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  2. Multiplicity (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Multiplicity_(mathematics)

    This means that 1 is a root of multiplicity 2, and −4 is a simple root (of multiplicity 1). The multiplicity of a root is the number of occurrences of this root in the complete factorization of the polynomial, by means of the fundamental theorem of algebra.

  3. Geometrical properties of polynomial roots - Wikipedia

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

    For polynomials with real or complex coefficients, it is not possible to express a lower bound of the root separation in terms of the degree and the absolute values of the coefficients only, because a small change on a single coefficient transforms a polynomial with multiple roots into a square-free polynomial with a small root separation, and ...

  4. 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 ...

  5. Polynomial root-finding algorithms - 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 ...

  6. Fundamental theorem of algebra - Wikipedia

    en.wikipedia.org/wiki/Fundamental_theorem_of_algebra

    The n complex numbers , …, are the roots of the polynomial. If a root appears in several factors, it is a multiple root, and the number of its occurrences is, by definition, the multiplicity of the root.

  7. Polynomial - Wikipedia

    en.wikipedia.org/wiki/Polynomial

    If P is a nonzero polynomial, there is a highest power m such that (x − a) m divides P, which is called the multiplicity of a as a root of P. The number of roots of a nonzero polynomial P , counted with their respective multiplicities, cannot exceed the degree of P , [ 25 ] and equals this degree if all complex roots are considered (this is a ...

  8. Root-finding algorithm - Wikipedia

    en.wikipedia.org/wiki/Root-finding_algorithm

    Graeffe's method – Algorithm for finding polynomial roots; Lill's method – Graphical method for the real roots of a polynomial; MPSolve – Software for approximating the roots of a polynomial with arbitrarily high precision; Multiplicity (mathematics) – Number of times an object must be counted for making true a general formula

  9. Eigenvalues and eigenvectors - Wikipedia

    en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors

    The algebraic multiplicity μ A (λ i) of the eigenvalue is its multiplicity as a root of the characteristic polynomial, that is, the largest integer k such that (λ − λ i) k divides evenly that polynomial. [9] [25] [26] Suppose a matrix A has dimension n and d ≤ n distinct eigenvalues.