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The algorithm is not self-correcting; each iteration must be performed with the desired number of correct digits for π 's final result. One iteration of this algorithm is equivalent to two iterations of the Gauss–Legendre algorithm. A proof of these algorithms can be found here: [6]
This approach to the factorial takes total time (): one logarithm comes from the number of bits in the factorial, a second comes from the multiplication algorithm, and a third comes from the divide and conquer. [88]
An alternative version uses the fact that the Poisson distribution converges to a normal distribution by the Central Limit Theorem. [5]Since the Poisson distribution with parameter converges to a normal distribution with mean and variance , their density functions will be approximately the same:
function factorial (n is a non-negative integer) if n is 0 then return 1 [by the convention that 0! = 1] else if n is in lookup-table then return lookup-table-value-for-n else let x = factorial(n – 1) times n [recursively invoke factorial with the parameter 1 less than n] store x in lookup-table in the n th slot [remember the result of n! for ...
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.
A classic example of recursion is the definition of the factorial function, given here in Python code: def factorial ( n ): if n > 0 : return n * factorial ( n - 1 ) else : return 1 The function calls itself recursively on a smaller version of the input (n - 1) and multiplies the result of the recursive call by n , until reaching the base case ...
In order to achieve this speed-up, the number field sieve has to perform computations and factorizations in number fields. This results in many rather complicated aspects of the algorithm, as compared to the simpler rational sieve. The size of the input to the algorithm is log 2 n or the number of bits in the binary representation of n.
The discrete logarithm algorithm and the factoring algorithm are instances of the period-finding algorithm, and all three are instances of the hidden subgroup problem. On a quantum computer, to factor an integer N {\displaystyle N} , Shor's algorithm runs in polynomial time , meaning the time taken is polynomial in log N {\displaystyle \log ...