Root-finding Algorithm - Finding Roots of Polynomials

Finding Roots of Polynomials

Much attention has been given to the special case that the function f is a polynomial; there exist root-finding algorithms exploiting the polynomial nature of f. For a univariate polynomial of degree less than five, there are closed form solutions such as the quadratic formula which produce all roots. However, even this degree-two solution should be used with care to ensure numerical stability. Even more care must be taken with the degree-three and degree-four solutions because of their complexity. Higher-degree polynomials have no such general solution, according to the Abel–Ruffini theorem (1824, 1799).

Birge-Vieta's method combines Horner's method of polynomial evaluation with Newton-Raphson to provide a computational speed-up.

For real roots, Sturm's theorem and Descartes' rule of signs provide guides to locating and separating roots. This plus interval arithmetic combined with Newton's method yields robust and fast algorithms.

The algorithm for isolating the roots, using Descartes' rule of signs and Vincent's theorem, had been originally called modified Uspensky's algorithm by its inventors Collins and Akritas. After going through names like "Collins-Akritas method" and "Descartes' method" (too confusing if ones considers Fourier's article), it was finally François Boulier, of Lille University, who gave it the name Vincent-Collins-Akritas (VCA) method, p. 24, based on the fact that "Uspensky's method" does not exist and neither does "Descartes' method". This algorithm has been improved by Rouillier and Zimmerman, and the resulting implementation is, to the date, the fastest bisection method. It has the same worst case complexity as Sturm algorithm, but is almost always much faster. It is the default algorithm of Maple root-finding function fsolve. Another method based on Vincent's theorem is the Vincent–Akritas–Strzeboński (VAS) method; it has been shown that the VAS (continued fractions) method is faster than the fastest implementation of the VCA (bisection) method, a fact that was independently confirmed elsewhere; more precisely, for the Mignotte polynomials of high degree VAS is about 50,000 times faster than the fastest implementation of VCA. VAS is the default algorithm for root isolation in Mathematica, Sage, SymPy, Xcas. See Budan's theorem for a description of the historical background of these methods. For a comparison between Sturm's method and VAS use the functions realroot(poly) and time(realroot(poly)) of Xcas. By default, to isolate the real roots of poly realroot uses the VAS method; to use Sturm's method write realroot(sturm, poly). See also the External links for a pointer to an iPhone/iPod/iPad application that does the same thing.

One possibility is to form the companion matrix of the polynomial. Since the eigenvalues of this matrix coincide with the roots of the polynomial, one can use any eigenvalue algorithm to find the roots of the polynomial. For instance the classical Bernoulli's method to find the root greater in modulus, if it exists, turns out to be the power method applied to the companion matrix. The inverse power method, which finds some smallest root first, is what drives the Jenkins–Traub method and gives it its numerical stability and fast convergence even in the presence of multiple or clustered roots.

Laguerre's method, as well as Halley's method, use second order derivatives and complex arithmetics, including the complex square root, to exhibit cubic convergence for simple roots, dominating the quadratic convergence displayed by Newton's method.

Bairstow's method uses Newton's method to find quadratic factors of a polynomial with real coefficients. It can determine both real and complex roots of a real polynomial using only real arithmetic.

The simple Durand–Kerner and the slightly more complicated Aberth method simultaneously finds all the roots using only simple complex number arithmetic.

The splitting circle method is useful for finding the roots of polynomials of high degree to arbitrary precision; it has almost optimal complexity in this setting. Another method with this style is the Dandelin–Gräffe method (actually due to Lobachevsky) which factors the polynomial.

Wilkinson's polynomial illustrates that high precision may be necessary when computing the roots of a polynomial given its coefficients: the problem of finding the roots from the coefficients is in general ill-conditioned.