These polynomials are defined by the recursive functions (4) n + 1 P n + 1 x = 2 n + 1 x P n x - n P n - 1 x P 0 x = 1 P 1 x = x For our purposes, we normalized them to be bounded by 0.1 and 0.9 in interval [-1,1].
If a solution that satisfies the integrality constraint cannot be found, the problem will be divided into two sub-problems by defining new upper and lower bounds followed by call to the recursive functions.
Since the recursive functions are just those which are effectively computable, by the Church-Turing thesis, we may also call these sets computably enumerable.
Briefly, the primitive recursive functions are those that can be formed from the basic functions: by using the operations of composition and primitive recursion: It is known that the Turing computable functions are exactly the recursive functions.
This is what Gödel did to extend the primitive recursive functions to the recursive functions.
In fact the Turing-computable functions are just the recursive functions, described below.
With this definition, the Recursive functions are exactly the same as the set of partial functions computable by the Lambda calculus, by Kleene Formal systems, by Markov algorithms, by Post machines, and by Turing machines.
The recursive functions, which form a class of computable functions, take their name from the process of "recurrence" or "recursion".
Then, the algorithm calculates F root (the value of root node) by recursive function.
As the error probability of each base may differ, P m (n i ) follows the Poisson binomial distribution and can be calculated exactly by a recursive function in O (n i 2 ) time [ 16].
