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In mathematics, more specifically in the theory of Monte Carlo methods, variance reduction is a procedure used to increase the precision of the estimates obtained for a given simulation or computational effort. [1] Every output random variable from the simulation is associated with a variance which limits the precision of the simulation results.
The main idea is to express an integral involving an integer parameter (e.g. power) of a function, represented by I n, in terms of an integral that involves a lower value of the parameter (lower power) of that function, for example I n-1 or I n-2. This makes the reduction formula a type of recurrence relation. In other words, the reduction ...
Algorithms for calculating variance play a major role in computational statistics.A key difficulty in the design of good algorithms for this problem is that formulas for the variance may involve sums of squares, which can lead to numerical instability as well as to arithmetic overflow when dealing with large values.
The term redex, short for reducible expression, refers to subterms that can be reduced by one of the reduction rules. For example, (λx.M) N is a β-redex in expressing the substitution of N for x in M. The expression to which a redex reduces is called its reduct; the reduct of (λx.M) N is M[x := N]. [b] If x is not free in M, λx.
Example of a reduction from the boolean satisfiability problem (A ∨ B) ∧ (¬A ∨ ¬B ∨ ¬C) ∧ (¬A ∨ B ∨ C) to a vertex cover problem.The blue vertices form a minimum vertex cover, and the blue vertices in the gray oval correspond to a satisfying truth assignment for the original formula.
In modular arithmetic, Barrett reduction is an algorithm designed to optimize the calculation of [1] without needing a fast division algorithm. It replaces divisions with multiplications, and can be used when n {\displaystyle n} is constant and a < n 2 {\displaystyle a<n^{2}} .
The simplest example given by Thimbleby of a possible problem when using an immediate-execution calculator is 4 × (−5). As a written formula the value of this is −20 because the minus sign is intended to indicate a negative number, rather than a subtraction, and this is the way that it would be interpreted by a formula calculator.
In computational complexity theory, a polynomial-time reduction is a method for solving one problem using another. One shows that if a hypothetical subroutine solving the second problem exists, then the first problem can be solved by transforming or reducing it to inputs for the second problem and calling the subroutine one or more times.