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Double Exponential
Double exponential may refer to: * A double exponential function ** Double exponential time, a task with time complexity roughly proportional to such a function ** 2-EXPTIME, the complexity class of decision problems solvable in double-exponential time by a deterministic Turing machine. * Double exponential distribution, which may refer to: ** Laplace distribution In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. It is also sometimes called the double exponential distribution, because it can be thought of as two exponen ..., a bilateral exponential distribution ** Gumbel distribution, an iterated exponential distribution * Double exponential integration, most commonly tanh-sinh quadrature * Double exponential smoothing {{mathematical disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Double Exponential Function
A double exponential function is a constant raised to the power of an exponential function. The general formula is f(x) = a^=a^ (where ''a''>1 and ''b''>1), which grows much more quickly than an exponential function. For example, if ''a'' = ''b'' = 10: *''f''(x) = 1010x *''f''(0) = 10 *''f''(1) = 1010 *''f''(2) = 10100 = googol *''f''(3) = 101000 *''f''(100) = 1010100 = googolplex. Factorials grow faster than exponential functions, but much more slowly than double exponential functions. However, tetration and the Ackermann function grow faster. See Big O notation for a comparison of the rate of growth of various functions. The inverse of the double exponential function is the double logarithm log(log(''x'')). The complex double exponential function is entire, because it is the composition of two entire functions f(x)=a^x=e^ and g(x)=b^x=e^. Double exponential sequences A sequence of positive integers (or real numbers) is said to have ''double exponential rate of growth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Time Complexity
In theoretical computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, supposing that each elementary operation takes a fixed amount of time to perform. Thus, the amount of time taken and the number of elementary operations performed by the algorithm are taken to be related by a constant factor. Since an algorithm's running time may vary among different inputs of the same size, one commonly considers the worst-case time complexity, which is the maximum amount of time required for inputs of a given size. Less common, and usually specified explicitly, is the average-case complexity, which is the average of the time taken on inputs of a given size (this makes sense because there are only a finite number of possible inputs of a given size). In both cases, the time complexity is gene ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
:en:2-EXPTIME
In computational complexity theory, the complexity class 2-EXPTIME (sometimes called 2-EXP, sometimes also written 2EXPTIME) is the set of all decision problems solvable by a deterministic Turing machine in O(22''p''(''n'')) time, where ''p''(''n'') is a polynomial function of ''n''. In terms of DTIME, : \mathsf = \bigcup_ \mathsf \left( 2^ \right) . Comparison with other complexity classes We know : P ⊆ NP ⊆ PSPACE ⊆ EXPTIME ⊆ NEXPTIME ⊆ EXPSPACE ⊆ 2-EXPTIME ⊆ ELEMENTARY. 2-EXPTIME can also be reformulated as the space class AEXPSPACE, the problems that can be solved by an alternating Turing machine in exponential space. This is one way to see that EXPSPACE ⊆ 2-EXPTIME, since an alternating Turing machine is at least as powerful as a deterministic Turing machine. 2-EXPTIME is one class in a hierarchy of complexity classes with increasingly higher time bounds. The class 3-EXPTIME is defined similarly to 2-EXPTIME but with a triply exponential time bound 2^. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complexity Class
In computational complexity theory, a complexity class is a set (mathematics), set of computational problems "of related resource-based computational complexity, complexity". The two most commonly analyzed resources are time complexity, time and space complexity, memory. In general, a complexity class is defined in terms of a type of computational problem, a model of computation, and a bounded resource like time complexity, time or space complexity, memory. In particular, most complexity classes consist of decision problems that are solvable with a Turing machine, and are differentiated by their time or space (memory) requirements. For instance, the class P (complexity), P is the set of decision problems solvable by a deterministic Turing machine in polynomial time. There are, however, many complexity classes defined in terms of other types of problems (e.g. Counting problem (complexity), counting problems and function problems) and using other models of computation (e.g. probabil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decision Problems
In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question on a set of input values. An example of a decision problem is deciding whether a given natural number is prime. Another example is the problem, "given two numbers ''x'' and ''y'', does ''x'' evenly divide ''y''?" A decision procedure for a decision problem is an algorithmic method that answers the yes-no question on all inputs, and a decision problem is called decidable if there is a decision procedure for it. For example, the decision problem "given two numbers ''x'' and ''y'', does ''x'' evenly divide ''y''?" is decidable since there is a decision procedure called long division that gives the steps for determining whether ''x'' evenly divides ''y'' and the correct answer, ''YES'' or ''NO'', accordingly. Some of the most important problems in mathematics are undecidable, e.g. the halting problem. The field of computational compl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplace Distribution
In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. It is also sometimes called the double exponential distribution, because it can be thought of as two exponential distributions (with an additional location parameter) spliced together along the x-axis, although the term is also sometimes used to refer to the Gumbel distribution. The difference between two Independent identically-distributed random variables, independent identically distributed exponential random variables is governed by a Laplace distribution, as is a Brownian motion evaluated at an exponentially distributed random time. Increments of Laplace motion or a variance gamma process evaluated over the time scale also have a Laplace distribution. Definitions Probability density function A random variable has a \operatorname(\mu, b) distribution if its probability density function is : f(x \mid \mu, b) = \frac \exp\left( -\frac \rig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gumbel Distribution
In probability theory and statistics, the Gumbel distribution (also known as the type-I generalized extreme value distribution) is used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions. This distribution might be used to represent the distribution of the maximum level of a river in a particular year if there was a list of maximum values for the past ten years. It is useful in predicting the chance that an extreme earthquake, flood or other natural disaster will occur. The potential applicability of the Gumbel distribution to represent the distribution of maxima relates to extreme value theory, which indicates that it is likely to be useful if the distribution of the underlying sample data is of the normal or exponential type. The Gumbel distribution is a particular case of the generalized extreme value distribution (also known as the Fisher–Tippett distribution). It is also known as the ''log-Weibull distribution'' and the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tanh-sinh Quadrature
Tanh-sinh quadrature is a method for numerical integration introduced by Hidetoshi Takahashi and Masatake Mori in 1974. It is especially applied where singularities or infinite derivatives exist at one or both endpoints. The method uses hyperbolic functions in the change of variables :x = \tanh\left(\frac\pi\sinh t\right)\, to transform an integral on the interval ''x'' ∈ (−1, 1) to an integral on the entire real line ''t'' ∈ (−∞, ∞), the two integrals having the same value. After this transformation, the integrand decays with a double exponential rate, and thus, this method is also known as the double exponential (DE) formula. For a given step size h, the integral is approximated by the sum :\int_^1 f(x) \, dx \approx \sum_^\infty w_k f(x_k), with the abscissas :x_k = \tanh\left(\frac\pi\sinh kh\right) and the weights :w_k = \frac. Use The Tanh-Sinh method is quite insensitive to endpoint behavior. Should singularities or infinite derivatives e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
