Special functions are important in mathematical modeling (the application of mathematical methods to real-world problems in science and engineering). They arise in eigenfunction expansions of solutions to physical problems, and in integral transform methods for solving such problems. In comparison to methods based on discretization and iteration, special functions can reduce the computational time and space required to generate solutions, and even when not directly applicable to a physical problem of interest, they can offer a means for providing evidence of correctness through the consideration of special, simplified cases of the problem. Special functions are important also in statistics, where they occur in the definitions of the normal, chi squared, beta, gamma, Student's, and other underlying probability distribution functions.
The purpose of this project is to provide high-quality computer software for generating numerical values of special functions, also known as higher transcendental functions, in cases where current software libraries lack coverage. Because the bulk of mathematical modeling is still done with Fortran programs, the emphasis is on the development of callable subroutines that are appropriate for inclusion in Fortran libraries. The development of algorithms and accompanying software requires specialized mathematical skills that are not always available in computer software companies. Basic developments almost always originate in a university or government research setting. ACMD is continuing the long tradition of accomplishments in special functions that was begun in the 1950s at NBS.
Dr. Rappoport visited ACMD under a scientific exchange agreement. He has contributed to theoretical and computational developments in special functions and integral transforms. His software for certain special cases of the Macdonald function is in use at the Computing Center of the Russian Academy of Sciences. These cases arise in boundary-value problems associated with wedge-shaped or cone-shaped geometries, and in the Kontorovich-Lebedev integral transform. Since analogous software is unavailable in western countries, the immediate goal of this project is to adapt the Russian software for widespread dissemination and use.
A major accomplishment has been the submittal of a funding proposal to the Civilian Research and Development Foundation, which has been chartered by Congress to support collaborative research with scientists from the Former Soviet Union. The proposal would support the work of Drs. Lozier and Rappoport on Macdonald-function software. They will discuss the planned work at the International Congress on Computational and Applied Mathematics in July, 1996.