Some applications of computer algebra and interval mathematics

Abstract

This thesis contains some applications of Computer Algebra to unconstrained optimization and some applications of Interval Mathematics to the problem of simultaneously bounding the simple zeros of polynomials. Chapter 1 contains a brief introduction to Computer Algebra and Interval Mathematics, and several of the fundamental results from Interval Mathematics which are used in Chapters 4 and 5. Chapter 2 contains a survey of those features of the symbol manipulation package ALgLIB[Shew-85] which it is necessary to understand in order to use ALgLIB as explained in Chapter 3. Chapter 3 contains a description of Sisser's method [Sis-82a] for unconstrained minimization and several modifications thereof which are implemented using the pseudo-code of Dennis and Schnabel [DenS-83], and ALgLIB, Chapter 3 also contains numerical results corresponding to Sisser's method and its modifications for 7 examples. Chapter 4 contains a new algorithm PRSS for the simultaneous estimation of polynomial zeros and the corresponding interval form IRSS for simultaneously bounding real polynomial zeros. Comparisons are made with some related existing algorithms. Numerical results of the comparisons are also given in this chapter. Chapter 5 contains an application of an idea due to Neumaier [Neu-85] to the problem of constructing interval versions of point iterative procedures for the estimation of simple zeros of analytic functions. In particular, interval versions of some point iterative procedures for the simultaneous estimation of simple (complex) polynomial zeros are described. Finally, numerical results are given to show the efficiency of the new algorithm.