The algorithm proceeds by successive subtractions in two loops: IF the test B ≥ A yields " yes " ( or true ) ( more accurately the number b in location B is greater than or equal to the number a in location A ) THEN the algorithm specifies B ← B − A ( meaning the number b − a replaces the old b ).

In mathematics and computer science, an algorithm ( originating from al-Khwārizmī, the famous Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī ) is a step-by-step procedure for calculations.

More precisely, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function.

While there is no generally accepted formal definition of " algorithm ," an informal definition could be " a set of rules that precisely defines a sequence of operations.

" For some people, a program is only an algorithm if it stops eventually ; for others, a program is only an algorithm if it stops before a given number of calculation steps.

A prototypical example of an algorithm is Euclid's algorithm to determine the maximum common divisor of two integers ; an example ( there are others ) is described by the flow chart above and as an example in a later section.

The concept of algorithm is also used to define the notion of decidability.

In logic, the time that an algorithm requires to complete cannot be measured, as it is not apparently related with our customary physical dimension.

Gurevich: "... Turing's informal argument in favor of his thesis justifies a stronger thesis: every algorithm can be simulated by a Turing machine ... according to Savage, an algorithm is a computational process defined by a Turing machine ".

Typically, when an algorithm is associated with processing information, data is read from an input source, written to an output device, and / or stored for further processing.

Stored data is regarded as part of the internal state of the entity performing the algorithm.

Because an algorithm is a precise list of precise steps, the order of computation will always be critical to the functioning of the algorithm.

In computer systems, an algorithm is basically an instance of logic written in software by software developers to be effective for the intended " target " computer ( s ), in order for the target machines to produce output from given input ( perhaps null ).

is the length of time taken to perform the algorithm.

Simulation of an algorithm: computer ( computor ) language: Knuth advises the reader that " the best way to learn an algorithm is to try it.

This is a serious problem with Algorithm I and is due to catastrophic cancellation in the subtraction of two similar numbers at the final stage of the algorithm.

A stable one-pass algorithm exists, similar to the one above, that computes:

He also was the original author of rzip, which uses a similar algorithm to rsync.

In automatic systems this can be done using a binary search algorithm or interpolation search ; manual searching may be performed using a roughly similar procedure, though this will often be done unconsciously.

A sorting algorithm ( similar to quicksort ):

A limitation of modeling fractals is that resemblance of a fractal model to a natural phenomenon does not prove that the phenomenon being modeled is formed by a process similar to the modeling algorithm.

The well-known radix-2 Cooley – Tukey algorithm, for N a power of 2, can compute the same result with only ( N / 2 ) log < sub > 2 </ sub > N complex multiplies ( again, ignoring simplifications of multiplications by 1 and similar ) and N log < sub > 2 </ sub > N complex additions.

A similar fact takes place for particular computational tasks, like the search problem, for which Grover's algorithm is optimal.

Path vector routing is discussed in RFC 1322 ; the path vector routing algorithm is somewhat similar to the distance vector algorithm in the sense that each border router advertises the destinations it can reach to its neighboring router.

When the critical section is longer than a few source code lines or involves lengthy looping, an embedded / real-time algorithm must resort to using mechanisms identical or similar to those available on general-purpose operating systems, such as semaphores and OS-supervised interprocess messaging.

SNOBOL4 pattern-matching uses a backtracking algorithm similar to that used in the logic programming language Prolog, which provides pattern-like constructs via DCGs.

The Christofides algorithm follows a similar outline but combines the minimum spanning tree with a solution of another problem, minimum-weight perfect matching.

It fundamentally improves on the Soundex algorithm by using information about variations and inconsistencies in English spelling and pronunciation to produce a more accurate encoding, which does a better job of matching words and names which sound similar.

It is similar to the μ-law algorithm used in North America and Japan.

It is similar to the A-law algorithm used in regions where digital telecommunication signals are carried on E-1 circuits, e. g. Europe.

This algorithm is guaranteed to converge to an image, and it should appear similar to the original image.

In modern Z-buffer systems, similar benefits can be gained through rough front-to-back sorting ( approaching the ' reverse painters algorithm '), early Z-reject ( in conjunction with hierarchical Z ), and less common deferred rendering techniques possible on programmable GPUs.

Compare this to heuristic, a similar concept used in mathematical discourse, psychology, and computer science, particularly in algorithm design.

