Error Control with Polynomial Approximations
J. L. Schonfelder, Mehdi Dastoori Razaz · IMA Journal of Numerical Analysis · 1981
A detailed analysis is given of the accumulation of errors which may occur in evaluating a polynomial approximation to a given function. Both backward recursion using untransformed Chebyshev expansions and the much faster nested multiplication using the transformed simple polynomial form are treated. Two types of arithmetic are dealt with covering most current machines. For the case of polynomials with coefficients of the same sign or strictly alternating signs, a situation which is of considerable practical importance in polynomial approximation of mathematical functions, we show that controlling relative error requires the ratios |y|max/|y|min and |y′|max/|y|min to be kept small. Experimental verification of these effects is given based on expansions available in the literature or produced by the authors for the Bessel function I0(x).