Step by step Tau method—Part I. Piecewise polynomial approximations

Abstract

Lánczos remarked that approximations obtained with the Tau method using a Legendre polynomial perturbation term defined in a finite interval J , give accurate estimations at the end point of J . This fact, coupled with a recursive technique for the generation of Tau approximations described by the author elsewhere (Ortiz, 1969, 1974), is used to construct a step by step formulation of the Tau method in which the error is minimized at the matching point of successive steps. This formulation is applied to the construction of accurate piecewise polynomial approximations with an almost equioscillant error and various degrees of smoothness at the breaking points. A technique based on the mapping of a master element Tau approximation defined over a finite interval of variable length is used in order to simplify the computational process. Numerical examples and an estimation of the step size in relation to the size of the error in the equation are also discussed.