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.

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.

Another polynomial viewpoint is exploited by the Winograd algorithm, which factorizes into cyclotomic polynomials — these often have coefficients of 1, 0, or − 1, and therefore require few ( if any ) multiplications, so Winograd can be used to obtain minimal-multiplication FFTs and is often used to find efficient algorithms for small factors.

: This formula is often used to compute least common multiples: one first computes the gcd with Euclid's algorithm and then divides the product of the given numbers by their gcd.

The use of a metasyntactic variable is helpful in freeing a programmer from creating a logically named variable, which is often useful when creating or teaching examples of an algorithm.

If the desired output values are often incorrect ( because of human error or sensor errors ), then the learning algorithm should not attempt to find a function that exactly matches the training examples.

A programmer who needs to implement a specific algorithm, especially an unfamiliar one, will often start with a pseudocode description, and then " translate " that description into the target programming language and modify it to interact correctly with the rest of the program.

::* This is often computed using the extended Euclidean algorithm.

This makes 2-4 trees an important tool for understanding the logic behind red – black trees, and this is why many introductory algorithm texts introduce 2-4 trees just before red – black trees, even though 2-4 trees are not often used in practice.

In real-time computing, the worst-case execution time is often of particular concern since it is important to know how much time might be needed in the worst case to guarantee that the algorithm will always finish on time.

This algorithm is often used in routing and as a subroutine in other graph algorithms.

The algorithm provides no cryptographic security, and is often cited as a canonical example of weak encryption.

* Link-state routing protocols are based on algorithms to find shortest paths in a graph ( the most often used algorithm is Dijkstra's algorithm ).

A key is often easier to protect ( it's typically a small piece of information ) than an encryption algorithm, and easier to change if compromised.

The name " Ford – Fulkerson " is often also used for the Edmonds – Karp algorithm, which is a specialization of Ford – Fulkerson.

Computer scientists and mathematicians often pronounce it as star ( as, for example, in the A * search algorithm or C *- algebra ).

Because the algorithm is very simple, it is often implemented in either the firmware or the graphics hardware of modern graphics cards.

The algorithm that evaluates final board positions is termed the " evaluation function ", and these algorithms are often vastly different between different chess programs.

However, it is often used as a subroutine in another sorting algorithm, radix sort, that can handle larger keys more efficiently.

The running time for multiplying rectangular matrices ( one m × p-matrix with one p × n-matrix ) is O ( mnp ), however, more efficient algorithms exist, such as Strassen's algorithm, devised by Volker Strassen in 1969 and often referred to as " fast matrix multiplication ".

* Data Encryption Algorithm, the block cipher algorithm defined in ( and often referred to as ) the Data Encryption Standard

Nowadays these are often considered as a single hypothesis, the Church – Turing thesis, which states that any function that is computable by an algorithm is a computable function.

Written in prose but much closer to the high-level language of a computer program, the following is the more formal coding of the algorithm in pseudocode or pidgin code:

A more numerically stable two-pass algorithm first computes the sample means, and then the covariance:

A slightly more accurate compensated version performs the full naive algorithm on the residuals.

ALGOL ( short for ALGOrithmic Language ) is a family of imperative computer programming languages originally developed in the mid 1950s which greatly influenced many other languages and was the standard method for algorithm description used by the ACM, in textbooks, and academic works for the next 30 years and more.

Bruce Schneier notes that while Blowfish is still in use, he recommends using the more recent Twofish algorithm instead.

For this reason cryptologists tend to look at indicators that an algorithm or key length shows signs of potential vulnerability, to move to longer key sizes or more difficult algorithms.

Jürgen Schmidhuber ( 2000 ) constructed a limit-computable " Super Omega " which in a sense is much more random than the original limit-computable Omega, as one cannot significantly compress the Super Omega by any enumerating non-halting algorithm.

The basic concepts are moderately straightforward, but intractable to calculate ; and a single elegant algorithm or approach has been elusive for more general purpose renderers.

An Earley parser is an example of such an algorithm, while the widely used LR and LL parsers are simpler algorithms that deal only with more restrictive subsets of context-free grammars.

Church subsequently modified his methods to include use of Herbrand – Gödel recursion and then proved ( 1936 ) that the Entscheidungsproblem is unsolvable: There is no generalized " effective calculation " ( method, algorithm ) that can determine whether or not a formula in either the recursive-or λ-calculus is " valid " ( more precisely: no method to show that a well formed formula has a " normal form ").

Problems are nonetheless still common when the algorithm has to encompass more than one language.

A key distinction between analysis of algorithms and computational complexity theory is that the former is devoted to analyzing the amount of resources needed by a particular algorithm to solve a problem, whereas the latter asks a more general question about all possible algorithms that could be used to solve the same problem.

The situation is further complicated by the traditional uses of the terms parallel and distributed algorithm that do not quite match the above definitions of parallel and distributed systems ; see the section Theoretical foundations below for more detailed discussion.

It was noted by Biham and Shamir that DES is surprisingly resistant to differential cryptanalysis, in the sense that even small modifications to the algorithm would make it much more susceptible.

Typically, the problem area is complex enough that a more simple traditional algorithm cannot provide a proper solution.

If implemented using remainders of Euclidean division rather than subtractions, Euclid's algorithm computes the GCD of large numbers efficiently: it never requires more division steps than five times the number of digits ( base 10 ) of the smaller integer.

It was once believed that real-input DFTs could be more efficiently computed by means of the discrete Hartley transform ( DHT ), but it was subsequently argued that a specialized real-input DFT algorithm ( FFT ) can typically be found that requires fewer operations than the corresponding DHT algorithm ( FHT ) for the same number of inputs.

A much more efficient method is the Euclidean algorithm, which uses a division algorithm such as long division in combination with the observation that the gcd of two numbers also divides their difference.

If the arguments are both greater than zero then the algorithm can be written in more elementary terms as follows: