Computation Of The Real And Complex Roots Of Algebraic And Transcendental Equations

Origin and History of the Problem: The science of algebra arose In its effort to solve equations. Since the main objects in algebra have been the discussion of equalities and the transformation of forms into simpler equivalent ones, that science may well be called the Science of Equations. The solution of an equation containing one unknown quantity consists in the determination of Its value or values, these being called roots. An algebraic equation of degree n has n roots, while transcendental equations have an infinite number of roots. There has existed for years the problem of finding values, as exact as possible, or as close as one wishes of these roots. This problem still has not been satisfactorily solved. It is the object of researchers to continue work to the end of calculating all of the real and complex roots of algebraic and transcendental equations simultaneously by a single method. Thus, Lagrange, at the beginning of his manuscript on The Solution of Numerical Equations (1767), published his first memoirs which were followed by works of Harriot, Ougtred, Pell, and others. Descartes has assisted in these findings by his rule of signs, which though fundamentally important, lacks sufficient accuracy. The greatest analysis initiated by Newton and terminated by Lagrange made a decisive step from Descartes\u27 theory by using the squares of the differences. This method was simple in theory but necessitated tiring and sometimes indefinite conclusions. Fourier attained approximately the same conclusions in his efforts. In 1820, he published a rule which he had formulated and used for several years. Though his rule lacked satisfactory proof, his findings at least aided Sturm with his theories advanced several years later. This theorem concerns only a derivative and the function itself which is somewhat similar to the highest common factor

The location of roots of equations with particular reference to the generalized eigenvalue problem

A survey is presented of algorithms which are in current use for the solution of a single algebraic or transcendental equation in one unknown, together with an appraisal of their practical performance. The first part of the thesis consists of an account of the theoretical basis of a number of iterative methods and an examination of the problems to be overcome in order to achieve a successful computer implementation. In the selection of specific programs for testing, the emphasis has been placed on methods which are suitable for use, in conjunction with determinant evaluation, for the solution of standard eigenvalue problems and generalized problems of the form A(λ)x = O, where the elements of A are linear or non-linear functions of λ. The principal requirements for such purposes are that: 1. the algorithm should not be restricted to polynomial equations 2. derivative evaluation should not be required. Examples of eigenvalue problems arising from engineering applications illustrate the potential difficulties of determining roots. Particular attention is given to the problem of calculating a number of roots in cases where a priori estimates for each root are not available. The discussion is extended to give a brief account of possible approaches to the problem of locating complex roots. Interpolation methods are found to be particularly versatile and can be recommended for their accuracy and efficiency. It is also suggested that such algorithms may often be employed as search strategies in the absence of good initial estimates of the roots. Mention is also made of those features of practical implementation which were found to be particularly useful, together with a list of some outstanding difficulties, associated principally with the automatic computation of several roots of an equation

Localised nonlinear modes in the PT-symmetric double-delta well Gross-Pitaevskii equation

We construct exact localised solutions of the PT-symmetric Gross-Pitaevskii equation with an attractive cubic nonlinearity. The trapping potential has the form of two $\delta$-function wells, where one well loses particles while the other one is fed with atoms at an equal rate. The parameters of the constructed solutions are expressible in terms of the roots of a system of two transcendental algebraic equations. We also furnish a simple analytical treatment of the linear Schr\"odinger equation with the PT-symmetric double-$\delta$ potential.Comment: To appear in Proceedings of the 15 Conference on Pseudo-Hermitian Hamiltonians in Quantum Physics, May 18-23 2015, Palermo, Italy (Springer Proceedings in Physics, 2016

What is a closed-form number?

If a student asks for an antiderivative of exp(x^2), there is a standard reply: the answer is not an elementary function. But if a student asks for a closed-form expression for the real root of x = cos(x), there is no standard reply. We propose a definition of a closed-form expression for a number (as opposed to a *function*) that we hope will become standard. With our definition, the question of whether the root of x = cos(x) has a closed form is, perhaps surprisingly, still open. We show that Schanuel's conjecture in transcendental number theory resolves questions like this, and we also sketch some connections with Tarski's problem of the decidability of the first-order theory of the reals with exponentiation. Many (hopefully accessible) open problems are described.Comment: 11 pages; submitted to Amer. Math. Monthl

The method of finite-product extraction and an application to Wiener-Hopf theory

Copyright @ The Author, 2011. The publisher version of the article can be accessed at the link below.In this work we describe a simple method for finding approximate representations for special functions which are entire transcendental functions that can be represented by infinite products. This method replaces the infinite product by a finite polynomial and Gamma functions. This approximate representation is shown in the case of Bessel functions to be very accurate over a large range of parameter values. These approximate expressions can be useful for finding the roots of a transcendental equation and the Wiener-Hopf factorization of functions involving such Bessel functions.The method is shown to be potentially useful for other transcendental andWiener-Hopf problems, which involve other entire functions that have infinite product representations

Analysis Of A Vector-Borne Diseases Model With A Two-Lag Delay Differential Equation

We are concerned with the stability analysis of equilibrium solutions for a two-lag delay differential equation which models the spread of vector-borne diseases, where the lags are incubation periods in humans and vectors. We show that there are some values of transmission and recovery rates for which the disease dies out and others for which the disease spreads into an endemic. The proofs of the main stability results are based on the linearization method and the analysis of roots of transcendental equations. We then simulate numerical solutions using MATLAB. We observe that the solution could possess chaotic and sometimes unbounded behaviors