Truncation Errors in Two Che by she v Series Approximations

However, in attempting to find a suitable polynomial approximation to a general function fix), the integral occurring in equation (1.3) cannot be evaluated explicitly, and recourse has to be made to approximate methods for evaluating a„ . The most widely used method is the "curve-fitting" method described by Lanczos [1], and, in greater detail, by Clenshaw and Curtis [2]. There are two variations of the method which we shall call the "practical" and "classical" methods, respectively. Suppose we wish to approximate to fix) by a polynomial of degree N. In the "practical" method, we construct a polynomial SLv(x) by collocation with fix) at the (N + 1) points x< = cos iiri/N), i = 0(1 )N, which are the zeros of the polynomial [TN+iix) — 7V_i(x)]. In the "classical" method, we construct a polynomial <fv(x) by collocation with/(x) at the (AT + 1) points