Theory of Polynomial Real Root-Finding

Sturm’s theorem was a significant breakthrough in early development of numerical methods for finding real roots of polynomials. Introduced in 1829 by Jacques Sturm, it provided the first robust, systematic algorithm to determine the number and location of distinct real roots of a polynomial within a given interval. While modern, more efficient methods using Descartes’ rule of signs have been developed, Sturm’s theorem remains useful in algorithms such as the b2studios algorithm for polynomials with only real distinct roots....

Finding the real roots of polynomials is a fundamental problem in mathematics and has applications across various fields, including engineering, physics, and computer science. While seemingly straightforward, numerical methods for polynomial root finding can be surprisingly subtle and prone to various pitfalls. This article explores some of these common caveats and discusses strategies for mitigating their effects. Instability in Numerical Methods Several numerical methods exist for finding polynomial roots, such as the bisection method, the Newton-Raphson method, the Durand-Kerner method, and methods based on companion matrices and eigenvalue computations....

Descartes’ Rule of Signs: An Overview Descartes’ Rule of Signs is a theorem that estimates the possible number of positive and negative real roots of a polynomial based on its coefficients’ sign changes. It provides an upper bound on real roots but does not locate them. The Rule For a polynomial $ P(x) = a_nx^n + \dots + a_0 $: Positive Real Roots: Count sign changes in $ P(x) $’s coefficients (ignoring zeroes)....