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Most root-finding algorithms can find some real roots, but cannot certify having found all the roots. Methods for finding all complex roots, such as Aberth method can provide the real roots. However, because of the numerical instability of polynomials (see Wilkinson's polynomial ), they may need arbitrary-precision arithmetic for deciding which ...
A quantum algorithm for solving this problem exists. This algorithm is, like the factor-finding algorithm, due to Peter Shor and both are implemented by creating a superposition through using Hadamard gates, followed by implementing as a quantum transform, followed finally by a quantum Fourier transform.
Factorization depends on the base field. For example, the fundamental theorem of algebra, which states that every polynomial with complex coefficients has complex roots, implies that a polynomial with integer coefficients can be factored (with root-finding algorithms) into linear factors over the complex field C.
A general-purpose factoring algorithm, also known as a Category 2, Second Category, or Kraitchik family algorithm, has a running time which depends solely on the size of the integer to be factored. This is the type of algorithm used to factor RSA numbers. Most general-purpose factoring algorithms are based on the congruence of squares method.
The Abel–Ruffini theorem shows that there are no general root formulas in terms of radicals for polynomials of degree five or higher. Using relations between roots. It may occur that one knows some relationship between the roots of a polynomial and its coefficients. Using this knowledge may help factoring the polynomial and finding its roots.
Algorithm: SFF (Square-Free Factorization) Input: A monic polynomial f in F q [x] where q = p m Output: Square-free factorization of f R ← 1 # Make w be the product (without multiplicity) of all factors of f that have # multiplicity not divisible by p c ← gcd(f, f′) w ← f/c # Step 1: Identify all factors in w i ← 1 while w ≠ 1 do y ...
The quantity = is known as the discriminant of the quadratic equation. If the coefficients , , and are real numbers then when >, the equation has two distinct real roots; when =, the equation has one repeated real root; and when <, the equation has no real roots but has two distinct complex roots, which are complex conjugates of each other.
In numerical analysis, Bairstow's method is an efficient algorithm for finding the roots of a real polynomial of arbitrary degree. The algorithm first appeared in the appendix of the 1920 book Applied Aerodynamics by Leonard Bairstow. [non-primary source needed] The algorithm finds the roots in complex conjugate pairs using only real arithmetic.