Quasi-analytical root-finding for non-polynomial functions

A method is presented for the calculation of roots of non-polynomial functions, motivated by the requirement to generate quadrature rules based on non-polynomial orthogonal functions. The approach uses a combination of local Taylor expansions and Sturm’s theorem for roots of a polynomial which together give a means of efficiently generating estimates of zeros which can be polished using Newton’s method. The technique is tested on a number of realistic problems including some chosen to be highly oscillatory and to have large variations in amplitude, both of which features pose particular challenges to root–finding methods.