On figures generated by normalized Tau approximation error curves

Abstract

We show that the error of approximations of the solution y(x) of a linear system of ordinary differential equations, generated by using the Tau method, attain the same order as the best constant E_n(y):

$$O\left( {r^n /2^n v_1^n \left( {n + 1} \right)!} \right) ,$$

where n is the degree of the Tau approximation y_n(x), r is the semi-amplitude of the interval and v₁ is related to the coefficients of the system.

We also show that the normalized error curves of successive Tau approximations of y(x) and of its derivative y′(x) are bounded by the simple curves

$$\pm L^* \sqrt {r^2 - x^2 } , - r \leqslant x \leqslant r ,$$

where L is a constant vector. These curves are reminiscent of curves already discussed by Ortiz and Rivlin in connection with intersections of orthogonal polynomials.