This article is concerned with finding scalar, real or complex roots, approximated as floating point numbers. Finding integer roots or exact algebraic roots are separate problems, whose algorithms have little in common with those discussed here. See Diophantine equation for integer roots.
Finding a root of f(x) − g(x) = 0 is the same as solving the equation f(x) = g(x). Here, x is called the unknown in the equation. Conversely, any equation can take the canonical form f(x) = 0, so equation solving is the same thing as computing (or finding) a root of a function.
Numerical root-finding methods use iteration, producing a sequence of numbers that hopefully converge towards the root as a limit. They require one or more initial guesses of the root as starting values, then each iteration of the algorithm produces a successively more accurate approximation to the root. Since the iteration must be stopped at some point these methods produce an approximation to the root, not an exact solution. Many methods compute subsequent values by evaluating an auxiliary function on the preceding values. The limit is thus a fixed point of the auxiliary function, which is chosen for having the roots of the original equation as fixed points.
The behaviour of root-finding algorithms is studied in numerical analysis. Algorithms perform best when they take advantage of known characteristics of the given function, so different algorithms are used to solve different types of equations. Desirable characteristics include a rapid rate of convergence, ability to separate close roots, robustness against failures of differentiability, and low propagation rate of rounding errors.
- 1 Bracketing methods
- 2 Open methods
- 3 Combinations of methods
- 4 Finding roots of polynomials
- 5 See also
- 6 References
- 7 External links
Bracketing methods calculate the endpoints of a series of successively smaller intervals containing a root. This allows them to provide absolute error bounds on a root's location when the function is known to be continuous. Bracketing methods require two initial points, one on either side of the root.
The simplest root-finding algorithm is the bisection method. It works when f is a continuous function and it requires previous knowledge of two initial guesses, a and b, such that f(a) and f(b) have opposite signs. Although it is reliable, it converges slowly, gaining one bit of accuracy with each iteration.
False position (regula falsi)
The false position method, also called the regula falsi method, is like the secant method. However, instead of retaining the last two points, it makes sure to keep one point on either side of the root. The false position method can be faster than the bisection method and will never diverge like the secant method, but fails to converge under some naive implementations. Ridders' method is a variant on the false-position method that also evaluates the function at the midpoint of the interval, giving faster convergence with similar robustness.
Regula falsi is an interpolation method because it approximates the function with a line between two points. Higher degree polynomials can also be used to approximate the function and its root, while bracketing the root. For example, Muller's method can be easily modified so that rather than always keeping the last 3 points, it tracks the last two points to bracket the root and the best current approximation. Such methods combine good average performance with absolute bounds on the worst-case performance.
Newton's method (and similar derivative-based methods)
Newton's method assumes the function f to have a continuous derivative. Newton's method may not converge if started too far away from a root. However, when it does converge, it is faster than the bisection method, and is usually quadratic. Newton's method is also important because it readily generalizes to higher-dimensional problems. Newton-like methods with higher orders of convergence are the Householder's methods. The first one after Newton's method is Halley's method with cubic order of convergence.
Replacing the derivative in Newton's method with a finite difference, we get the secant method. This method does not require the computation (nor the existence) of a derivative, but the price is slower convergence (the order is approximately 1.6). A generalization of the secant method in higher dimensions is Broyden's method.
The secant method also arises if one approximates the unknown function f by linear interpolation. When quadratic interpolation is used instead, one arrives at Muller's method. It converges faster than the secant method. A particular feature of this method is that the iterates xn may become complex.
Sidi's method allows for interpolation with an arbitrarily high degree polynomial. The higher the degree of the interpolating polynomial, the faster the convergence. Sidi's method allows for convergence with an order arbitrarily close to 2.
The appearance of complex values in interpolation methods can be avoided by interpolating the inverse of f, resulting in the inverse quadratic interpolation method. Again, convergence is asymptotically faster than the secant method, but inverse quadratic interpolation often behaves poorly when the iterates are not close to the root.
Combinations of methods
Brent's method is a combination of the bisection method, the secant method and inverse quadratic interpolation. At every iteration, Brent's method decides which method out of these three is likely to do best, and proceeds by doing a step according to that method. This gives a robust and fast method, which therefore enjoys considerable popularity.
