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A recursive formulation of collocation in terms of canonical polynomials. (English) Zbl 0797.65056
For boundary value problems of linear differential operators with polynomial coefficients two related but analytically different techniques are discussed: the collocation method and Ortiz’s recursive formulation of the tau method. First the authors show that it is possible to simulate with the tau method collocation approximants for any desired degree in terms of shifted canonical polynomials.
In the linear case a collocation approximants of order $$N$$ by this approach is shown to require $$O(N)$$ arithmetic operations while obtaining the same approximants by the direct approach involves $$O(N^ 3)$$. Several linear and nonlinear numerical examples are given to illustrate the new approach.

##### MSC:
 65L10 Numerical solution of boundary value problems involving ordinary differential equations 65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations 34B05 Linear boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations
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