Lanczos $\tau$-method optimal algorithm in APS for approximating the mathematical functions

Abstract

A new procedure is constructed by means of APS in APLAN language. The procedure solves the initial-value problem for linear differential equations of order $k$ with polynomial coefficients and regular singularity in the initialization point in the interval $[a,b]$ and computes the algebraic polynomial $y_n$ of given order $n$. A new algorithm of Lanczos $\tau$-method is built for this procedure, the solution existence $y_n$ of the initial-value problem proved on this algorithm and also is proved the optimality by precision of order $k$ derivative of the initial-value problem solution.

10 pages. Based on a talk given at the Workshop on Symbolic Calculations and Exact Methods in Mathematical Physics, Kiev, June 20-26, 2005