divide and conquer algorithm

In mathematics and computer science an algorithm is a finite set of well-defined instructions for accomplishing some task which, given an initial state, will terminate in a corresponding recognizable end-state. Algorithms can be implemented by computer programs. Informally, the concept of an algorithm is often illustrated by the example of a recipe, although many algorithms are much more complex; algorithms often have steps that repeat (iterate) or require decisions (such as logic or comparison). The concept of algorithms was formalized in 1936 by Alan Turing's Turing machines and Alonzo Church's lambda c ...

Including:

• Algorithm - Formalization of algorithms
• Algorithm - Implementation
• Algorithm - Example
• Algorithm - History
• Algorithm - Classes
• Algorithm - Classification by implementation
• Algorithm - Classification by design paradigm
• Algorithm - Classification by field of study
• Algorithm - Classification by complexity
• Algorithm - Sources
• Algorithm - Legal issues

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There are many ways to classify algorithms, and the merits of each classification have been the subject of ongoing debate. Algorithm - Classification by implementation. One way to classify algorithms is by implementation means. Recursion vs. Iteration: A recursive algorithm is one that invokes (makes reference to) itself repeatedly until a certain condition matches, which is a method common to functional programming. Iterative algorithms use repetitive constructs like loops and p ...

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Algorithm, Algorithm - Formalization of algorithms, Algorithm - Implementation, Algorithm - Example, Algorithm - History, Algorithm - Classes, Algorithm - Classification by implementation, Algorithm - Classification by design paradigm, Algorithm - Classification by field of study, Algorithm - Classification by complexity, Algorithm - Sources, Algorithm - Legal issues

Read more here: » Algorithm: Encyclopedia II - Algorithm - Classes

Main articles: Prime-factor FFT algorithm, Bruun's FFT algorithm, Rader's FFT algorithm, Bluestein's FFT algorithm. There are other FFT algorithms distinct from Cooley-Tukey. For n = n1n2 with coprime n1 and n2, one can use the Prime-Factor (Good-Thomas) algorithm (PFA), based on the Chinese Remainder Theorem, to factorize the DFT similarly to Cooley-Tukey but with ...

See also:

Fast Fourier transform, Fast Fourier transform - The Cooley-Tukey algorithm, Fast Fourier transform - Other FFT algorithms, Fast Fourier transform - FFT algorithms specialized for real and/or symmetric data, Fast Fourier transform - Accuracy and approximations, Fast Fourier transform - Multidimensional FFT algorithms

Read more here: » Fast Fourier transform: Encyclopedia II - Fast Fourier transform - Other FFT algorithms

There are many ways to classify algorithms, and the merits of each classification have been the subject of ongoing debate. Algorithm - Classification by implementation. One way to classify algorithms is by implementation means. Recursion or iteration: A recursive algorithm is one that invokes (makes reference to) itself repeatedly until a certain condition matches, which is a method common to functional programming. Iterative algorithms use repetitive constructs like loops and so ...

See also:

Algorithm, Algorithm - History, Algorithm - Formalization of algorithms, Algorithm - Implementation, Algorithm - Example, Algorithm - Algorithm analysis, Algorithm - Classes, Algorithm - Classification by implementation, Algorithm - Classification by design paradigm, Algorithm - Classification by field of study, Algorithm - Classification by complexity, Algorithm - Legal issues

Read more here: » Algorithm: Encyclopedia II - Algorithm - Classes

There are many ways to classify algorithms, and the merits of each classification have been the subject of ongoing debate. Algorithm - Classification by implementation. One way to classify algorithms is by implementation means. Recursion or iteration: A recursive algorithm is one that invokes (makes reference to) itself repeatedly until a certain condition matches, which is a method common to functional programming. Iterative algorithms use repetitive constructs like loops and so ...

