|
[[Ridders' method]] is a hybrid method that uses the value of function at the midpoint of the interval to perform an exponential interpolation to the root. This gives a fast convergence with a guaranteed convergence of at most twice the number of iterations as the bisection method.
== Roots of polynomials {{anchor|Polynomials}} ==
{{excerpt|Polynomial root-finding algorithms}}
Finding roots of [[polynomial]] is a long-standing problem that has been the object of much research throughout history. A testament to this is that up until the 19th century [[algebra]] meant essentially [[theory of equations|theory of polynomial equations]].
Finding the root of a [[linear polynomial]] (degree one) is easy and needs only one division. For [[quadratic polynomial]]s (degree two), the [[quadratic formula]] produces a solution, but its numerical evaluation may require some care for ensuring [[numerical stability]]. For degrees three and four, there are closed-form solutions in terms of [[radical expression|radicals]], which are generally not convenient for numerical evaluation, as being too complicated and involving the computation of several [[nth root|{{mvar|n}}th roots]] whose computation is not easier than the direct computation of the roots of the polynomial (for example the expression of the real roots of a [[cubic polynomial]] may involve non-real [[cube root]]s). For polynomials of degree five or higher [[Abel–Ruffini theorem]] asserts that there is, in general, no radical expression of the roots.
So, except for very low degrees, root finding of polynomials consists of finding approximations of the roots. By the [[fundamental theorem of algebra]], one knows that a polynomial of degree {{mvar|n}} has at most {{mvar|n}} real or complex roots, and this number is reached for almost all polynomials.
It follows that the problem of root finding for polynomials may be split in three different subproblems;
* Finding one root
* Finding all roots
* Finding roots in a specific region of the [[complex plane]], typically the real roots or the real roots in a given interval (for example, when roots represents a physical quantity, only the real positive ones are interesting).
For finding one root, [[Newton's method]] and other general [[iterative method]]s work generally well.
For finding all the roots, the oldest method is to start by finding a single root. When a root {{mvar|r}} has been found, it can be removed from the polynomial by dividing out the binomial {{math|''x'' – ''r''}}. The resulting polynomial contains the remaining roots, which can be found by iterating on this process. However, except for low degrees, this does not work well because of the [[numerical instability]]: [[Wilkinson's polynomial]] shows that a very small modification of one coefficient may change dramatically not only the value of the roots, but also their nature (real or complex). Also, even with a good approximation, when one evaluates a polynomial at an approximate root, one may get a result that is far to be close to zero. For example, if a polynomial of degree 20 (the degree of Wilkinson's polynomial) has a root close to 10, the derivative of the polynomial at the root may be of the order of <math>10^{20};</math> this implies that an error of <math>10^{-10}</math> on the value of the root may produce a value of the polynomial at the approximate root that is of the order of <math>10^{10}.</math>
For avoiding these problems, methods have been elaborated, which compute all roots simultaneously, to any desired accuracy. Presently the most efficient method is [[Aberth method]]. A [[free software|free]] implementation is available under the name of [[MPSolve]]. This is a reference implementation, which can find routinely the roots of polynomials of degree larger than 1,000, with more than 1,000 significant decimal digits.
The methods for computing all roots may be used for computing real roots. However, it may be difficult to decide whether a root with a small imaginary part is real or not. Moreover, as the number of the real roots is, on the average, the logarithm of the degree, it is a waste of computer resources to compute the non-real roots when one is interested in real roots.
The oldest method for computing the number of real roots, and the number of roots in an interval results from [[Sturm's theorem]], but the methods based on [[Descartes' rule of signs]] and its extensions—[[Budan's theorem|Budan's]] and [[Vincent's theorem]]s—are generally more efficient. For root finding, all proceed by reducing the size of the intervals in which roots are searched until getting intervals containing zero or one root. Then the intervals containing one root may be further reduced for getting a quadratic convergence of [[Newton's method]] to the isolated roots. The main [[computer algebra system]]s ([[Maple (software)|Maple]], [[Mathematica]], [[SageMath]], [[PARI/GP]]) have each a variant of this method as the default algorithm for the real roots of a polynomial.
Another class of methods is based on converting the problem of finding polynomial roots to the problem of finding [[eigenvalues]] of the [[companion matrix]] of the polynomial.<ref>{{cite web | title=Polynomial roots - MATLAB roots | website=MathWorks | date=2021-03-01 | url=https://www.mathworks.com/help/matlab/ref/roots.html | access-date=2021-09-20}}</ref> 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 algorithm]] and gives it its numerical stability. Additionally, it is insensitive to multiple roots and has fast convergence with order <math>1+\varphi\approx 2.6</math> (where <math>\varphi</math> is the [[golden ratio]]) even in the presence of clustered roots. This fast convergence comes with a cost of three polynomial evaluations per step, resulting in a residual of {{math|''O''({{!}}''f''(''x''){{!}}<sup>2+3''φ''</sup>)}}, that is a slower convergence than with three steps of Newton's method.
=== Finding one root ===
The most widely used method for computing a root is [[Newton's method]], which consists of the iterations of the computation of
:<math>x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)},</math>
by starting from a well-chosen value <math>x_0.</math>
If {{mvar|f}} is a polynomial, the computation is faster when using [[Horner's method]] or [[Polynomial evaluation#Evaluation with preprocessing|evaluation with preprocessing]] for computing the polynomial and its derivative in each iteration.
