Implementing Families of Implicit Chebyshev Methods with Exact Coefficients for the Numerical Integration of First- and Second-Order Differential Equations

摘要

A method is presented for the generation of exact numerical coefficients found in two families of implicit Chebyshev methods for the numerical integration of first- and second-order ordinary differential equations. The exact coefficients are represented as algebraic numbers, those numbers that solve polynomial equations with rational coefficients, and are written in nested radical from when possible. In addition to the integration weights, the intra-nodal locations required by the implicit methods are computed. These exact coefficients provide a convenient way to obtain accurate floating-point weights for an integration method of given degree with respect to a specific floating-point precision.