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Ternary Search
A ternary search algorithm is a technique in computer science for finding the minimum or maximum of a unimodal function. The function Assume we are looking for a maximum of f(x) and that we know the maximum lies somewhere between A and B. For the algorithm to be applicable, there must be some value x such that * for all a, b with A \leq a * if f(m_1) = f(m_2), then the search should be conducted in _1; m_2/math>, but this case can be attributed to any of the previous two (in order to simplify the code). Sooner or later the length of the segment will be a little less than a predetermined constant, and the process can be stopped. choice points m_1 and m_2: * m_1 = l + (r - l) / 3 * m_2 = r - (r - l) / 3 ; Run time order : T(n) = T(2n/3) + O(1) = \Theta(\log n) (by the Master Theorem) Recursive algorithm def ternary_search(f, left, right, absolute_precision) -> float: """Left and right are the current bounds; the maximum is between them. """ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Computer Science
Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, applied disciplines (including the design and implementation of Computer architecture, hardware and Software engineering, software). Algorithms and data structures are central to computer science. The theory of computation concerns abstract models of computation and general classes of computational problem, problems that can be solved using them. The fields of cryptography and computer security involve studying the means for secure communication and preventing security vulnerabilities. Computer graphics (computer science), Computer graphics and computational geometry address the generation of images. Programming language theory considers different ways to describe computational processes, and database theory concerns the management of re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Maxima And Minima
In mathematical analysis, the maximum and minimum of a function are, respectively, the greatest and least value taken by the function. Known generically as extremum, they may be defined either within a given range (the ''local'' or ''relative'' extrema) or on the entire domain (the ''global'' or ''absolute'' extrema) of a function. Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions. As defined in set theory, the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers, have no minimum or maximum. In statistics, the corresponding concept is the sample maximum and minimum. Definition A real-valued function ''f'' defined on a domain ''X'' has a global (or absolute) maximum point at ''x''∗, if for all ''x'' in ''X''. Similarly, the function has a global (or absolute) minimum point at ''x''� ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Unimodality
In mathematics, unimodality means possessing a unique mode. More generally, unimodality means there is only a single highest value, somehow defined, of some mathematical object. Unimodal probability distribution In statistics, a unimodal probability distribution or unimodal distribution is a probability distribution which has a single peak. The term "mode" in this context refers to any peak of the distribution, not just to the strict definition of mode which is usual in statistics. If there is a single mode, the distribution function is called "unimodal". If it has more modes it is "bimodal" (2), "trimodal" (3), etc., or in general, "multimodal". Figure 1 illustrates normal distributions, which are unimodal. Other examples of unimodal distributions include Cauchy distribution, Student's ''t''-distribution, chi-squared distribution and exponential distribution. Among discrete distributions, the binomial distribution and Poisson distribution can be seen as unimodal, thoug ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Master Theorem (analysis Of Algorithms)
Master, master's or masters may refer to: Ranks or titles In education: *Master (college), head of a college *Master's degree, a postgraduate or sometimes undergraduate degree in the specified discipline * Schoolmaster or master, presiding officer of a school In military: * Master (naval), a former naval rank * Master mariner, a licensed mariner who is qualified to be a sea captain in the merchant marine *Master or shipmaster, the sea captain of a merchant vessel * Master-at-arms, a naval police officer, often addressed as "Master" in the Royal Navy In orders and organizations: *Master craftsman, in the Medieval guilds In other: * Master (form of address), an English honorific for boys and young men * Master (judiciary), a judicial official in the courts of common law jurisdictions * Master (Peerage of Scotland), the male heir-apparent or heir-presumptive to a title in the Peerage of Scotland * Master of ceremonies, or MC (emcee), the host of an official public or private stag ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Newton's Method In Optimization
In calculus, Newton's method (also called Newton–Raphson) is an iterative method for finding the roots of a differentiable function f, which are solutions to the equation f(x)=0. However, to optimize a twice-differentiable f, our goal is to find the roots of f'. We can therefore use Newton's method on its derivative f' to find solutions to f'(x)=0, also known as the critical points of f. These solutions may be minima, maxima, or saddle points; see section "Several variables" in Critical point (mathematics) and also section "Geometric interpretation" in this article. This is relevant in optimization, which aims to find (global) minima of the function f. Newton's method The central problem of optimization is minimization of functions. Let us first consider the case of univariate functions, i.e., functions of a single real variable. We will later consider the more general and more practically useful multivariate case. Given a twice differentiable function f:\mathbb\to \math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Golden-section Search
The golden-section search is a technique for finding an extremum (minimum or maximum) of a function inside a specified interval. For a strictly unimodal function with an extremum inside the interval, it will find that extremum, while for an interval containing multiple extrema (possibly including the interval boundaries), it will converge to one of them. If the only extremum on the interval is on a boundary of the interval, it will converge to that boundary point. The method operates by successively narrowing the range of values on the specified interval, which makes it relatively slow, but very robust. The technique derives its name from the fact that the algorithm maintains the function values for four points whose three interval widths are in the ratio ''φ'':1:''φ'', where ''φ'' is the golden ratio. These ratios are maintained for each iteration and are maximally efficient. Excepting boundary points, when searching for a minimum, the central point is always less than or equal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Binary Search Algorithm
In computer science, binary search, also known as half-interval search, logarithmic search, or binary chop, is a search algorithm that finds the position of a target value within a sorted array. Binary search compares the target value to the middle element of the array. If they are not equal, the half in which the target cannot lie is eliminated and the search continues on the remaining half, again taking the middle element to compare to the target value, and repeating this until the target value is found. If the search ends with the remaining half being empty, the target is not in the array. Binary search runs in logarithmic time in the worst case, making O(\log n) comparisons, where n is the number of elements in the array. Binary search is faster than linear search except for small arrays. However, the array must be sorted first to be able to apply binary search. There are specialized data structures designed for fast searching, such as hash tables, that can be searched ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Interpolation Search
Interpolation search is an algorithm for searching for a key in an array that has been ordered by numerical values assigned to the keys (''key values''). It was first described by W. W. Peterson in 1957. Interpolation search resembles the method by which people search a telephone directory for a name (the key value by which the book's entries are ordered): in each step the algorithm calculates where in the remaining search space the sought item might be, based on the key values at the bounds of the search space and the value of the sought key, usually via a linear interpolation. The key value actually found at this estimated position is then compared to the key value being sought. If it is not equal, then depending on the comparison, the remaining search space is reduced to the part before or after the estimated position. This method will only work if calculations on the size of differences between key values are sensible. By comparison, binary search always chooses the middl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Exponential Search
In computer science, an exponential search (also called doubling search or galloping search or Struzik search) is an algorithm, created by Jon Bentley and Andrew Chi-Chih Yao in 1976, for searching sorted, unbounded/infinite lists. There are numerous ways to implement this, with the most common being to determine a range that the search key resides in and performing a binary search within that range. This takes O(\log i) time, where i is the position of the search key in the list, if the search key is in the list, or the position where the search key should be, if the search key is not in the list. Exponential search can also be used to search in bounded lists. Exponential search can even out-perform more traditional searches for bounded lists, such as binary search, when the element being searched for is near the beginning of the array. This is because exponential search will run in ''O(\log i)'' time, where i is the index of the element being searched for in the list, whereas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Linear Search
In computer science, linear search or sequential search is a method for finding an element within a list. It sequentially checks each element of the list until a match is found or the whole list has been searched. A linear search runs in linear time in the worst case, and makes at most comparisons, where is the length of the list. If each element is equally likely to be searched, then linear search has an average case of comparisons, but the average case can be affected if the search probabilities for each element vary. Linear search is rarely practical because other search algorithms and schemes, such as the binary search algorithm and hash tables, allow significantly faster searching for all but short lists. Algorithm A linear search sequentially checks each element of the list until it finds an element that matches the target value. If the algorithm reaches the end of the list, the search terminates unsuccessfully. Basic algorithm Given a list of elements with values ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Articles With Example Python (programming Language) Code
Article often refers to: * Article (grammar), a grammatical element used to indicate definiteness or indefiniteness * Article (publishing), a piece of nonfictional prose that is an independent part of a publication Article(s) may also refer to: Government and law * Elements of treaties of the European Union * Articles of association, the regulations governing a company, used in India, the UK and other countries; called articles of incorporation in the US * Articles of clerkship, the contract accepted to become an articled clerk * Articles of Confederation, the predecessor to the current United States Constitution * Article of impeachment, a formal document and charge used for impeachment in the United States * Article of manufacture, in the United States patent law, a category of things that may be patented * Articles of organization, for limited liability organizations, a US equivalent of articles of association Other uses * Article element , in HTML * "Articles", a song ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |