Polynomial root-finding

Finding the roots of polynomials is a long-standing problem that has been extensively studied throughout the history and substantially influenced the development of mathematics. It involves determining either a numerical approximation or a closed-form expression of the roots of a univariate polynomial, i.e., determining approximate or closed form solutions of in the equation
where are either real or complex numbers.
Efforts to understand and solve polynomial equations led to the development of important mathematical concepts, including irrational and complex numbers, as well as foundational structures in modern algebra such as fields, rings, and groups.
Despite being historically important, finding the roots of higher degree polynomials no longer play a central role in mathematics and computational mathematics, with one major exception in computer algebra.

Overview

Closed-form formulas

Closed-form formulas for polynomial roots exist only when the degree of the polynomial is less than 5. The quadratic formula has been known since antiquity, and the cubic and quartic formulas were discovered in full generality during the 16th century.
When the degree of polynomial is at least 5, a closed-form expression for the roots by the polynomial coefficients does not exist in general, if we only uses additions, subtractions, multiplications, divisions, and radicals in the formula. This is due to the celebrated Abel-Ruffini theorem. On the other hand, the fundamental theorem of algebra shows that all nonconstant polynomials have at least one root. Therefore, root-finding algorithms consists of finding numerical solutions in most cases.

Numerical algorithms

Root-finding algorithms can be broadly categorized according to the goal of the computation. Some methods aim to find a single root, while others are designed to find all complex roots at once. In certain cases, the objective may be to find roots within a specific region of the complex plane. It is often desirable and even necessary to select algorithms specific to the computational task due to efficiency and accuracy reasons. See Root Finding Methods for a summary of the existing methods available in each case.

History

Closed-form formulas

The root-finding problem of polynomials was first recognized by the Sumerians and then the Babylonians. Since then, the search for closed-form formulas for polynomial equations lasted for thousands of years.

The quadratics

The Babylonions and Egyptians were able to solve specific quadratic equations in the second millennium BCE, and their solutions essentially correspond to the quadratic formula.
However, it took 2 millennia of effort to state the quadratic formula in an explicit form similar to the modern formulation, provided by Indian Mathematician Brahmagupta in his book Brāhmasphuṭasiddhānta 625 CE. The full recognition of the quadratic formula requires the introduction of complex numbers, which took another a millennia.

The cubics and the quartics

The first breakthrough in a closed-form formula of polynomials with degree higher than two took place in Italy. In the early 16th century, the Italian mathematician Scipione del Ferro found a closed-form formula for cubic equations of the form, where are nonnegative numbers. Later, Niccolò Tartaglia also discovered methods to solve such cubic equations, and Gerolamo Cardano summarized and published their work in his book Ars Magna in 1545.
Meanwhile, Cardano's student Lodovico Ferrari discovered the closed-form formula of the quartic equations in 1540. His solution is based on the closed-form formula of the cubic equations, thus had to wait until the cubic formula to be published.
In Ars Magna, Cardano noticed that Tartaglia's method sometimes involves extracting the square root of a negative number. In fact, this could happen even if the roots are real themselves. Later, the Italian mathematician Rafael Bombelli investigated further into these mathematical objects by giving an explicit arithmetic rules in his book Algebra published in 1569. These mathematical objects are now known as the complex numbers, which are foundational in mathematics, physics, and engineering.

Insolvability of the quintics

Since the discovery of cubic and quartic formulas, solving quintic equations in a closed form had been a major problem in algebra. The French lawyer Viete, who first formulated the root formula for cubics in modern language and applied trigonometric methods to root-solving, believed that his methods generalize to a closed-form formula in radicals for polynomial with arbitrary degree. Descartes also held the same opinion.
However, Lagrange noticed the flaws in these arguments in his 1771 paper Reflections on the Algebraic Theory of Equations, where he analyzed why the methods used to solve the cubics and quartics would not work to solve the quintics. His argument involves studying the permutation of the roots of polynomial equations. Nevertheless, Lagrange still believed that closed-form formula in radicals of the quintics exist. Gauss seems to have been the first prominent mathematician who suspected the insolvability of the quintics, stated in his 1799 doctoral dissertation.
The first serious attempt at proving the insolvability of the quintic was given by the Italian mathematician Paolo Ruffini. He published six versions of his proof between 1799 and 1813, yet his proof was not widely accepted as the writing was long and difficult to understand, and turned out to have a gap.
The first rigorous and accepted proof of the insolvability of the quintic was famously given by Niels Henrik Abel in 1824, which made essential use of the Galois theory of field extensions. In the paper, Abel proved that polynomials with degree more than 4 do not have a closed-form root formula by radicals in general. This puts an end in the search of closed form formulas of the roots of polynomials by radicals of the polynomial coefficients.

