Towards a symbolic system for floating-point error analysis

Combining symbolic algebra with numerical computation is a useful way of contributing to the solution of many scientific and engineering problems. Symbolic algebra systems have been used to produce FORTRAN expressions since the early 1970's. One of the difficulties, in practice, is producing efficient stable code from the large expressions generated by symbolic algebra systems.;We have merged ideas from both the numerical and symbolic domains to develop a prototype for a symbolic error-analysis system. The differential error-propagation model has been extended to handle non-elementary operations. This provides the potential of analyzing problems at different levels of abstraction. We have also made progress towards combining expression reformulation with error analysis. The objective of our error-analysis system is to identify regions of numerical instability and suggest possible reformulations to improve the stability in these regions. We present an approach to analysis and reformulation based on a bottom-up breadth-first traversal of the computational graph and sparse matrix techniques. For the initial development, testing of these error analysis techniques has been restricted to a particular class of problems: rational polynomials. This class of functions is rich enough to provide examples of the main issues we wish to study without being overwhelming in its complexity.;We present the results of research undertaken towards the development of a symbolic system for floating-point error analysis. This research is based on a differential error-propagation model and Bauer's computational graphs. The differential error-propagation model is a first-order Taylor series expansion of the accumulated rounding error with respect to the local rounding errors. Previous work by Bauer; Miller; Tienari, Linnainmaa and Ukkonen; Larson and Sameh; Hulshof and van Hulzen; and Stoutemyer, among others, has been considered in the development of this system.