Natural language expressions of algorithms tend to be verbose and ambiguous, and are rarely used for complex or technical algorithms.

Programming languages are primarily intended for expressing algorithms in a form that can be executed by a computer, but are often used as a way to define or document algorithms.

Very similar algorithms can be used to compute the covariance.

The term is also used, especially in the description of algorithms, to mean associative array or " abstract array ", a theoretical computer science model ( an abstract data type or ADT ) intended to capture the essential properties of arrays.

The latter is more cumbersome to use, so it's only employed when necessary, for example in the analysis of arbitrary-precision arithmetic algorithms, like those used in cryptography.

Abstract data types are purely theoretical entities, used ( among other things ) to simplify the description of abstract algorithms, to classify and evaluate data structures, and to formally describe the type systems of programming languages.

The imperative style is often used when describing abstract algorithms.

Combinatorics is used frequently in computer science to obtain formulas and estimates in the analysis of algorithms.

Techniques such as Huffman coding are now used by computer-based algorithms to compress large data files into a more compact form for storage or transmission.

* By whether the same key is used for both encryption and decryption ( symmetric key algorithms ), or if a different key is used for each ( asymmetric key algorithms ).

* symmetric key algorithms ( Private-key cryptography ), where the same key is used for encryption and decryption, and

* asymmetric key algorithms ( Public-key cryptography ), where two different keys are used for encryption and decryption.

Many ciphers are based on publicly known algorithms or are open source, and so it is only the difficulty of obtaining the key that determines security of the system, provided that there is no analytic attack ( i. e., a ' structural weakness ' in the algorithms or protocols used ), and assuming that the key is not otherwise available ( such as via theft, extortion, or compromise of computer systems ).

Many rendering algorithms have been researched, and software used for rendering may employ a number of different techniques to obtain a final image.

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.

Paired with the problem are any number of algorithms that can be used to solve the problem, that is, to calculate the solution.

The checksum algorithms that are most used in practice, such as Fletcher's checksum, Adler-32, and cyclic redundancy checks ( CRCs ), address these weaknesses by considering not only the value of each word but also its position in the sequence.

* < tt > jcksum </ tt >, a Java library, that can be used by developers in Java applications to calculate checksums using different algorithms.

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 general consensus is that these public key algorithms are insecure at any key size if sufficiently large quantum computers capable of running Shor's algorithm become available.

Many of these algorithms use convolution with the filter 1 to slightly whiten or flatten the spectrum, thereby allowing traditional lossless compression to work more efficiently.

Compression algorithms can average a color across these similar areas to reduce space, in a manner similar to those used in JPEG image compression.

However, all of these algorithms can create distorted and even unusable grid elements.

For example, machines, unlike humans, can support a wide selection of learning algorithms, and controversy has arisen over the relative ethical merits of these options.

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.

Unfortunately, these algorithms require too many additions to be practical, at least on modern computers with hardware multipliers.

However, complex-data FFTs are so closely related to algorithms for related problems such as real-data FFTs, discrete cosine transforms, discrete Hartley transforms, and so on, that any improvement in one of these would immediately lead to improvements in the others ( Duhamel & Vetterli, 1990 ).

Even the " exact " FFT algorithms have errors when finite-precision floating-point arithmetic is used, but these errors are typically quite small ; most FFT algorithms, e. g. Cooley – Tukey, have excellent numerical properties as a consequence of the pairwise summation structure of the algorithms.

These algorithms model diffuse inter-reflection which is a very important part of global illumination ; however most of these ( excluding radiosity ) also model specular reflection, which makes them more accurate algorithms to solve the lighting equation and provide a more realistically illuminated scene.

Researchers applying these algorithms are automatically producing human competitive designs — but no human designer is required.

The most difficult integers to factor in practice using existing algorithms are those that are products of two large primes of similar size, and for this reason these are the integers used in cryptographic applications.

However, if there are complex interactions among features, then algorithms such as decision trees and neural networks work better, because they are specifically designed to discover these interactions.

The methods underlying these algorithms are described in the article systems of polynomial equations.

For this reason, many online gambling sites provide descriptions of their shuffling algorithms and the sources of randomness used to drive these algorithms, with some gambling sites also providing auditors ' reports of the performance of their systems.

Scientific applications generally require no more than 40 digits of, so the primary motivation for these computations is the human desire to break records, but the extensive calculations involved have been used to test supercomputers and high-precision multiplication algorithms.

The use of these algorithms also allows the authenticity of a message to be checked by creating a digital signature of the message using the private key, which can then be verified by using the public key.

However, other existing cryptographic algorithms do not appear to be broken by these algorithms.

Earlier rendering algorithms, such as Whitted-style ray tracing were capable of computing effects such as reflections, refractions, and shadows, but despite being highly global phenomena, these effects were not commonly referred to as " global illumination.

It is estimated that the total amount of the information that is stored on the world's storage devices could be further compressed with existing compression algorithms by a remaining average factor of 4. 5: 1.

DSP processor ICs are found in every type of modern electronic systems and products including, SDTV | HDTV sets, radios and mobile communication devices, Hi-Fi audio equipments, Dolby noise reduction algorithms, GSM mobile phones, mp3 multimedia players, camcorders and digital cameras, automobile control systems, noise cancelling headphones, digital spectrum analyzers, intelligent missile guidance, radar, GPS based cruise control systems and all kinds of image processing, video processing, audio processing and speech processing systems.

Modern vocoders that are used in communication equipment and in voice storage devices today are based on the following algorithms:

JTAG tool vendors provide various types of stimulus and sophisticated algorithms, not only to detect the failing nets, but also to isolate the faults to specific nets, devices, and pins.

Most general-purpose microprocessors and operating systems can execute DSP algorithms successfully, but are not suitable for use in portable devices such as mobile phones and PDAs because of power supply and space constraints.

( Internally, those devices use one of a variety of division algorithms ).

In the mid-1980s, Baxter's interest in music recording technology led him to wonder about hardware and software that was originally developed for military use, i. e. data-compression algorithms and large-capacity storage devices.

* New and emerging cryptographic algorithms and protocols targeting embedded devices

Their purpose, in part, is to eliminate the need for perpetual calendar tables, algorithms and computation devices.

More practically, the Chernoff bound is used in randomized algorithms ( or in computational devices such as quantum computers ) to determine a bound on the number of runs necessary to determine a value by majority agreement, up to a specified probability.

The artificial intelligence, which required both advances in the underlying algorithms and leaps in processing power both on mobile devices and the servers that share the workload, allows software to understand not just words but the intentions behind them.

As the GSM localisation is less accurate than GPS based systems, lots of devices have to be tracked and complex algorithms need to be used to extract high-quality data.

Since the 1980s, reversible circuits have attracted interest as components of quantum algorithms, and more recently in photonic and nano-computing technologies where some switching devices offer no signal gain.