See also:

Algorithm, Algorithm - History, Algorithm - Formalization of algorithms, Algorithm - Implementation, Algorithm - Example, Algorithm - Classes, Algorithm - Classification by implementation, Algorithm - Classification by design paradigm, Algorithm - Classification by field of study, Algorithm - Classification by complexity, Algorithm - Legal issues

Read more here: » Algorithm: Encyclopedia II - Algorithm - Classes

The most common application of binary search is to find a specific value in a sorted list. To cast this in the frame of the guessing game (see Example below), realize that we are now guessing the index, or numbered place, of the value in the list. The search begins by examining the value in the center of the list; because the values are sorted, it then knows whether the value occurs before or after the center value, and searches through the correct half in the same way. Here is simple pseudocode which determines the index of a given value in a sorted li ...

See also:

Binary search algorithm, Binary search algorithm - The algorithm, Binary search algorithm - Examples, Binary search algorithm - Language support, Binary search algorithm - Applications to complexity theory

Read more here: » Binary search algorithm: Encyclopedia II - Binary search algorithm - The algorithm

One of the simplest algorithms is to find the largest number in an (unsorted) list of numbers. The solution necessarily requires looking at every number in the list, but only once at each. From this follows a simple algorithm, which can be stated in English as Let us assume the first item is largest. Look at each of the remaining items in the list and make the following adjustment. a. If it is larger than the largest item we gathered so far, make a note of it. The latest noted item is the largest in the list when the process is complete. And here is a more formal c ...

See also:

Algorithm, Algorithm - History, Algorithm - Formalization of algorithms, Algorithm - Implementation, Algorithm - Example, Algorithm - Algorithm analysis, Algorithm - Classes, Algorithm - Classification by implementation, Algorithm - Classification by design paradigm, Algorithm - Classification by field of study, Algorithm - Classification by complexity, Algorithm - Legal issues

Read more here: » Algorithm: Encyclopedia II - Algorithm - Example

Some countries allow algorithms to be patented when embodied in software or in hardware. Patents have long been a controversial issue (see, for example, the software patent debate). Some countries do not allow certain algorithms, such as cryptographic algorithms, to be exported from that country. ...

See also:

Algorithm, Algorithm - History, Algorithm - Formalization of algorithms, Algorithm - Implementation, Algorithm - Example, Algorithm - Algorithm analysis, Algorithm - Classes, Algorithm - Classification by implementation, Algorithm - Classification by design paradigm, Algorithm - Classification by field of study, Algorithm - Classification by complexity, Algorithm - Legal issues

Read more here: » Algorithm: Encyclopedia II - Algorithm - Legal issues

Algorithms are essential to the way computers process information, because a computer program is essentially an algorithm that tells the computer what specific steps to perform (in what specific order) in order to carry out a specified task, such as calculating employees’ paychecks or printing students’ report cards. Thus, an algorithm can be considered to be any sequence of operations which can be performed by a Turing-complete system. Typically, when an algorithm is associated with processing information, data is read from an in ...

See also:

Algorithm, Algorithm - History, Algorithm - Formalization of algorithms, Algorithm - Implementation, Algorithm - Example, Algorithm - Classes, Algorithm - Classification by implementation, Algorithm - Classification by design paradigm, Algorithm - Classification by field of study, Algorithm - Classification by complexity, Algorithm - Legal issues

Read more here: » Algorithm: Encyclopedia II - Algorithm - Formalization of algorithms

An example of binary search in action is a simple guessing game in which a player has to guess a positive integer selected by another player between 1 and N, using only questions answered with yes or no. Supposing N is 16 and the number 11 is selected, the game might proceed as follows. Is the number greater than 8? (Yes) Is the number greater than 12? (No) Is the number greater than 10? (Yes) ...

See also:

Binary search algorithm, Binary search algorithm - The algorithm, Binary search algorithm - Examples, Binary search algorithm - Language support, Binary search algorithm - Applications to complexity theory

Read more here: » Binary search algorithm: Encyclopedia II - Binary search algorithm - Examples

Some countries allow algorithms to be patented when embodied in software or in hardware. Patents have long been a controversial issue (see, for example, the software patent debate). Some countries do not allow certain algorithms, such as cryptographic algorithms, to be exported from that country. ...