General solution using combinatorics

In 2025, Norman Wildberger and Dean Rubine introduced a general solution for arbitrary degree, involving a formal power series. The equation has a solution
This is a generalization of a solution for the quadratics using Catalan numbers, for which it reduces to. For the quintic, this is closely related to the Eisenstein series.

Numerical methods

Since finding a closed-form formula of higher degree polynomials is significantly harder than that of quadratic equations, the earliest attempts to solve cubic equations are either geometrical or numerical. Also, for practical purposes, numerical solutions are necessary.

Iterative methods

The earliest iterative approximation methods of root-finding were developed to compute square roots. In Heron of Alexandria's book Metrica, approximate values of square roots were computed by iteratively improving an initial estimate. Jamshīd al-Kāshī presented a generalized version of the method to compute th roots. A similar method was also found in Henry Briggs's publication Trigonometria Britannica in 1633. Franciscus Vieta also developed an approximation method that is almost identical to Newton's method.
Newton further generalized the method to compute the roots of arbitrary polynomials in De analysi per aequationes numero terminorum infinitas, now known as Newton's method. In 1690, Joseph Raphson published a refinement of Newton's method, presenting it in a form that more closely aligned with the modern version used today.
In 1879, the English mathematician Arthur Cayley noticed the difficulties in generalizing Newton's method to complex roots of polynomials with degree greater than 2 and complex initial values in his paper The Newton–Fourier imaginary problem. This opened the way to the study of the theory of iterations of rational functions.

Real-root isolation methods

A class of methods of finding numerical value of real roots is based on real-root isolation. The first example of such method is given by René Descartes in 1637. It counts the roots of a polynomial by examining sign changes in its coefficients. In 1807, the French mathematician François Budan de Boislaurent generalized Descarte's result into Budan's theorem which counts the real roots in a half-open interval (a, b]. However, both methods are not suitable as an effective algorithm.
The first complete real-root isolation algorithm was given by Jacques Charles François Sturm in 1829, known as the Sturm's theorem.
In 1836, Alexandre Joseph Hidulphe Vincent proposed a method for isolating real roots of polynomials using continued fractions, a result now known as Vincent's theorem. The work was largely forgotten until it was rediscovered over a century later by J. V. Uspensky, who included it in his 1948 textbook Theory of Equations. The theorem was subsequently brought to wider academic attention by the American mathematician Alkiviadis G. Akritas, who recognized its significance while studying Uspensky's account. The first implimentation of real-root isolation method by modern computer is given by G.E. Collins and Alkiviadis G. Akritas in 1976, where they proved an effective version of Vincent's theorem. Variants of the algorithm were subsequently studied.

Mechanical methods

Before electronic computers were invented, people used mechanical computers to automate the polynomial-root solving problems. In 1758, the Hungarian scientist J.A. De Segner proposed a design of root-solving machine in his paper, which operates by drawing the graph of the polynomial on a plane and find the roots as the intersections of the graph with x-axis. In 1770, the English mathematician Jack Rowning investigated the possibility of drawing the graph of polynomials via local motions.
In 1845, the English mathematician Francis Bashforth proposed to use trigonometric methods to simplify the root finding problem. Given a polynomial, substitute. Since can be written as a linear combination of , the polynomial can be reformulated into the following form
Such curves can be drawn by a harmonic analyzer. The first harmonic analyzer was built by Lord Kelvin in 1872, while Bashforth envisioned such a machine in his paper 27 years earlier.
The Spanish engineer and mathematician Leonardo Torres Quevedo built several machines for solving real and complex roots of polynomials between 1893-1900. His machine employs a logarithmic algorithm, and has a mechanical component called the Endless principle to the value of from with high accuracy. This allow him to achieve high accuracy in polynomial root-finding: the machine computes the roots of deg 8 polynomials with an accuracy of.