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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 ...
A quadratic with two real roots, for example, will have exactly two angles that satisfy the above conditions. For complex roots, one must also find a series of similar triangles, but with the vertices of the root path displaced from the polynomial path by a distance equal to the imaginary part of the root. In this case, the root path will not ...
In numerical analysis, the Weierstrass method or Durand–Kerner method, discovered by Karl Weierstrass in 1891 and rediscovered independently by Durand in 1960 and Kerner in 1966, is a root-finding algorithm for solving polynomial equations. [1]
A system of polynomial equations (sometimes simply a polynomial system) is a set of simultaneous equations f 1 = 0, ..., f h = 0 where the f i are polynomials in several variables, say x 1, ..., x n, over some field k.
MPSolve – Software for approximating the roots of a polynomial with arbitrarily high precision; Multiplicity (mathematics) – Number of times an object must be counted for making true a general formula; n th root algorithm; System of polynomial equations – Roots of multiple multivariate polynomials
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 roots , of the quadratic polynomial () = + + satisfy + =, =. The first of these equations can be used to find the minimum (or maximum) of P ; see Quadratic equation § Vieta's formulas .
This polynomial is further reduced to = + + which is shown in blue and yields a zero of −5. The final root of the original polynomial may be found by either using the final zero as an initial guess for Newton's method, or by reducing () and solving the linear equation. As can be seen, the expected roots of −8, −5, −3, 2, 3, and 7 were ...
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