Root-Finding-Algorithms

The behaviour of general root-finding algorithms is studied in numerical analysis. How-ever, for polynomials, root-finding study belongs generally to computer algebra, sincealgebraic properties of polynomials are fundamental for the most efficient algorithms.The efficiency of an algorithm may depend dramatically on the characteristics of thegiven functions. For example, many algorithms use the derivative of the input function,while others work on every continuous function. In general, numerical algorithms are notguaranteed to find all the roots of a function, so failing to find a root does not prove thatthere is no root. However, for polynomials, there are specific algorithms that use algebraicproperties for certifying that no root is missed, and locating the roots in separate intervals(or disks for complex roots) that are small enough to ensure the convergence of numericalmethods (typically Newton’s method) to the unique root so located.