See also:

Algorithm, Algorithm - History, Algorithm - Formalization of algorithms, Algorithm - Implementation, Algorithm - Example, Algorithm - Classes, Algorithm - Classification by implementation, Algorithm - Classification by design paradigm, Algorithm - Classification by field of study, Algorithm - Classification by complexity, Algorithm - Legal issues

Read more here: » Algorithm: Encyclopedia II - Algorithm - Legal issues

One of the simplest algorithms is to find the largest number in an (unsorted) list of numbers. The solution necessarily requires looking at every number in the list, but only once at each. From this follows a simple algorithm, which can be stated in English as Let us assume the first item is largest. Look at each of the remaining items in the list and make the following adjustment. a. If it is larger than the largest item we gathered so far, make a note of it. The latest noted item is the largest in the list when the process is complete. And here is a more formal c ...

See also:

Algorithm, Algorithm - History, Algorithm - Formalization of algorithms, Algorithm - Implementation, Algorithm - Example, Algorithm - Classes, Algorithm - Classification by implementation, Algorithm - Classification by design paradigm, Algorithm - Classification by field of study, Algorithm - Classification by complexity, Algorithm - Legal issues

Read more here: » Algorithm: Encyclopedia II - Algorithm - Example

Algorithms are essential to the way computers process information, because a computer program is essentially an algorithm that tells the computer what specific steps to perform (in what specific order) in order to carry out a specified task, such as calculating employees’ paychecks or printing students’ report cards. Thus, an algorithm can be considered to be any sequence of operations which can be performed by a Turing-complete system. Typically, when an algorithm is associated with processing information, data is read from an in ...

See also:

Algorithm, Algorithm - History, Algorithm - Formalization of algorithms, Algorithm - Implementation, Algorithm - Example, Algorithm - Algorithm analysis, Algorithm - Classes, Algorithm - Classification by implementation, Algorithm - Classification by design paradigm, Algorithm - Classification by field of study, Algorithm - Classification by complexity, Algorithm - Legal issues

Read more here: » Algorithm: Encyclopedia II - Algorithm - Formalization of algorithms

The word algorithm comes from the name of the 9th century Persian mathematician Abu Abdullah Muhammad bin Musa al-Khwarizmi. The word algorism originally referred only to the rules of performing arithmetic using Hindu-Arabic numerals but evolved via European Latin translation of al-Khwarizmi's name into algorithm by the 18th century. The word evolved to include all definite procedures for ...

See also:

Algorithm, Algorithm - History, Algorithm - Formalization of algorithms, Algorithm - Implementation, Algorithm - Example, Algorithm - Classes, Algorithm - Classification by implementation, Algorithm - Classification by design paradigm, Algorithm - Classification by field of study, Algorithm - Classification by complexity, Algorithm - Legal issues

Read more here: » Algorithm: Encyclopedia II - Algorithm - History

All of the FFT algorithms discussed so far compute the DFT exactly (in exact arithmetic, i.e. neglecting floating-point errors). A few "FFT" algorithms have been proposed, however, that compute the DFT approximately, with an error that can be made arbitrarily small at the expense of increased computations. Such algorithms trade the approximation error for increased speed or other properties. For example, an approximate FFT algorithm by Edelman et al. (1999) achieves lower communication requirements for parallel computing with the help ...

See also:

Fast Fourier transform, Fast Fourier transform - The Cooley-Tukey algorithm, Fast Fourier transform - Other FFT algorithms, Fast Fourier transform - FFT algorithms specialized for real and/or symmetric data, Fast Fourier transform - Accuracy and approximations, Fast Fourier transform - Multidimensional FFT algorithms

Read more here: » Fast Fourier transform: Encyclopedia II - Fast Fourier transform - Accuracy and approximations

Algorithms are essential to the way computers process information, because a computer program is essentially an algorithm that tells the computer what specific steps to perform (in what specific order) in order to carry out a specified task, such as calculating employees’ paychecks or printing students’ report cards. Thus, an algorithm can be considered to be any sequence of operations which can be performed by a Turing-complete system. Typically, when an algorithm is associated with processing information, data is read from an in ...

