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The goal of the quantum subroutine of Shor's algorithm is, given coprime integers and < <, to find the order of modulo , which is the smallest positive integer such that (). To achieve this, Shor's algorithm uses a quantum circuit involving two registers.
In mathematics, and more specifically in numerical analysis, Householder's methods are a class of root-finding algorithms that are used for functions of one real variable with continuous derivatives up to some order d + 1.
To find the probabilities of the order statistics, three values are first needed, namely p 1 = P ( X < x ) = F ( x ) − f ( x ) , p 2 = P ( X = x ) = f ( x ) , and p 3 = P ( X > x ) = 1 − F ( x ) . {\displaystyle p_{1}=P(X<x)=F(x)-f(x),\ p_{2}=P(X=x)=f(x),{\text{ and }}p_{3}=P(X>x)=1-F(x).}
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For finite sets, total orders may be identified with linear sequences of objects, where the "≤" relation is true whenever the first object precedes the second object in the order; a comparison sorting algorithm may be used to convert a total order into a sequence in this way.
The order of a group G is denoted by ord(G) or | G |, and the order of an element a is denoted by ord(a) or | a |, instead of ( ), where the brackets denote the generated group. Lagrange's theorem states that for any subgroup H of a finite group G , the order of the subgroup divides the order of the group; that is, | H | is a divisor of | G | .
In numerical analysis, Ridders' method is a root-finding algorithm based on the false position method and the use of an exponential function to successively approximate a root of a continuous function (). The method is due to C. Ridders.
In numerical analysis, Halley's method is a root-finding algorithm used for functions of one real variable with a continuous second derivative. Edmond Halley was an English mathematician and astronomer who introduced the method now called by his name. The algorithm is second in the class of Householder's methods, after Newton's method.