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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
DOI: https://doi.org/10.1090/S0025-5718-97-00816-8
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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Additional Information

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

DOI: https://doi.org/10.1090/S0025-5718-97-00816-8
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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