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.

For example, the subprogram in Euclid's algorithm to compute the remainder would execute much faster if the programmer had a " modulus " ( division ) instruction available rather than just subtraction ( or worse: just Minsky's " decrement ").

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:

This algorithm is often more numerically reliable than the naïve algorithm for large sets of data, although it can be worse if much of the data is very close to but not precisely equal to the mean and some are quite far away from it.

Take as an example a program that looks up a specific entry in a sorted list of size n. Suppose this program were implemented on Computer A, a state-of-the-art machine, using a linear search algorithm, and on Computer B, a much slower machine, using a binary search algorithm.

In addition, as the same algorithm is used for background correction and elimination of lamp noise, the background corrected signals show a much better signal-to-noise ratio compared to the uncorrected signals, which is also in contrast to LS AAS.

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.

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.

An arbitrary PID has much the same " structural properties " of a Euclidean domain ( or, indeed, even of the ring of integers ), but knowing an explicit algorithm for Euclidean division, and thus also for greatest common divisor computation, gives a concreteness which is useful for algorithmic applications.

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.

In conjunction with an appropriate codec using MPEG or various manufacturers proprietary algorithms, an ISDN BRI can be used to send stereo bi-directional audio coded at 128 kbit / s with 20 Hz – 20 kHz audio bandwidth, although commonly the G. 722 algorithm is used with a single 64 kbit / s B channel to send much lower latency mono audio at the expense of audio quality.

Propositional logic is much less expressive but highly consistent and complete and can efficiently produce inferences with minimal algorithm complexity.

Any lossless compression algorithm that makes some files shorter must necessarily make some files longer, but it is not necessary that those files become very much longer.

Because symmetric key algorithms are nearly always much less computationally intensive than asymmetric ones, it is common to exchange a key using a key-exchange algorithm, then transmit data using that key and a symmetric key algorithm.

Once all parties have obtained the session key, they can use a much faster symmetric algorithm to encrypt and decrypt messages.

Large-scale quantum computers could be able to solve certain problems much faster than any classical computer by using the best currently known algorithms, like integer factorization using Shor's algorithm or the simulation of quantum many-body systems.

Since a paper describing the algorithm had been published in August 1977, prior to the December 1977 filing date of the patent application, regulations in much of the rest of the world precluded patents elsewhere and only the US patent was granted.

Bacon describes a cycle-collection algorithm for reference counting systems with some similarities to tracing systems, including the same theoretical time bounds, but that takes advantage of reference count information to run much more quickly and with less cache damage.

A much more efficient algorithm than Hashiguchi's procedure was devised by Kirsten in 2005.

In order to analyze how much time and space a given algorithm requires, computer scientists express the time or space required to solve the problem as a function of the size of the input problem.

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.

We can run the algorithm a constant number of times and take a majority vote to achieve any desired probability of correctness less than 1, using the Chernoff bound.

Otherwise, if the sought key is less than the middle element's key, then the algorithm repeats its action on the sub-array to the left of the middle element or, if the input key is greater, on the sub-array to the right.

That a rational number must have a finite or recurring decimal expansion can be seen to be a consequence of the long division algorithm, in that there are only q-1 possible nonzero remainders on division by q, so that the recurring pattern will have a period less than q.

* earley3. py A stand-alone implementation of the algorithm in less than 150 lines of code, including generation of the parsing-forest and samples.

Since the algorithm ensures that the remainders decrease steadily with every step, r < sub > k − 1 </ sub > is less than its predecessor r < sub > k − 2 </ sub >.

Some FFTs other than Cooley – Tukey, such as the Rader-Brenner algorithm, are intrinsically less stable.

Fischer and Rabin's work also implies that Presburger arithmetic can be used to define formulas which correctly calculate any algorithm as long as the inputs are less than relatively large bounds.

Other algorithms can speed up a task less dramatically — for example, Grover's search algorithm which gives a quadratic speed-up over the best possible classical algorithm.

" The nomad " allegedly found this decryption algorithm through so-called reverse engineering of a Xing DVD-player, where the keys were more or less openly accessible.

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.

" On the other hand, if keeping the algorithm secret is not important, but only the keys used with the algorithm must be secret, then disclosure of the keys simply requires the simpler, less costly process of generating and distributing new keys.

Note that, if the language is infinite, the enumerating algorithm provided can be chosen so that it avoids repetitions, since we can test whether the string produced for number n is " already " produced for a number which is less than n. If it already is produced, use the output for input n + 1 instead ( recursively ), but again, test whether it is " new ".

The Gibbs sampling algorithm is a special case of the Metropolis – Hastings algorithm which is usually faster and easier to use but is less generally applicable.

Adoption of technical standards for digital signatures have lagged behind much of the legislation, delaying a more or less unified engineering position on interoperability, algorithm choice, key lengths, and so on what the engineering is attempting to provide.

The algorithm will be initially presented only for the octant in which the segment goes down and to the right ( x < sub > 0 </ sub >≤ x < sub > 1 </ sub > and y < sub > 0 </ sub >≤ y < sub > 1 </ sub >), and its horizontal projection is longer than the vertical projection ( the line has a negative slope whose absolute value is less than 1.

The Preliminary Credibility Assessment Screening System, or PCASS, captures less physiological information than a polygraph, and uses an algorithm, not the judgment of a polygraph examiner, to render a decision whether it believes the person is being deceptive or not.

The painter's algorithm is another common solution which, though less efficient, can also handle non-opaque scene elements.