# Mathematical Aspects of Scientific Software

## Abstract

The goal is to survey the impact of scientific software on mathematics. Three types of impacts are identified and several topics from each are discussed in some depth. First is the impact on the structure of mathematics through its role as the scientific tool for problem solving. Scientific software leads to new assessments of what algorithms are, how well they work, and what a solution really is. Second is the initiation of new mathematical endeavors. Numerical computation is already very widely known, we discuss the important future roles of symbolic and geometric computation. Finally, there are particular mathematical problems that arise from scientific software. Examples discussed include round-off errors and the validation of computations, mapping problems and algorithms into machines, and adaptive methods. There is considerable discussion of the shortcommings of mathematics in providing an adequate model for the scientific analysis of scientific software.

## Keywords

Adaptive Algorithm Scientific Software Mapping Problem Computational Structure Machine Structure## Preview

Unable to display preview. Download preview PDF.

## References

- Aho, A.V., J.E. Hoperoft and J.D. Ullman,
*The Design and Analysis of Computer Algorithms*, Addison Wesley (1974).Google Scholar - Berman, F., The mapping problem in parallel computation, this volume, (1988).Google Scholar
- Davis, P.J., Fidelity in mathematical discourse: Is one plus one really two?,
*Amer. Math. Monthly*,**79**(1972), 252–263.MathSciNetCrossRefMATHGoogle Scholar - Feinerman, R.P. and D.T. Newman,
*Polynomial Approximation*, Wilkins and Wilkins (1974).Google Scholar - Houstis, C.E., E.N. Houstis and J.R. Rice, Partitioning PDE computations: Methods and performance evaluation,
*J. Parallel Computing*,**5**(1987), 141–163.MathSciNetCrossRefMATHGoogle Scholar - Klerer, M. and J. Reinfelds,
*Interactive Systems for Experimental Applied Mathematics*, Academic Press (1968).Google Scholar - Rice, J.R., NAPSS-like systems: problems and prospects,
*Proc. Nat. Computer Conf*. (1973), 43–47.Google Scholar - Rice, J.R., An educational adaptive quadrature algorithm,
*SIGNUM Newsletter*,**8**(April 1973)Google Scholar - Rice, J.R., A metalgorithm for adaptive quadrature,
*J. Assoc. Comp. Mach.*,**22**(1975), 61–82.MathSciNetCrossRefMATHGoogle Scholar - Traub, J.F. and H. Wozniakowski, A
*General Theory of Optimal Algorithms*, Academic Press (1980).Google Scholar - Traub, J.F.,
*Analytic Computational Complexity*, Academic Press (1976).Google Scholar