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  2. Polynomial root-finding - Wikipedia

    en.wikipedia.org/wiki/Polynomial_root-finding

    Finding the root of a linear polynomial (degree one) is easy and needs only one division: the general equation + = has solution = /. For quadratic polynomials (degree two), the quadratic formula produces a solution, but its numerical evaluation may require some care for ensuring numerical stability.

  3. Durand–Kerner method - Wikipedia

    en.wikipedia.org/wiki/Durand–Kerner_method

    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]

  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. 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.

  6. Lill's method - Wikipedia

    en.wikipedia.org/wiki/Lill's_method

    Finding roots of 3x 2 + 5x − 2. Lill's method can be used with Thales's theorem to find the real roots of a quadratic polynomial. In this example with 3x 2 + 5x − 2, the polynomial's line segments are first drawn in black, as above. A circle is drawn with the straight line segment joining the start and end points forming a diameter.

  7. 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 ...

  8. 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 ...

  9. Newton's method - Wikipedia

    en.wikipedia.org/wiki/Newton's_method

    An illustration of Newton's method. In numerical analysis, the Newton–Raphson method, also known simply as Newton's method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valued function.

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