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  2. Durand–Kerner method - Wikipedia

    en.wikipedia.org/wiki/Durand–Kerner_method

    Since every polynomial can be reduced modulo ƒ(X) to a polynomial of degree n − 1 or lower, the space of residue classes can be identified with the space of polynomials of degree bounded by n − 1. A problem specific basis can be taken from Lagrange interpolation as the set of n polynomials

  3. Polynomial root-finding - Wikipedia

    en.wikipedia.org/wiki/Polynomial_root-finding

    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 ...

  4. System of polynomial equations - Wikipedia

    en.wikipedia.org/wiki/System_of_polynomial_equations

    For example, if a system contains , a system over the rational numbers is obtained by adding the equation r 2 22 = 0 and replacing by r 2 in the other equations. In the case of a finite field, the same transformation allows always supposing that the field k has a prime order.

  5. Graeffe's method - Wikipedia

    en.wikipedia.org/wiki/Graeffe's_method

    Graeffe's method works best for polynomials with simple real roots, though it can be adapted for polynomials with complex roots and coefficients, and roots with higher multiplicity. For instance, it has been observed [ 2 ] that for a root x ℓ + 1 = x ℓ + 2 = ⋯ = x ℓ + d {\displaystyle x_{\ell +1}=x_{\ell +2}=\dots =x_{\ell +d}} with ...

  6. Root-finding algorithm - Wikipedia

    en.wikipedia.org/wiki/Root-finding_algorithm

    This consists in using the last computed approximate values of the root for approximating the function by a polynomial of low degree, which takes the same values at these approximate roots. Then the root of the polynomial is computed and used as a new approximate value of the root of the function, and the process is iterated.

  7. Laguerre's method - Wikipedia

    en.wikipedia.org/wiki/Laguerre's_method

    Even if the "drastic set of assumptions" does not work well for some particular polynomial p(x), then p(x) can be transformed into a related polynomial r for which the assumptions are viable; e.g. by first shifting the origin towards a suitable complex number w, giving a second polynomial q(x) = p(x − w), that give distinct roots clearly distinct magnitudes, if necessary (which it will be if ...

  8. Horner's method - Wikipedia

    en.wikipedia.org/wiki/Horner's_method

    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 ...

  9. Bairstow's method - Wikipedia

    en.wikipedia.org/wiki/Bairstow's_method

    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.

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