Minimax approximations to the zeros of \(P_ n(x)\) and Gauss-Legendre quadrature (Q1899984)

This paper is concerned with the numerical development of some minimax trigonometric approximations to the positive zeros of the \(n\)-th Legendre polynomial \(P_n(x)\). One of the approximation formulas yields at least 4.2 significant decimal digits of accuracy for any \(n \geq 2\), 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.