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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 ...
Bairstow's approach is to use Newton's method to adjust the coefficients u and v in the quadratic + + until its roots are also roots of the polynomial being solved. The roots of the quadratic may then be determined, and the polynomial may be divided by the quadratic to eliminate those roots.
Return to step 1 but use the polynomial and the initial guess . These two steps are repeated until all real zeros are found for the polynomial. If the approximated zeros are not precise enough, the obtained values can be used as initial guesses for Newton's method but using the full polynomial rather than the reduced polynomials.
For univariate polynomials over the rationals (or more generally over a field of characteristic zero), Yun's algorithm exploits this to efficiently factorize the polynomial into square-free factors, that is, factors that are not a multiple of a square, performing a sequence of GCD computations starting with gcd(f(x), f '(x)). To factorize the ...
This can be seen in the following tables, the left of which shows Newton's method applied to the above f(x) = x + x 4/3 and the right of which shows Newton's method applied to f(x) = x + x 2. The quadratic convergence in iteration shown on the right is illustrated by the orders of magnitude in the distance from the iterate to the true root (0,1 ...
In mathematics, particularly computational algebra, Berlekamp's algorithm is a well-known method for factoring polynomials over finite fields (also known as Galois fields). The algorithm consists mainly of matrix reduction and polynomial GCD computations. It was invented by Elwyn Berlekamp in 1967.
The polynomial x 2 + cx + d, where a + b = c and ab = d, can be factorized into (x + a)(x + b).. In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind.
An irreducible polynomial F(x) of degree m over GF(p), where p is prime, is a primitive polynomial if the smallest positive integer n such that F(x) divides x n − 1 is n = p m − 1. A primitive polynomial of degree m has m different roots in GF(p m), which all have order p m − 1, meaning that any of them generates the multiplicative group ...