Symbolic Computation Group

Given a bivariate function f(x,y), we consider the problem of generating, in an automated manner, a routine for efficient numerical evaluation at any point in a specified rectangular region of the x-y plane. The goals for the generated routine are: accuracy (for a specified fixed precision), effciency, and the generated routine must be "purely numerical" in the sense that it can be translated into a language such as C and can be compiled for inclusion in a numerical library. For example, we are able to generate numerical evaluation routines for the various Bessel functions (with the order treated as a real-valued variable), and other bivariate functions, with significantly increased speed of evaluation compared with current implementations. We exploit the capabilities of a computer algebra system to achieve the desired level of automation. A fundamental tool in our method is the natural tensor product series developed in a doctoral thesis by Frederick W. Chapman in 2003. Using this technique, f(x,y) is approximated by an interpolation series such that each term in the series is a tensor product c_i g_i(x) h_i(y). The efficiency of approximation achieved by this method derives from the fact that the univariate basis functions are cross-sections of the original bivariate function. The bivariate approximation problem is thereby reduced to a sequence of univariate approximation problems which can be handled by various well-known techniques. Assuming that the univariate basis functions are analytic with isolated singularities, we apply singularity-handling techniques to ensure the e?ciency of the univariate approximations.