Accurate Evaluation of Polynomials in Legendre Basis

Abstract EN

This paper presents a compensated algorithm for accurate evaluation of a polynomial in Legendre basis.

Since the coefficients of the evaluated polynomial are fractions, we propose to store these coefficients in two floating point numbers, such as double-double format, to reduce the effect of the coefficients’ perturbation.

The proposed algorithm is obtained by applying error-free transformation to improve the Clenshaw algorithm.

It can yield a full working precision accuracy for the ill-conditioned polynomial evaluation.

Forward error analysis and numerical experiments illustrate the accuracy and efficiency of the algorithm.