Yau Method 4x4 Algorithms

[Brayan Jacobsen]

In contrast, a heuristic is an approach to problem-solving that may not be fully specified or may not guarantee correct or optimal results, especially in problem domains where there is no well-defined correct or optimal result.[3] For example, social media recommender systems rely on heuristics in such a way that, although widely characterized as "algorithms" in 21st-century popular media, cannot deliver correct results due to the nature of the problem.

As an effective method, an algorithm can be expressed within a finite amount of space and time[4] and in a well-defined formal language[5] for calculating a function.[6] Starting from an initial state and initial input (perhaps empty),[7] the instructions describe a computation that, when executed, proceeds through a finite[8] number of well-defined successive states, eventually producing "output"[9] and terminating at a final ending state. The transition from one state to the next is not necessarily deterministic; some algorithms, known as randomized algorithms, incorporate random input.[10]

Most algorithms are intended to be implemented as computer programs. However, algorithms are also implemented by other means, such as in a biological neural network (for example, the human brain implementing arithmetic or an insect looking for food), in an electrical circuit, or in a mechanical device.

Since antiquity, step-by-step procedures for solving mathematical problems have been attested. This includes in Babylonian mathematics (around 2500 BC),[16] Egyptian mathematics (around 1550 BC),[16] Indian mathematics (around 800 BC and later),[17][18] The Ifa Oracle (around 500 BC), Greek mathematics (around 240 BC),[19] and Arabic mathematics (around 800 AD).[20]

The first cryptographic algorithm for deciphering encrypted code was developed by Al-Kindi, a 9th-century Arab mathematician, in A Manuscript On Deciphering Cryptographic Messages. He gave the first description of cryptanalysis by frequency analysis, the earliest codebreaking algorithm.[20]

Telephone-switching networks of electromechanical relays (invented 1835) was behind the work of George Stibitz (1937), the inventor of the digital adding device. As he worked in Bell Laboratories, he observed the "burdensome' use of mechanical calculators with gears. "He went home one evening in 1937 intending to test his idea... When the tinkering was over, Stibitz had constructed a binary adding device".[28] The mathematician Martin Davis supported the particular importance of the electromechanical relay.[29]

Algorithms can be expressed in many kinds of notation, including natural languages, pseudocode, flowcharts, drakon-charts, programming languages or control tables (processed by interpreters). Natural language expressions of algorithms tend to be verbose and ambiguous and are rarely used for complex or technical algorithms. Pseudocode, flowcharts, drakon-charts, and control tables are structured ways to express algorithms that avoid many of the ambiguities common in statements based on natural language. Programming languages are primarily intended for expressing algorithms in a form that can be executed by a computer, but they are also often used as a way to define or document algorithms.

There is a wide variety of representations possible and one can express a given Turing machine program as a sequence of machine tables (see finite-state machine, state-transition table, and control table for more), as flowcharts and drakon-charts (see state diagram for more), or as a form of rudimentary machine code or assembly code called "sets of quadruples" (see Turing machine for more). Representations of algorithms can also be classified into three accepted levels of Turing machine description: high-level description, implementation description, and formal description.[32] A high-level description describes qualities of the algorithm itself, ignoring how it is implemented on the Turing machine.[32] An implementation description describes the general manner in which the machine moves its head and stores data in order to carry out the algorithm, but does not give exact states.[32] In the most detail, a formal description gives the exact state table and list of transitions of the Turing machine.[32]

The analysis, and study of algorithms is a discipline of computer science, and is often practiced abstractly without the use of a specific programming language or implementation. In this sense, algorithm analysis resembles other mathematical disciplines in that it focuses on the underlying properties of the algorithm and not on the specifics of any particular implementation. Usually, pseudocode is used for analysis as it is the simplest and most general representation. However, ultimately, most algorithms are usually implemented on particular hardware/software platforms and their algorithmic efficiency is eventually put to the test using real code. For the solution of a "one-off" problem, the efficiency of a particular algorithm may not have significant consequences (unless n is extremely large) but for algorithms designed for fast interactive, commercial or long life scientific usage it may be critical. Scaling from small n to large n frequently exposes inefficient algorithms that are otherwise benign.

Empirical testing is useful because it may uncover unexpected interactions that affect performance. Benchmarks may be used to compare before/after potential improvements to an algorithm after program optimization.Empirical tests cannot replace formal analysis, though, and are not trivial to perform in a fair manner.[34]

To illustrate the potential improvements possible even in well-established algorithms, a recent significant innovation, relating to FFT algorithms (used heavily in the field of image processing), can decrease processing time up to 1,000 times for applications like medical imaging.[35] In general, speed improvements depend on special properties of the problem, which are very common in practical applications.[36] Speedups of this magnitude enable computing devices that make extensive use of image processing (like digital cameras and medical equipment) to consume less power.

Algorithm design refers to a method or a mathematical process for problem-solving and engineering algorithms. The design of algorithms is part of many solution theories, such as divide-and-conquer or dynamic programming within operation research. Techniques for designing and implementing algorithm designs are also called algorithm design patterns,[37] with examples including the template method pattern and the decorator pattern. One of the most important aspects of algorithm design is resource (run-time, memory usage) efficiency; the big O notation is used to describe e.g., an algorithm's run-time growth as the size of its input increases.[38]

Algorithms, by themselves, are not usually patentable. In the United States, a claim consisting solely of simple manipulations of abstract concepts, numbers, or signals does not constitute "processes" (USPTO 2006), so algorithms are not patentable (as in Gottschalk v. Benson). However practical applications of algorithms are sometimes patentable. For example, in Diamond v. Diehr, the application of a simple feedback algorithm to aid in the curing of synthetic rubber was deemed patentable. The patenting of software is controversial,[43] and there are criticized patents involving algorithms, especially data compression algorithms, such as Unisys's LZW patent. Additionally, some cryptographic algorithms have export restrictions (see export of cryptography).

Another way of classifying algorithms is by their design methodology or paradigm. There is a certain number of paradigms, each different from the other. Furthermore, each of these categories includes many different types of algorithms. Some common paradigms are:

For optimization problems there is a more specific classification of algorithms; an algorithm for such problems may fall into one or more of the general categories described above as well as into one of the following:

One of the simplest algorithms is to find the largest number in a list of numbers of random order. Finding the solution requires looking at every number in the list. From this follows a simple algorithm, which can be stated in a high-level description in English prose, as:

(Quasi-)formal description: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:

How do you differentiate between an algorithm and a method? Why dont we call Newton's Method or Ford-Faulkerson method Algorithms? What are a properties of a good algorithm and what qualifies a method as an algorithm?

A procedure that has all of the characteristics of an algorithm except that it possible lacks finiteness may be called a computational method. Euclid originally presented not only an algorithm for the greatest common divisor of numbers, but also a very similar geometrical construction for the "greatest common measure" of the lengths of two line segments; this is a computational method that does not terminate if the given lengths are incommensurable. -- D.Knuth, TAOCP vol 1, Basic Concepts: Algorithms

The Newton Raphson method is not guaranteed to converge, not does it detect convergence failure. If you wrap the method up with convergence detection and termination at a finite epsilon or after a finite number of steps, you get an algorithm.

EDIT: On reflection, perhaps Pete is correct that algorithms terminate and methods may not (who am I to argue with Knuth?) However, I don't think that's a distinction that most people will make based only on your use of one word or the other.

In my opinion, a method is a more general concept than algorithm and can be more or less anything, e.g. writing data to a file. Just about anything that should happen due to an event or to some logical expression. Also, the meaning of the words "method" and "algorithm" can vary depending on in what context they are used. They might be used to describe the same thing.

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