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  2. System of polynomial equations - Wikipedia

    en.wikipedia.org/wiki/System_of_polynomial_equations

    Thus solving a polynomial system over a number field is reduced to solving another system over the rational numbers. For example, if a system contains 2 {\displaystyle {\sqrt {2}}} , a system over the rational numbers is obtained by adding the equation r 2 22 = 0 and replacing 2 {\displaystyle {\sqrt {2}}} by r 2 in the other equations.

  3. Lill's method - Wikipedia

    en.wikipedia.org/wiki/Lill's_method

    Finding roots −1/2, −1/ √ 2, and 1/ √ 2 of the cubic 4x 3 + 2x 2 − 2x − 1, showing how negative coefficients and extended segments are handled. Each number shown on a colored line is the negative of its slope and hence a real root of the polynomial. To employ the method, a diagram is drawn starting at the origin.

  4. Horner's method - Wikipedia

    en.wikipedia.org/wiki/Horner's_method

    Horner's method can be used to convert between different positional numeral systems – in which case x is the base of the number system, and the a i coefficients are the digits of the base-x representation of a given number – and can also be used if x is a matrix, in which case the gain in computational efficiency is even greater.

  5. Cubic equation - Wikipedia

    en.wikipedia.org/wiki/Cubic_equation

    If only one root, say r 1, is real, then r 2 and r 3 are complex conjugates, which implies that r 2 – r 3 is a purely imaginary number, and thus that (r 2 – r 3) 2 is real and negative. On the other hand, r 1 – r 2 and r 1 – r 3 are complex conjugates, and their product is real and positive. [23]

  6. Numerical algebraic geometry - Wikipedia

    en.wikipedia.org/wiki/Numerical_algebraic_geometry

    Solutions to polynomial systems computed using numerical algebraic geometric methods can be certified, meaning that the approximate solution is "correct".This can be achieved in several ways, either a priori using a certified tracker, [7] [8] or a posteriori by showing that the point is, say, in the basin of convergence for Newton's method.

  7. Triangular decomposition - Wikipedia

    en.wikipedia.org/wiki/Triangular_decomposition

    When the purpose is to describe the solution set of S in the algebraic closure of its coefficient field, those simpler systems are regular chains. If the coefficients of the polynomial systems S 1, ..., S e are real numbers, then the real solutions of S can be obtained by a triangular decomposition into regular semi-algebraic systems. In both ...

  8. Gröbner basis - Wikipedia

    en.wikipedia.org/wiki/Gröbner_basis

    This solving process is only theoretical, because it implies GCD computation and root-finding of polynomials with approximate coefficients, which are not practicable because of numeric instability. Therefore, other methods have been developed to solve polynomial systems through Gröbner bases (see System of polynomial equations for more details).

  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 ...