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 with negative height

where denotes the degree of . b) We establish a basic classification of the c.p.'s and their orders , as *primary* or *derived*, depending, respectively, on whether or such does not exist; and we state a classification of the indices , as *generic* , *singular* , and *indefinite* . 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 with any height , a recurrency formula which generates bases of the polynomial space and their multiple c.p.'s arising from distinct , , 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 . 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 , 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