See also:

Algorithm, Algorithm - Formalization of algorithms, Algorithm - Implementation, Algorithm - Example, Algorithm - History, Algorithm - Classes, Algorithm - Classification by implementation, Algorithm - Classification by design paradigm, Algorithm - Classification by field of study, Algorithm - Classification by complexity, Algorithm - Sources, Algorithm - Legal issues

Read more here: » Algorithm: Encyclopedia II - Algorithm - Formalization of algorithms

One of the simplest algorithms is to find the largest number in an (unsorted) list of numbers. The solution necessarily requires looking at every number in the list, but only once at each. From this follows a simple algorithm: Look at each item in the list. If it is larger than any that has been seen so far, make a note of it. The latest noted item is the largest in the list when the process is complete. And here is a more formal c ...

See also:

Algorithm, Algorithm - Formalization of algorithms, Algorithm - Implementation, Algorithm - Example, Algorithm - History, Algorithm - Classes, Algorithm - Classification by implementation, Algorithm - Classification by design paradigm, Algorithm - Classification by field of study, Algorithm - Classification by complexity, Algorithm - Sources, Algorithm - Legal issues

Read more here: » Algorithm: Encyclopedia II - Algorithm - Example

The word algorithm comes from the name of the 9th century Persian astrologer Abu Abdullah Muhammad bin Musa al-Khwarizmi. According to Carl B. Boyer, author of The History of Mathematics (1968) al-Khwarizmi wrote numerous books on arithmatics, and algebra. One of his books, "De numero Indorum" (Concerning the Art of Hindu Reckoning) was based on the Arabic translation of Brahmagupta where he described the Hindu digits of zero, 1,2,3, through the number 9 and the decimal place value that was a recent arrival from India. The word ...

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Algorithm, Algorithm - Formalization of algorithms, Algorithm - Implementation, Algorithm - Example, Algorithm - History, Algorithm - Classes, Algorithm - Classification by implementation, Algorithm - Classification by design paradigm, Algorithm - Classification by field of study, Algorithm - Classification by complexity, Algorithm - Sources, Algorithm - Legal issues

Read more here: » Algorithm: Encyclopedia II - Algorithm - History

In many applications, the input data for the DFT are purely real, in which case the outputs satisfy the symmetry and efficient FFT algorithms have been designed for this situation (see e.g. Sorensen, 1987). One approach consists of taking an ordinary algorithm (e.g. Cooley-Tukey) and removing the redundant parts of the computation, saving roughly a factor of two in time and memory. Alternatively, it is possible to express an even-length real-input DFT as a complex DFT of half the length (whose real and imaginary parts are t ...

See also:

Fast Fourier transform, Fast Fourier transform - The Cooley-Tukey algorithm, Fast Fourier transform - Other FFT algorithms, Fast Fourier transform - FFT algorithms specialized for real and/or symmetric data, Fast Fourier transform - Accuracy and approximations, Fast Fourier transform - Multidimensional FFT algorithms

Read more here: » Fast Fourier transform: Encyclopedia II - Fast Fourier transform - FFT algorithms specialized for real and/or symmetric data

Main article: Cooley-Tukey FFT algorithm. By far the most common FFT is the Cooley-Tukey algorithm. This is a divide and conquer algorithm that recursively breaks down a DFT of any composite size n = n1n2 into many smaller DFTs of sizes n1 and n2, along with O(n) multiplications by complex roots of unity traditionally called twiddle factors. This method (and the general idea of an FFT) was popularized by a publication of J. W. Co ...

See also:

Fast Fourier transform, Fast Fourier transform - The Cooley-Tukey algorithm, Fast Fourier transform - Other FFT algorithms, Fast Fourier transform - FFT algorithms specialized for real and/or symmetric data, Fast Fourier transform - Accuracy and approximations, Fast Fourier transform - Multidimensional FFT algorithms

Read more here: » Fast Fourier transform: Encyclopedia II - Fast Fourier transform - The Cooley-Tukey algorithm