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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. Zero of a function - Wikipedia

    en.wikipedia.org/wiki/Zero_of_a_function

    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 closed extension) counted with their multiplicities. [3]

  4. Bézout's theorem - Wikipedia

    en.wikipedia.org/wiki/Bézout's_theorem

    The concept of multiplicity is fundamental for Bézout's theorem, as it allows having an equality instead of a much weaker inequality. Intuitively, the multiplicity of a common zero of several polynomials is the number of zeros into which the common zero can split when the coefficients are slightly changed.

  5. Geometrical properties of polynomial roots - Wikipedia

    en.wikipedia.org/wiki/Geometrical_properties_of...

    If the coefficients a i of a random polynomial are independently and identically distributed with a mean of zero, most complex roots are on the unit circle or close to it. In particular, the real roots are mostly located near ±1, and, moreover, their expected number is, for a large degree, less than the natural logarithm of the degree.

  6. Newton's method - Wikipedia

    en.wikipedia.org/wiki/Newton's_method

    This is equivalent to finding the zeroes of a single vector-valued function :. In the formulation given above, the scalars x n are replaced by vectors x n and instead of dividing the function f ( x n ) by its derivative f ′ ( x n ) one instead has to left multiply the function F ( x n ) by the inverse of its k × k Jacobian matrix J F ( x n ) .

  7. Descartes' rule of signs - Wikipedia

    en.wikipedia.org/wiki/Descartes'_rule_of_signs

    To find the number of negative roots, change the signs of the coefficients of the terms with odd exponents, i.e., apply Descartes' rule of signs to the polynomial = + + This polynomial has two sign changes, as the sequence of signs is (−, +, +, −) , meaning that this second polynomial has two or zero positive roots; thus the original ...

  8. Zeros and poles - Wikipedia

    en.wikipedia.org/wiki/Zeros_and_poles

    Because of the order of zeros and poles being defined as a non-negative number n and the symmetry between them, it is often useful to consider a pole of order n as a zero of order –n and a zero of order n as a pole of order –n. In this case a point that is neither a pole nor a zero is viewed as a pole (or zero) of order 0.

  9. Quartic equation - Wikipedia

    en.wikipedia.org/wiki/Quartic_equation

    Multiplicity-3 (M3): when the general quartic equation can be expressed as () =, where and are two different real numbers. This is the only case that can never be reduced to a biquadratic equation. This is the only case that can never be reduced to a biquadratic equation.