Though the convergence is generally [[quadratic convergence|quadratic]], it may converge much slowly or even not converge at all. In particular, if the polynomial has no real root, and <math>x_0</math> is real, then Newton's method cannot converge. However, if the polynomial has a real root, which is larger than the larger real root of its derivative, then Newton's method converges quadratically to this largest root if <math>x_0</math> is larger than this larger root (there are easy ways for computing an upper bound of the roots, see [[Properties of polynomial roots]]). This is the starting point of [[Horner method]] for computing the roots.
When one root {{mvar|r}} has been found, one may use [[Euclidean division of polynomials|Euclidean division]] for removing the factor {{math|''x'' – ''r''}} from the polynomial. Computing a root of the resulting quotient, and repeating the process provides, in principle, a way for computing all roots. However, 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, one may, for each root that is found, restart Newton's method with the original polynomial, and this approximate root as starting value.
However, there is no warranty that this will allow finding all roots. In fact, the problem of finding the roots of a polynomial from its coefficients is in general highly [[ill-conditioned]]. This is illustrated by
[[Wilkinson's polynomial]]: the roots of this polynomial of degree 20 are the 20 first positive integers; changing the last bit of the 32-bit representation of one of its coefficient (equal to –210) produces a polynomial with only 10 real roots and 10 complex roots with imaginary parts larger than 0.6.
Closely related to Newton's method are [[Halley's method]] and [[Laguerre's method]]. Both use the polynomial and its two first derivations for an iterative process that has a [[cubic convergence]]. Combining two consecutive steps of these methods into a single test, one gets a [[rate of convergence]] of 9, at the cost of 6 polynomial evaluations (with Horner rule). On the other hand, combining three steps of Newtons method gives a rate of convergence of 8 at the cost of the same number of polynomial evaluation. This gives a slight advantage to these methods (less clear for Laguerre's method, as a square root has to be computed at each step).
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.
===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 [[multidimensional Newton's method]] to this task results in [[Bairstow's method]].
The real variant of [[Jenkins–Traub algorithm]] is an improvement of this method.
===Finding all roots at once===
The simple [[Durand–Kerner method|Durand–Kerner]] and the slightly more complicated [[Aberth method]] 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. The implementation of this method in the [[free software]] [[MPSolve]] is a reference for its efficiency and its accuracy.
Another method with this style is the [[Graeffe's method|Dandelin–Gräffe method]] (sometimes also ascribed to [[Nikolai Ivanovich Lobachevsky|Lobachevsky]]), which uses [[polynomial transformations]] to repeatedly and implicitly square the roots. This greatly magnifies variances in the roots. Applying [[Vieta's formulas|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 involve finding the coefficients of shifted and scaled versions of the polynomial. For large degrees, [[fast Fourier transform|FFT]]-based accelerated methods become viable.
For real roots, see next sections.
The [[Lehmer–Schur algorithm]] uses the [[Lehmer–Schur algorithm#Schur–Cohn test|Schur–Cohn test]] for circles; a variant, [[Lehmer–Schur algorithm#Wilf's global bisection algorithm|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.{{citation needed|date=November 2018}}
=== Real-root isolation ===
{{Main|Real-root isolation}}
Finding the real roots of a polynomial with real coefficients is a problem that has received much attention since the beginning of 19th century, and is still an active ___domain of research. Most root-finding algorithms can find some real roots, but cannot certify having found all the roots. Methods for finding all complex roots, such as [[Aberth method]] can provide the real roots. However, because of the numerical instability of polynomials (see [[Wilkinson's polynomial]]), they may need [[arbitrary-precision arithmetic]] for deciding which roots are real. Moreover, they compute all complex roots when only few are real.
It follows that the standard way of computing real roots is to compute first disjoint intervals, called ''isolating intervals'', such that each one contains exactly one real root, and together they contain all the roots. This computation is called ''real-root isolation''. Having isolating interval, one may use fast numerical methods, such as [[Newton's method]] for improving the precision of the result.
The oldest complete algorithm for real-root isolation results from [[Sturm's theorem]]. However, it appears to be much less efficient than the methods based on [[Descartes' rule of signs]] and [[Vincent's theorem]]. These methods divide into two main classes, one using [[continued fraction]]s and the other using bisection. Both method have been dramatically improved since the beginning of 21st century. With these improvements they reach a [[computational complexity]] that is similar to that of the best algorithms for computing all the roots (even when all roots are real).
These algorithms have been implemented and are available in [[Mathematica]] (continued fraction method) and [[Maple (software)|Maple]] (bisection method). Both implementations can routinely find the real roots of polynomials of degree higher than 1,000.
=== Finding multiple roots of polynomials ===
{{Main|Square-free factorization}}
Most root-finding algorithms behave badly when there are [[Multiplicity (mathematics)|multiple roots]] or very close roots. However, for polynomials whose coefficients are exactly given as [[integer]]s or [[rational number]]s, 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 [[polynomial greatest common divisor|greatest common divisor]] of the polynomial and its derivative.
The square-free factorization of a polynomial ''p'' is a factorization <math>p=p_1p_2^2\cdots p_k^k </math> where each <math>p_i</math> is either 1 or a polynomial without multiple roots, and two different <math>p_i</math> do not have any common root.
An efficient method to compute this factorization is [[Square-free factorization#Yun's algorithm|Yun's algorithm]].
== See also ==
| 