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The 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, [1] in principle, 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 ...
However, for polynomials specifically, the study of root-finding algorithms belongs to computer algebra, since algebraic properties of polynomials are fundamental for the most efficient algorithms. The efficiency and applicability of an algorithm may depend sensitively on the characteristics of the given functions.
MPSolve (Multiprecision Polynomial Solver) is a package for the approximation of the roots of a univariate polynomial. It uses the Aberth method, [1] combined with a careful use of multiprecision. [2] "Mpsolve takes advantage of sparsity, and has special hooks for polynomials that can be evaluated efficiently by straight-line programs" [3]
If x is a simple root of the polynomial , then Laguerre's method converges cubically whenever the initial guess, , is close enough to the root . On the other hand, when x 1 {\displaystyle \ x_{1}\ } is a multiple root convergence is merely linear, with the penalty of calculating values for the polynomial and its first and second derivatives at ...
The Aberth method, or Aberth–Ehrlich method or Ehrlich–Aberth method, named after Oliver Aberth [1] and Louis W. Ehrlich, [2] is a root-finding algorithm developed in 1967 for simultaneous approximation of all the roots of a univariate polynomial.
The Jenkins–Traub algorithm for polynomial zeros is a fast globally convergent iterative polynomial root-finding method published in 1970 by Michael A. Jenkins and Joseph F. Traub. They gave two variants, one for general polynomials with complex coefficients, commonly known as the "CPOLY" algorithm, and a more complicated variant for the ...
Bairstow's approach is to use Newton's method to adjust the coefficients u and v in the quadratic + + until its roots are also roots of the polynomial being solved. The roots of the quadratic may then be determined, and the polynomial may be divided by the quadratic to eliminate those roots.
The roots may be found using brute force: there are a finite number of x, so the polynomial can be evaluated for each element x i. If the polynomial evaluates to zero, then that element is a root. For the trivial case x = 0, only the coefficient λ 0 need be tested for zero. Below, the only concern will be for non-zero x i.