Yiu-Kwong Man

A Rational Approach to the Prelle-Singer Algorithm

We present an approach to computing the Darboux polynomials required in the Prelle?Singer algorithm which avoids algebraic extensions of the constant field, and describe a partial implementation in REDUCE in which the leading terms of the polynomials are obtained by a modified version of the method described by Christopher and Collins.

Computing closed form solutions of first order ODEs using the Prelle-Singer procedure

Abstract The Prelle-Singer procedure is an important method for formal solution of first order ODEs. Two different REDUCE implementations (PSODE versions 1 & 2) of this procedure are presented in this paper. The aim is to investigate which implementation is more efficient in solving different types of ODEs (such as exact, linear, separable, linear in coefficients, homogeneous or Bernoulli equations). The test pool is based on Kamkes collection of first order and first degree ODEs. Experimental results, timings and comparison of efficiency and solvability with the present REDUCE differential equation solver (ODESOLVE) and a MACSYMA implementation (ODEFI) of the Prelle-Singer procedure are provided. Discussion of technical difficulties and some illustrative examples are also included.

Fast polynomial dispersion computation and its application to indefinite summation

An algorithm for computing the dispersion of one or two polynomials is described, based on irreducible factorization. It is demonstrated that in practice it is faster than the “conventional” resultant-based algorithm, at least for small problems. It can be applied to algorithms for indefinite summation and closed-form solution of linear difference equations. A brief survey of existing mostly resultant-based dispersion algorithms is given and the complexity of the resultant involved is analysed. The effectiveness of the proposed algorithm applied to indefinite summation is demonstrated by some examples that are not easily summed by the standard facilities in several computer algebra systems.

First integrals of autonomous systems of differential equations and the Prelle-Singer procedure

The Prelle-Singer procedure for determining elementary first integrals of two-dimensional autonomous systems of ordinary differential equations is introduced and how it can be generalized to higher dimensions is discussed. The application of this procedure in several dynamical systems is reported.

A simple algorithm for computing partial fraction expansions with multiple poles

A simple algorithm for computing the partial fraction expansions of proper rational functions with multiple poles is presented. The main idea is to use the Heavisides cover-up technique to determine the numerators of the partial fractions and polynomial divisions to reduce the multiplicities of the poles involved successively, without the use of differentiation.

Solving the water jugs problem by an integer sequence approach

In this article, we present an integer sequence approach to solve the classic water jugs problem. The solution steps can be obtained easily by additions and subtractions only, which is suitable for manual calculation or programming by computer. This approach can be introduced to secondary and undergraduate students, and also to teachers and lecturers involved in teaching mathematical problem solving, recreational mathematics, or elementary number theory.

On computing closed forms for indefinite summations

A decision procedure for finding closed forms for indefinite summation of polynomials, rational functions, quasipolynomials and quasirational functions is presented. It is also extended to deal with some non-hypergeometric sums with rational inputs, which are not summable by means of Gospers algorithm. Discussion of its implementation, analysis of degree bounds and some illustrative examples are included.

On Computing the Measurable Amounts of the Two Jugs Problem

The two jugs problem is a classic problem in mathematics and computer sciences. In this paper, we introduce a new algorithm to solve the general two jugs problem. This algorithm has the advantage that one can easily apply it to determine all measurable amounts by the given jugs and provides us insight on the hidden relationship between the two different approaches to tackle the problem, namely the attempt to fill the smaller jug or the larger jug first. We also discuss how to implement this algorithm in Excel and visualize the generated outputs by means of staircase diagrams. Some basic properties of the measurable amounts by the jugs will also be described, together with some examples.

On Computing the Inverse of Vandermonde Matrix via Synthetic Divisions

A simple method for computing the inverse of Vandermonde matrix via synthetic divisions is introduced. It can be applied to compute each row of the inverse of Vandermonde matrix systematically and effectively. Some illustrative examples are provided.

Solving the Fagnano’s Problem via a Dynamic Geometry Approach

The Fagnano’s problem is a famous historical problem in plane geometry, which involves finding an inscribed triangle with minimal perimeter in a given acute triangle. We discuss how to solve this problem via a dynamic geometry approach and derive a simple formula for finding the perimeter of the orthic triangle, which is the solution of the Fagnano’s problem. Some illustrative examples are included.