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  2. Root-finding algorithm - Wikipedia

    en.wikipedia.org/wiki/Root-finding_algorithm

    In numerical analysis, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function f is a number x such that f ( x ) = 0 . As, generally, the zeros of a function cannot be computed exactly nor expressed in closed form , root-finding algorithms provide approximations to zeros.

  3. Aberth method - Wikipedia

    en.wikipedia.org/wiki/Aberth_method

    (Stieltjes also modeled the positions of zeros of polynomials as solutions to electrostatic problems.) Inside the formula of the Aberth method one can find elements of Newton's method and the Durand–Kerner method. Details for an efficient implementation, esp. on the choice of good initial approximations, can be found in Bini (1996). [3]

  4. Polynomial root-finding algorithms - Wikipedia

    en.wikipedia.org/wiki/Polynomial_root-finding...

    For finding all the roots, arguably the most reliable method is the Francis QR algorithm computing the eigenvalues of the companion matrix corresponding to the polynomial, implemented as the standard method [1] in MATLAB. The oldest method of finding all roots is to start by finding a single root.

  5. Durand–Kerner method - Wikipedia

    en.wikipedia.org/wiki/Durand–Kerner_method

    which may increasingly become a concern as the degree of the polynomial increases. If the coefficients are real and the polynomial has odd degree, then it must have at least one real root. To find this, use a real value of p 0 as the initial guess and make q 0 and r 0, etc., complex conjugate pairs.

  6. Zero of a function - Wikipedia

    en.wikipedia.org/wiki/Zero_of_a_function

    A root of a polynomial is a zero of the corresponding polynomial function. [1] The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree , and that the number of roots and the degree are equal when one considers the complex roots (or more generally, the roots in an algebraically ...

  7. Jenkins–Traub algorithm - Wikipedia

    en.wikipedia.org/wiki/Jenkins–Traub_algorithm

    The polynomial has all its zeros lying on two half-circles of different radii. Wilkinson recommends that it is desirable for stable deflation that smaller zeros be computed first. The second-stage shifts are chosen so that the zeros on the smaller half circle are found first. After deflation the polynomial with the zeros on the half circle is ...

  8. Rational root theorem - Wikipedia

    en.wikipedia.org/wiki/Rational_root_theorem

    The theorem is used to find all rational roots of a polynomial, if any. It gives a finite number of possible fractions which can be checked to see if they are roots. If a rational root x = r is found, a linear polynomial ( x – r ) can be factored out of the polynomial using polynomial long division , resulting in a polynomial of lower degree ...

  9. 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 ...