the polynomial space and their multiple c.p.'s arising from distinct $x^{n}$, $n\in N$, so that no complementary algorithmic construction is needed; the (primary) c.p.'s so produced are classified as generic or singular, depending on the index $n$. d) We establish the general properties of the multiplicity relations of the primary c.p.'s and of their associated indices. It becomes clear that Ortiz's formula generates, for $h\ge 0$, the generic c.p.'s in terms of the singular and derived c.p.'s, while singular and derived c.p.'s and the multiples of distinct indices are constructed by the algorithm.

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  • 1. Lanczos, C.: Trigonometric interpolation of empirical and analytical functions, J. Math. Phys. 17, (1938), 123-199.
  • 2. Lanczos, C.: ``Applied Analysis'', Prentice Hall, New Jersey, (1956). MR 18:823c
  • 3. Llorente, P., Ortiz, E. L.: Sur quelques aspects algébriques d'une méthode d'approximation de M. Lanczos, Math. Notae, 21, (1968), 17-23. MR 39:7823
  • 4. Ortiz, E.L.: The Tau Method, SIAM J. Numer. Anal. 6, (1969), 480-491. MR 41:2934
  • 5. Ortiz, E.L.: ``Canonical polynomials in Lanczos Tau Method'', in Studies in Numerical Analysis (B.P.K. Scaife, Ed.), Academic Press, London, (1974). MR 57:14478
  • 6. Chaves, T. (Costa, Therezinha S.), Ortiz, E.L.: On the numerical solution of two point boundary valued problems for linear differential equations, Z. Angew Math. Mech., 48, (1968), 415-418. MR 39:7824
  • 7. Ortiz, E.L.: On the numerical solution of non-linear and functional-differential equations with the Tau Method, in Numerical Treatment of Differential Equations in Applications (Ansorg, R., Törnig, W. eds.), 127-139, Berlin, Springer-Verlag (1978). MR 80c:65180
  • 8. Namasivayam, S., Ortiz, E.L.: Best approximation and the numerical solution of partial differential equations with the Tau Method, Portugaliae Mathematica 40, (1981), 97-119. MR 88a:65118
  • 9. Liu, K.M., Ortiz, E.L.: Numerical solution of ordinary and partial functional-differential eigenvalue problems with the Tau Method, Computing 41, (1989), 205-217. MR 90c:65156
  • 10. Liu, K.M., Ortiz E.L.: Numerical solution of eigenvalue problems for partial differential equations with the Tau-lines Method, Comp. and Maths. with Appls. 12B, (1986), 1153-1168. MR 88e:65130
  • 11. Ortiz, E.L., Samara, H.: An operational approach to the Tau Method for the numerical solution of non-linear differential equations, Computing 27, (1981), 15-25. MR 83b:65079
  • 12. Freilich, J.H., Ortiz E.L.: Numerical solution of systems of ordinary differential equations with the Tau Method: An error analysis, Math. Comp., 39, (1982), 467-479. MR 84k:65066
  • 13. Hosseini, M., Abadi Ali, Ortiz, E.L.: A Tau Method based on non-uniform space-time elements for the numerical simulation of solutions, Computers Math. Applic., vol. 22, No 9 (1991), 7-19.
  • 14. Khajah, H.G., Ortiz E.L.: Numerical approximation of solutions of functional equations using the Tau Method, Applied Numerical Mathematics 9 (1992), 461-474. MR 92k:65187
  • 15. El-Daou, M.K., Ortiz, E.L., Samara, H.: A unified approach to the Tau Method and Chebyshev series expansion techniques, Computers Math. Applic., vol. 25, No 3 (1993), 73-82. MR 94b:65106
  • 16. Bunchaft, M.E. Froes: GPC (Gera Polinômios Canônicos) Programa em linguagem Fortran, Projeto Final de Programação, P.U.C. (1994).
  • 17. Bunchaft, M.E. Froes, Costa, Therezinha S.: Software for a method of determination of a basis of primary canonical polynomials in the Tau Method (to be submitted for publication in Appl. Math. Modelling).

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Additional Information

M. E. Froes Bunchaft
Affiliation: Departamento de Ciências da Computação, Instituto de Matemática, Universidade Federal da Bahia, Salvador (Bahia), Brasil

Keywords: Initial value problems, boundary value problems, ordinary differential equations, approximation of functions, Tau Method, Lanczos-Ortiz's canonical polynomials
Received by editor(s): May 23, 1995
Received by editor(s) in revised form: April 12, 1996
Additional Notes: This paper is a modified version of part of the author’s thesis at Pontifícia Universidade Católica, Departamento de Informática, PUC-Rio
Article copyright: © Copyright 1997 American Mathematical Society

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