ONLINE POLYNOMIAL ROOT CALCULATOR
This online tool calculates the roots of a polynomial.
I use this tool to calculate the roots to plot Bode diagrams. I hope it helps to you too. Enjoy...
Quadratic equations can be solved by a process known in American English as factoring and in other varieties of English as factorizing, by completing the square, by using the quadratic formula, or by graphing.
Which part of the quadratic formula tells you whether the quadratic equation can be solved by factoring? −b, b2 − 4ac, 2a Use the part of the quadratic formula that you chose above and find its value given the following quadratic equation?
Bisection Method: The simplest root-finding algorithm is the bisection method.
False Position (Regula Falsi): The false position method, also called the regula falsi method makes sure to keep one point on either side of the root.
Newton's Method (and similar derivative-based methods) assumes the function f to have a continuous derivative. The Householder's methods and The Halley's method with cubic order of convergence.
Secant Method replace the derivative in Newton's method with a finite difference. A generalization of the secant method in higher dimensions is Broyden's method.
Interpolation: quadratic interpolation at Muller's method. Sidi's method is an arbitrarily high degree polynomial.
Inverse Interpolation convergence is asymptotically faster than the secant method, but inverse quadratic interpolation. Combinations of Methods
Brent's method is
Finding roots of polynomials
For a univariate polynomial, higher-degree polynomials have no such general solution, according to the Abel–Ruffini theorem (1824, 1799).
Finding one root at a time
The general idea is to find a root of the polynomial and then apply Horner's method to remove the corresponding factor according to the Ruffini rule.
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 using Newton's method. One can use Horner's method twice to efficiently evaluate first derivative, Birge-Vieta's method.
Closely method are Halley's method and Laguerre's method. The Laguerre method will leave the real axis.
finding eigenvaluesof the companion matrix of the polynomial. The classical Bernoulli's method to find the root of greatest modulus. Horner evaluations is slower than Newton's method.
Finding roots in pairs the multi-dimensional Newton's method to this task results in Bairstow's method. 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 method similar to the Fast Fourier Transform.
The Dandelin–Gräffe method (!Lobachevsky) Applying Viete's formulas.
Exclusion and Enclosure methods
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.
Method based on the Budan-Fourier theorem or Sturm chains using Budans theorem is based on Sturm chains is computationally more involved.
The algorithm for isolating the roots, using Descartes' rule of signs and Vincent's theorem, had been originally called modified Uspensky's algorithm 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 Sturm algorithm, Maple root-finding Vincent–Akritas–Strzeboński (VAS) for the Mignotte in Mathematica, Sage, SymPy, Xcas.
Finding multiple roots of polynomials with Square-free factorization. An efficient method to compute this factorization is Yun's algorithm.
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