Finding roots of polynomials
Much attention has been given to the special case that the function f is a polynomial, and there are several root-finding algorithms for polynomials. For univariate polynomials of degrees one (linear polynomial) through four (quartic polynomial), there are closed-form solutions which produce all roots. Linear polynomials are easy to solve, but using the quadratic formula to solve quadratic (second degree) equations may require some care to ensure numerical stability. The closed-form solutions for degrees three (cubic polynomial) and four (quartic polynomial) are complicated and require much more care; consequently, numerical methods such as Bairstow's method may be easier to use. Fifth-degree (quintic) and higher-degree polynomials do not have a general algebraic solution according to the Abel–Ruffini theorem (1824, 1799).
Finding one root at a time
This iterative scheme is numerically unstable; the approximation errors accumulate during the successive factorizations, so that the last roots are determined with a polynomial that deviates widely from a factor of the original polynomial. To reduce this error, it is advisable to find the roots in increasing order of magnitude.
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.
The most simple method to find a single root fast is Newton's method. One can use Horner's method twice to efficiently evaluate the value of the polynomial function and its first derivative; this combination is called Birge–Vieta's method. This method provides quadratic convergence for simple roots at the cost of two polynomial evaluations per step.
Closely related to Newton's method are Halley's method and Laguerre's method. Using one additional Horner evaluation, the value of the second derivative is used to obtain methods of cubic convergence order for simple roots. If one starts from a point x close to a root and uses the same complexity of 6 function evaluations, these methods perform two steps with a residual of O(|f(x)|9), compared to three steps of Newton's method with a reduction O(|f(x)|8), giving a slight advantage to these methods.
When applying these methods to polynomials with real coefficients and real starting points, Newton's and Halley's method stay inside the real number line. One has to choose complex starting points to find complex roots. In contrast, the Laguerre method with a square root in its evaluation will leave the real axis of its own accord.
Another class of methods is based on translating the problem of finding polynomial roots to the problem of finding eigenvalues of the companion matrix of the polynomial. In principle, one can use any eigenvalue algorithm to find the roots of the polynomial. However, for efficiency reasons one prefers methods that employ the structure of the matrix, that is, can be implemented in matrix-free form. Among these methods are the power method, whose application to the transpose of the companion matrix is the classical Bernoulli's method to find the root of greatest modulus. The inverse power method with shifts, which finds some smallest root first, is what drives the complex (cpoly) variant of the Jenkins–Traub method and gives it its numerical stability. Additionally, it is insensitive to multiple roots and has fast convergence with order even in the presence of clustered roots. This fast convergence comes with a cost of three Horner evaluations per step, resulting in a residual of O(|f(x)|2+3Φ), which is slower than three steps of Newton's method.
Finding roots in pairs
If the given polynomial only has real coefficients, one may wish to avoid computations with complex numbers. To that effect, one has to find quadratic factors for pairs of conjugate complex roots. The application of the multi-dimensional Newton's method to this task results in Bairstow's method. In the framework of inverse power iterations of the companion matrix, the double shift method of Francis results in the real (rpoly) variant of the Jenkins–Traub method.
Finding all roots at once
The simple Durand–Kerner and the slightly more complicated Aberth methods simultaneously find all of the roots using only simple complex number arithmetic. Accelerated algorithms for multi-point evaluation and interpolation similar to the fast Fourier transform can help speed them up for large degrees of the polynomial. It is advisable to choose an asymmetric, but evenly distributed set of initial points.
Another method with this style is the Dandelin–Gräffe method (sometimes also ascribed to Lobachevsky), which uses polynomial transformations to repeatedly and implicitly square the roots. This greatly magnifies variances in the roots. Applying Viète's formulas, one obtains easy approximations for the modulus of the roots, and with some more effort, for the roots themselves.
Exclusion and enclosure methods
Several fast tests exist that tell if a segment of the real line or a region of the complex plane contains no roots. By bounding the modulus of the roots and recursively subdividing the initial region indicated by these bounds, one can isolate small regions that may contain roots and then apply other methods to locate them exactly.
All these methods require to find the coefficients of shifted and scaled versions of the polynomial. For large degrees, FFT-based accelerated methods become viable.
For real roots, Sturm's theorem and Descartes' rule of signs with its extension in the Budan–Fourier theorem provide guides to locating and separating roots. This plus interval arithmetic combined with Newton's method yields robust and fast algorithms.
The Lehmer–Schur algorithm uses the Schur–Cohn test for circles, Wilf's global bisection algorithm uses a winding number computation for rectangular regions in the complex plane.
