Accurate Roots Of Polynomials In Floating Point Arithmetic

In floating-point arithmetic, the iterative process for solving the nonlinear equationmay cause rounding error accumulation, which eventually leads to great relativeforward error or even non-convergence of the iterative method. With the rapiddevelopment of computer, high-precision numerical methods get more and moreattention of scientists. This paper bases on error-free transformation in floating pointarithmetic, and the compensated Horner algorithm which accurate evaluation of theexponent polynomial function and its derivatives are applied to iterative methods forsolving polynomial equations that we propose a series of effective, high-precisioniterative methods. The main contents are as follows:We describe Chord iterative, Secant iterative and Newton iterative methods forsolving the problem of finding a real simple root of polynomial equations. We analyzetheir forward error bounds in floating-point arithmetic and give three correspondinghigh-precision algorithms by applied compensated Horner algorithm and its derivativesalgorithm. The designed high-precision iteration algorithms and the original iterationalgorithms are compared in some numerical experiments to prove the effectiveness ofthe high-precision algorithms.Then, we introduce the Muller iteration and Newton iteration for finding a complexsimple root of polynomial equations. We analyze their forward error in complexfloating-point arithmetic and give two corresponding high-precision algorithms whichapplying compensated Horner algorithm in complex floating-point arithmetic. Then, wegive an improved high-precision Newton iteration algorithm which using compensationHorner derivative algorithm in floating-point arithmetic. Some numerical experimentsare given to prove the effectiveness of the high-precision algorithmsFor finding a multiple root of polynomial equations, we introduce two improvedNewton iterative methods that the multiplicity is known or unknown and a fourth-orderiterative method. We also give three corresponding high-precision algorithms by usingcompensated Horner algorithm and compensated Horner derivative algorithm. However,we only analyze the improved Newton iterative which the multiplicity is unknownforward error in floating-point arithmetic. Some numerical experiments are also given.Finally, we sum up the research efforts in this thesis and gives prospects of thefurther research in the fields.