An odd or even continuous function on a symmetric interval [-a,a] can be evaluated in two different ways, each using only one uniform polynomial approximation. It is of practical importance to know which method of evaluation takes fewer arithmetic operations. This is a special case of a more general problem, which is concerned with the optimal subdivision of the interval of evaluation of a function f into sub-intervals, on each of which f has a uniform polynomial approximation. In the first three chapters a method of computing the number of arithmetic operations for evaluation is developed. Expansions in Chebyshev polynomials are studied, with emphasis on the practical problem of computing coefficients, and then it is shown how the expansion in Chebyshev polynomials may be used to obtain truncation error bounds for the uniform polynomial approximation. From these bounds the required degree for the approximation and the required number of multiplications for evaluation may be easily determined. Tables of computed results are given. In Chapter 4 theoretical results are developed from the theory of Lagrange interpolation and these results are in agreement with the computed results obtained previously. In the problem of evaluation of even and odd functions on [-a,a] , use of the uniform polynomial approximation on [-a,a] is advantageous unless the rate of increase of the derivative of f is rapid. In the general case of evaluation of a continuous function, use of approximations on sub-intervals becomes more advantageous the more rapidly the derivatives of f increase. / Science, Faculty of / Mathematics, Department of / Graduate

by Yuen Wong. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1990. / Chapter Chapter 1 --- Introduction --- p.1 / Chapter Chapter 2 --- Basic Results --- p.5 / Chapter §2.1 --- General Results --- p.5 / Chapter §2.2 --- Williamson's Method --- p.8 / Chapter Chapter 3 --- Algorithms And Subroutines --- p.15 / Chapter §3.1 --- Introduction --- p.15 / Chapter §3.2 --- Increasing Determinant Method --- p.15 / Chapter §3.3 --- Williamson's Method - Direct Computation --- p.21 / Chapter §3.4 --- Williamson's Method - Increasing Determinant --- p.26 / Chapter Chapter 4 --- Comparisons And Recommendations On Algorithms --- p.32 / Chapter §4.1 --- Introduction --- p.32 / Chapter §4.2 --- Comparisons And Recommendations On IMPROV(N) --- p.32 / Chapter §4.3 --- Comparisons And Recommendations On GENHA(N) --- p.34 / Chapter §4.4 --- Comparisons And Recommendations On VTID(N) --- p.35 / Chapter §4.5 --- Summary --- p.37 / Chapter Chapter 5 --- Applications Of Hadamard Matrices --- p.38 / Chapter §5.1 --- Hadamard Matrices And Balanced Incomplete Block Designs' --- p.38 / Chapter §5.2 --- Hadamard Matrices And Optimal Weighing Designs --- p.43 / Chapter Chapter 6 --- Conclusion --- p.51 / References --- p.52 / Appendices --- p.53

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This work is a study of an information theoretic model which is used to develop a complexity measure of an algorithm. The measure is defined to reflect the computational cost and structure of the given algorithm. In this study computational costs are expressed as the execution times of the algorithm, where the algorithm is coded as a program in a machine independent language, and analysed in terms of its representation as a pseudograph. It is shown that this measure aids in deciding which sections of the algorithm should be optimized, segmented or expressed as subprograms. The model proposed is designed to yield a measure which reflects both the program flow and computational cost. Such a measure allows an 'optimal' algorithm to be selected from a set of algorithms, all of which solve the given problem. This selection is made with a more meaningful criterion for decision than simply execution cost. The measure can also be used to further analyse a given algorithm and point to where code optimization techniques should be applied. However it does not yield a method of generating equivalent algorithms. / Science, Faculty of / Computer Science, Department of / Graduate

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Thesis (Ph. D.)--University of Wisconsin--Madison, 1968. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliography.

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Thesis (PhD)--Stellenbosch University, 1975. / ENGLISH ABSTRACT: Some problems of applied mathematics, for instance in the fields of aerodynamics or electron optics, involve certain singular integrals which do not exist classically. The problems can, however, be solved pLovided that such integrals are interpreted as finite-part integrals. Although the concept of a finite-part integral has existed for about fifty years, it was possible to define it rigorously only by means of distribution theory, developed about twenty-five years ago. But, to the best of our knowledge, no quadrature formula for the numerical eva= luation of finite-part integrals ha~ been given in the literature. The main concern of this thesis is the study and discussion of.two kinds of quadrature formulae for evaluating finite-part integrals in= volving an algebraic singularity. Apart from a historical introduction, the first chapter contains some physical examples of finite-part integrals and their definition based on distribution theory. The second chapter treats the most im= portant properties of finite-part integrals; in particular we study their behaviour under the most common rules for ordinary integrals. In chapters three and four we derive a quadrature formula for equispaced stations and one which is optimal in the sense of the Gauss-type quadra= ture. In connection with the latter formula, we also study a new class of orthogonal polynomials. In the fifth and.last chapter we give a derivative-free error bound for the equispaced quadrature formula. The error quantities which are independent of the integrand were computed for the equispaced quadrature formula and are also given. In the case of some examples, we compare the computed error bounds with the actual errors. ~esides this theoretical investigation df finite-part integrals, we also computed - for several orders of the algebraic singularity the coefficients for both of the aforesaid quadrature formulae, in which the number of stations ranges from three up to twenty. In the case of the equispaced quadrature fortnu1a,we give the weights and - for int~ger order of the singularity - the coefficients for a numerical derivative of the integrand function. For the Gauss-type quadrature, we give the stations, the corresponding weights and the coefficients of the orthogonal polynomials. These data are being published in a separate report [18] which also contains detailed instructions on the use of the tables.