The splitting circle method uses FFT-based polynomial transformations to find large-degree factors corresponding to clusters of roots. The precision of the factorization is maximized using a Newton-type iteration. This method is useful for finding the roots of polynomials of high degree to arbitrary precision; it has almost optimal complexity in this setting.
Method based on the Budan–Fourier theorem or Sturm chains
The methods in this class give for polynomials with rational coefficients, and when carried out in rational arithmetic, provably complete enclosures of all real roots by rational intervals. The test of an interval for real roots using Budan's theorem is computationally simple but may yield false positive results. However, these will be reliably detected in the following transformation and refinement of the interval. The test based on Sturm chains is computationally more involved but gives always the exact number of real roots in the interval.
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 "Uspensky's method" not existing and neither does "Descartes' method". This algorithm has been improved by Rouillier and Zimmerman, and the resulting implementation is, to date,[when?] the fastest bisection method. It has the same worst case complexity as the 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, which 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, SageMath, 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.
Finding multiple roots of polynomials
Most root-finding algorithms behave badly when there are multiple roots or very close roots. However, for polynomials whose coefficients are exactly given as integers or rational numbers, there is an efficient method to factorize them into factors that have only simple roots and whose coefficients are also exactly given. This method, called square-free factorization, is based on the multiple roots of a polynomial being the roots of the greatest common divisor of the polynomial and its derivative.
The square-free factorization of a polynomial p is a factorization where each is either 1 or a polynomial without multiple roots, and two different do not have any common root.
An efficient method to compute this factorization is Yun's algorithm.
- nth root algorithm
- Broyden's method
- Multiplicity (mathematics)
- Polynomial greatest common divisor
- Graeffe's method
- Cryptographically secure pseudorandom number generator — a class of functions designed specifically to be unsolvable by root-finding algorithms.
- System of polynomial equations — root-finding algorithms in the multivariate case
- GNU Scientific Library
- Collins, George E.; Alkiviadis G. Akritas (1976). Polynomial Real Root Isolation Using Descartes' Rule of Signs. SYMSAC '76, Proceedings of the third ACM symposium on Symbolic and algebraic computation. Yorktown Heights, NY, USA: ACM. pp. 272–275.
- Fourier, Jean Baptiste Joseph (1820). "Sur l'usage du théorème de Descartes dans la recherche des limites des racines" (PDF). Bulletin des Sciences, par la Société Philomatique de Paris: 156–165.
- Boulier, François (2010). Systèmes polynomiaux : que signifie " résoudre " ? (PDF). Université Lille 1.
- Akritas, Alkiviadis G. (1986). There's no "Uspensky's Method". In: Proceedings of the fifth ACM Symposium on Symbolic and Algebraic Computation (SYMSAC '86, Waterloo, Ontario, Canada), pp. 88–90.
- Akritas, Alkiviadis G. (2008). There is no "Descartes' method" (PDF). In: M.J.Wester and M. Beaudin (Eds), Computer Algebra in Education, AullonaPress, USA, pp. 19–35.
- F. Rouillier and P. Zimmerman, Efficient isolation of polynomial's real roots, Journal of Computational and Applied Mathematics 162 (2004)
- Akritas, Alkiviadis G.; A.W. Strzeboński; P.S. Vigklas (2008). "Improving the performance of the continued fractions method using new bounds of positive roots" (PDF). Nonlinear Analysis: Modelling and Control. 13: 265–279.
- Akritas, Alkiviadis G.; Adam W. Strzeboński (2005). "A Comparative Study of Two Real Root Isolation Methods" (PDF). Nonlinear Analysis: Modelling and Control. 10 (4): 297–304.
- Tsigaridas, P.E.; I.Z. Emiris (2006). "Univariate polynomial real root isolation: Continued fractions revisited". LNCS. 4168: 817–828. doi:10.1007/11841036_72.
- Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). "Chapter 9. Root Finding and Nonlinear Sets of Equations". Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8.
- Animations for Fixed Point Iteration
- GAMS: Roots of polynomials with real coefficients
- Free online polynomial root finder for both real and complex coefficients
- Kehagias, Spyros: RealRoots, a free App for iPhone, iPod Touch and iPad to compare Sturm's method and VAS https://itunes.apple.com/gr/app/realroots/id483609988?mt=8