Minimax approximations to the zeros of Pn( x) and Gauss-Legendre quadrature

Abstract

This paper is concerned with the numerical development of some minimax trigonometric approximations to the positive zeros of the nth Legendre polynomial P n ( x). One of the approximation formulas we derive yields at least 4.2 significant decimal digits of accuracy for any n ⩾ 2, and can be used to furnish initial guesses in an iterative method for the computation of the zeros of P n ( x) to nearly full machine accuracy. This approach avoids some of the computational complexity associated with the selection of appropriate initial guesses for use in a special 5th order scheme previously developed by the first author for the numerical computation of the abscissas required in the n-point Gauss-Legendre quadrature rule.