Desigualdades que garantem a convergência do método de Newton-Raphson para os zeros do polinômio ultraesférico no caso principal

Abstract

The n points of Gauss-Gegenbauer quadrature are the zeros of the ultraspherical polynomial of degree n. The traditional and most-widely used eigensystem method computes the points as the eigenvalues of a symmetric tridiagonal matrix whose eigenvectors can be used to compute the corresponding weights. Alternatively the Newton-Raphson method can provide such points and weights using some properties of ultraspherical polynomials. In this work we show that if certain initial guesses are used, the Newton-Raphson method is in fact convergent for zeros of ultraspherical polynomials in the case 0 << 1. As a result weobtain some inequalities for zeros of ultraspherical polynomials. In addition, we compare the accuracy and computation time of both methods: eigensystem and Newton-Raphson.