Real Roots of Polynomials with Real Coefficients

Horner;Newton;Sturm;Budan-Fourier;Descartes;real roots;real polynomials

Hewitt, Christina Maria ; Michael Singer, Committee Chair,Agnes Szanto, Committee Member,Ernie Stitzinger, Committee Member,Hewitt, Christina Maria ; Michael Singer ; Committee Chair ; Agnes Szanto ; Committee Member ; Ernie Stitzinger ; Committee Member

University:North Carolina State University

关键词: Horner;    Newton;    Sturm;    Budan-Fourier;    Descartes;    real roots;    real polynomials;   

Others  :  https://repository.lib.ncsu.edu/bitstream/handle/1840.16/2348/etd.pdf?sequence=1&isAllowed=y

美国|英语

来源: null

【 摘 要 】

Polynomial equations are used throughout mathematics.When solving polynomials many questions arise such as: Are there any real roots? If so, how many?Where are they located?Are these roots positive or negative?Depending on the problem being solved sometimes a rough estimate for the interval where a root is located is enough.There are many methods that can be used to answer these questions.We will focus on Descartes' Rule of Signs, the Budan-Fourier theorem and Sturm's theorem.Descartes' Rule of Signs traditionally is used to determine the possible number of positive real roots of a polynomial.This method can be modified to also find the possible negative roots for a polynomial.The Budan-Fourier theorem takes advantage of the derivatives of a polynomial to determine the number of possible number of roots.While Sturm's theorem uses a blend of derivatives and the Euclidean Algorithm to determine the exact number of roots.In some cases, an interval where a root of the polynomial exists is not enough.Two methods, Horner and Newton's methods, to numerically approximate roots up to a given precision are also discussed.We will also give a real world application that uses Sturm's theorem to solve a problem.