An algorithm for the rapid numerical evaluation of Bessel functions of real orders and arguments. Bremer, James

We describe a method for the rapid numerical evaluation of the Bessel functions of the first and second kinds of nonnegative real orders and positive arguments. Our algorithm makes use of the well-known observation that although the Bessel functions themselves are expensive to represent via piecewise polynomial expansions, the logarithms of certain solutions of Bessel's equation are not. We exploit this observation by numerically precomputing the logarithms of carefully chosen Bessel functions and representing them with piecewise bivariate Chebyshev expansions. Our scheme is able to evaluate Bessel functions of orders between 0 and $1\sep,000\sep,000\sep,000$ at essentially any positive real argument. In that regime, it is competitive with existing methods for the rapid evaluation of Bessel functions and has several advantages over them. First, our approach is quite general and can be readily applied to many other special functions which satisfy second order ordinary differential equations. Second, by calcula...

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