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A general-purpose factoring algorithm, also known as a Category 2, Second Category, or Kraitchik family algorithm, [10] 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.
For example, the problem of factoring "Given a positive integer n, find a nontrivial prime factor of n." is a computational problem that has a solution, as there are many known integer factorization algorithms. A computational problem can be viewed as a set of instances or cases together with a, possibly empty, set of solutions for every ...
Fermat's factorization method, named after Pierre de Fermat, is based on the representation of an odd integer as the difference of two squares: =. That difference is algebraically factorable as (+) (); if neither factor equals one, it is a proper factorization of N.
An optimal strategy for choosing these polynomials is not known; one simple method is to pick a degree d for a polynomial, consider the expansion of n in base m (allowing digits between −m and m) for a number of different m of order n 1/d, and pick f(x) as the polynomial with the smallest coefficients and g(x) as x − m.
The problem that we are trying to solve is: given an odd composite number, find its integer factors. To achieve this, Shor's algorithm consists of two parts: A classical reduction of the factoring problem to the problem of order-finding.
The system Q(Rx) = b is solved by Rx = Q T b = c, and the system Rx = c is solved by 'back substitution'. The number of additions and multiplications required is about twice that of using the LU solver, but no more digits are required in inexact arithmetic because the QR decomposition is numerically stable .
Conversely, the factor theorem asserts that, if r is a root of P(x) = 0, then P(x) may be factored as = (), where Q(x) is the quotient of Euclidean division of P(x) = 0 by the linear (degree one) factor x – r. If the coefficients of P(x) are real or complex numbers, the fundamental theorem of algebra asserts that P(x) has a real or complex ...
In linear least squares problem one seeks a solution x of an over-determined system Ax = l, such that quadratic norm of the residual vector Ax-l is minimum. This may be accomplished by solving by Cholesky decomposition normal equations N x = A T l {\displaystyle \mathbf {Nx} =\mathbf {A} ^{\mathsf {T}}\mathbf {l} } , where N = A T A ...