Search results
Results from the WOW.Com Content Network
Distinct-degree factorization algorithm tests every d not greater than half the degree of the input polynomial. Rabin's algorithm takes advantage that the factors are not needed for considering fewer d. Otherwise, it is similar to distinct-degree factorization algorithm. It is based on the following fact.
Modern algorithms and computers can quickly factor univariate polynomials of degree more than 1000 having coefficients with thousands of digits. [3] For this purpose, even for factoring over the rational numbers and number fields , a fundamental step is a factorization of a polynomial over a finite field .
The Cantor–Zassenhaus algorithm takes as input a square-free polynomial (i.e. one with no repeated factors) of degree n with coefficients in a finite field whose irreducible polynomial factors are all of equal degree (algorithms exist for efficiently factoring arbitrary polynomials into a product of polynomials satisfying these conditions, for instance, () / ((), ′ ()) is a squarefree ...
Similarly, the product of the factors a − r 2 b is a square in Z[r 2], with a "square root" which also can be computed. It should be remarked that the use of Gaussian elimination does not give the optimal run time of the algorithm. Instead, sparse matrix solving algorithms such as Block Lanczos or Block Wiedemann are used.
Therefore, the algorithm of square-free factorization is basic in computer algebra. Over a field of characteristic 0, the quotient of f {\displaystyle f} by its greatest common divisor (GCD) with its derivative is the product of the a i {\displaystyle a_{i}} in the above square-free decomposition.
The first algorithm for polynomial decomposition was published in 1985, [6] though it had been discovered in 1976, [7] and implemented in the Macsyma/Maxima computer algebra system. [8] That algorithm takes exponential time in worst case, but works independently of the characteristic of the underlying field .
The Lenstra elliptic-curve factorization or the elliptic-curve factorization method (ECM) is a fast, sub-exponential running time, algorithm for integer factorization, which employs elliptic curves. For general-purpose factoring, ECM is the third-fastest known factoring method.
Polynomial Matrix Spectral Factorization or Matrix Fejer–Riesz Theorem is a tool used to study the matrix decomposition of polynomial matrices. Polynomial matrices are widely studied in the fields of systems theory and control theory and have seen other uses relating to stable polynomials .