SIAM Journal on Mathematical Analysis
Accurate computation of weights in classical Gauss-Christoffel quadrature rules
Journal of Computational Physics
On the computation of the Gauss-Legendre quadrature formula with a given precision
Journal of Computational and Applied Mathematics - Numerical evaluation of integrals
Some recent results on the zeros of Bessel functions and orthogonal polynomials
Journal of Computational and Applied Mathematics - Special issue on orthogonal polynomials, special functions and their applications
The Zeros of Special Functions from a Fixed Point Method
SIAM Journal on Numerical Analysis
Computing the Zeros and Turning Points of Solutions of Second Order Homogeneous Linear ODEs
SIAM Journal on Numerical Analysis
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Journal of Approximation Theory
Numerical Methods for Special Functions
Numerical Methods for Special Functions
New inequalities from classical Sturm theorems
Journal of Approximation Theory
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A fourth order fixed point method to compute the zeros of solutions of second order homogeneous linear ODEs is obtained from the approximate integration of the Riccati equation associated with the ODE. The method requires the evaluation of the logarithmic derivative of the function and also uses the coefficients of the ODE. An algorithm to compute with certainty all the zeros in an interval is given which provides a fast, reliable, and accurate method of computation. The method is illustrated by the computation of the zeros of Gauss hypergeometric functions (including Jacobi polynomials) and confluent hypergeometric functions (Laguerre polynomials, Hermite polynomials, and Bessel functions included) among others. The examples show that typically 4 or 5 iterations per root are enough to provide more than 100 digits of accuracy, without requiring a priori estimations of the roots.