* Midpoint circle algorithm, a similar algorithm for drawing circles

The resulting algorithm is formally similar, except that every dot product is replaced by a nonlinear kernel function.

A pattern-based modeling approach can thus be seen as an image construction algorithm, where the patterns of the training image are used, and tiled next to each other such that a new image with similar characteristics / features is generated.

Automata theory also studies if there exist any effective algorithm or not to solve problems similar to the following list.

The current algorithm with the lowest known exponent k is a generalization of the Coppersmith – Winograd algorithm that has an asymptotic complexity of O ( n < sup > 2. 3727 </ sup >) thanks to Vassilevska Williams .< ref > The original algorithm was presented by Don Coppersmith and Shmuel Winograd in 1990, has an asymptotic complexity of O ( n < sup > 2. 376 </ sup >).</ ref > This algorithm, and the Coppersmith-Winograd algorithm on which it is based, are similar to Strassen's algorithm: a way is devised for multiplying two k × k-matrices with fewer than k < sup > 3 </ sup > multiplications, and this technique is applied recursively.

The asks for an algorithm that takes as input a statement of a first-order logic ( possibly with a finite number of axioms beyond the usual axioms of first-order logic ) and answers " Yes " or " No " according to whether the statement is universally valid, i. e., valid in every structure satisfying the axioms.

There are many ways in which the flood-fill algorithm can be structured, but they all make use of a queue or stack data structure, explicitly or implicitly.

The data structure used depends on both the graph structure and the algorithm used for manipulating the graph.

# Determine the structure of the learned function and corresponding learning algorithm.

This SMILES is unique for each structure, although dependent on the canonicalization algorithm used to generate it, and is termed the Canonical SMILES.

No currently existing algorithm is yet able to consistently predict a proteins ' tertiary or quaternary structure given only its primary structure ; learning how to accurately predict the tertiary and quaternary structure of any protein given only its amino acid sequence and the pertinent cellular conditions would be a monumental achievement.

* Visitor separates an algorithm from an object structure by moving the hierarchy of methods into one object.

# Say we have an algorithm that finds an MST ( which we will call A ) based on the structure of the graph and the order of the edges when ordered by weight.

* SCHEMA ( bioinformatics ), an algorithm used in protein engineering to identify fragments of proteins that can be recombined without disturbing the integrity of the proteins ' three-dimensional structure

There is also an ever growing connection between Bayesian methods and simulation-based Monte Carlo techniques since complex models cannot be processed in closed form by a Bayesian analysis, while a graphical model structure may allow for efficient simulation algorithms like the Gibbs sampling and other Metropolis – Hastings algorithm schemes.

Depending on the physical structure of the sensor, a color filter array may be used which requires a demosaicing / interpolation algorithm.

Provided that the edges are either already sorted or can be sorted in linear time ( for example with counting sort or radix sort ), the algorithm can use more sophisticated disjoint-set data structure to run in O ( E α ( V )) time, where α is the extremely slowly growing inverse of the single-valued Ackermann function.

Often a carefully chosen data structure will allow the most efficient algorithm to be used.

The algorithm uses a queue data structure to store intermediate results as it traverses the graph, as follows:

Depth-first search ( DFS ) is an algorithm for traversing or searching a tree, tree structure, or graph.

RC5 also consists of a number of modular additions and eXclusive OR ( XOR ) s. The general structure of the algorithm is a Feistel-like network.

For example, given sequence data structures, e. g. singly linked list, vector etc., and algorithms to operate on them, e. g., etc., a direct approach would implement each algorithm specifically for each data structure, giving combinations to implement.

However, in the generic programming approach, each data structure returns a model of an iterator concept ( a simple value type which can be dereferenced to retrieve the current value, or changed to point to another value in the sequence ) and each algorithm is instead written generically with arguments of such iterators, e. g. a pair of iterators pointing to the beginning and end of the subsequence to process.

This limits which data structures a given algorithm can be applied to and such complexity requirements are a major determinant of data structure choice.

While it is possible to solve any of these problems using the simplex algorithm, each specialization has more efficient algorithms designed to take advantage of its special structure.

The rsync utility uses an algorithm invented by the Australian computer programmer Andrew Tridgell for efficiently transmitting a structure ( such as a file ) across a communications link when the receiving computer already has a similar, but not identical, version of the same structure.