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algorism (W3)

ALGORISM (1256)

augorime, augrim, augrime

[ FEW: 19,73a Huwarizm; Gdf: 1,220c algorisme; GdfC: Ø; TL: 1,669 augorisme; DEAF: Ø; DMF: algorisme; TLF: Ø; OED: algorism n.; MED: augrim n.; DMLBS: 61b algorismus ]
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A step-by-step procedure for solving a problem in a finite number of steps that often involves repetition of an operation.
Algorithm is an alteration of "algorism", possibly influenced by arithmetic. It comes to us from the Arabic name of a ninth century Persian mathematician and textbook author, via Old French and Medieval Latin: Muhammad ibn-Musa "al-Khwarizmi" was from the Khwarizm region, an area south of the Aral Sea. "Al-Khwarizmi" wrote a book titled Kitab al jabr wa'l-muqabala ("Rules of restoring and equating") which is the source of the word "algebra".

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algorithm (W3)

Greedy algorithm

Smallest algorithms: A006457, A006458, A006459

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... the etymology of the word "algorithm". This derives not from the Greek "algiros" (for pain) as many think, but from the name of a ninth century Arabic mathematician, "Mohammed al-Khowarizimi". ["al-Khowarizini" means "from the city of Khowarizm", the Arabic name of a city in Russia]. It used to refer to arithmetic procedures like long division and square root extraction (which before calculators, all schoolchildren had to learn by heart). Today, in the age of the computer revolution, it has of course taken on a new meaning.
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• Arbitrary precision computation Last update: Feb. 16 2001
• FFT based multiplication of large numbers Last update: Aug. 31 2001
• Binary splitting method Last update: Jan. 11 2001
• Computation of inverse and square root in high precision Last update: Feb. 02 2001
• Newton's iteration Last update: Oct. 4 2001
• N-th digit computation: How to compute directly the n-th digit of p, log 2, ¼. Includes a new algorithm to compute directly the n-th decimal digit of p. Last update: Feb. 12 2003

• What is a computer algorithm?
• Why is the Google algorithm so important?
• How Routing Algorithms Work

algorithm

Algorithms are sets of rules according to which search engines rank web pages. Figuring out the algorithms is a major part of SEO. The thinking is that if you understand how they calculate relevance, you can make specific pages on your site super relevant for specific search terms. For more on algorithms and SEO in general, please refer to the Search Engine Yearbook.

algorithm-based software

Data mining software typically used for statistical analysis.

algorithmic search

A term often used in the financial community to distinguish between algorithm-based search engines like Google and payment-driven search engines like Overture. The term is slightly confusing. All search engines - including Overture - has some form of algorithm working behind the scenes.

Bing Optimization
This article deals with the differences in the algorithms used by Bing and Google.

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De Casteljau's algorithm (W3)

In the mathematical field of numerical analysis, "De Casteljau's algorithm" is a recursive method to evaluate polynomials in Bernstein form or Bézier curves, named after its inventor "Paul de Casteljau". "De Casteljau's algorithm" can also be used to split a single Bézier curve into two Bézier curves at an arbitrary parameter value.

Although the algorithm is slower for most architectures when compared with the direct approach, it is more numerically stable.

Contents

• 1 Definition
• 2 Example implementation
• 3 Notes
• 4 Example
• 5 Bézier curve
• 6 Geometric interpretation
• 7 References
• 8 See also

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DSA (W3)

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euclidean algorithm
Euclid's algorithm (W3)

Euclid's algorithm

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Fermat's Algorithm (W3)

Pierre de Fermat (17 August 1601 or 1607/8 - 12 January 1665) was a French lawyer at the Parlement of Toulouse, France, and an amateur mathematician who is given credit for early developments that led to infinitesimal calculus, including his adequality. In particular, he is recognized for his discovery of an original method of finding the greatest and the smallest ordinates of curved lines, which is analogous to that of the then unknown differential calculus, as well as his research into number theory. He made notable contributions to analytic geometry, probability, and optics. He is best known for Fermat's Last Theorem, which he described in a note at the margin of a copy of Diophantus' Arithmetica.
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IDEA (W3)

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Integration (W3)

HISTORY OF INTEGRATION
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Over 2000 years ago, Archimedes (287-212 BC) found formulas for the surface areas and volumes of solids such as the sphere, the cone, and the paraboloid. His method of integration was remarkably modern considering that he did not have algebra, the function concept, or even the decimal representation of numbers.

Leibniz (1646-1716) and Newton (1642-1727) independently discovered calculus. Their key idea was that differentiation and integration undo each other. Using this symbolic connection, they were able to solve an enormous number of important problems in mathematics, physics, and astronomy.

Fourier (1768-1830) studied heat conduction with a series of trigonometric terms to represent functions. Fourier series and integral transforms have applications today in fields as far apart as medicine, linguistics, and music.

Gauss (1777-1855) made the first table of integrals, and with many others continued to apply integrals in the mathematical and physical sciences.
Cauchy (1789-1857) took integrals to the complex domain. Riemann (1826-1866) and Lebesgue (1875-1941) put definite integration on a firm logical foundation.

Liouville (1809-1882) created a framework for constructive integration by finding out when indefinite integrals of elementary functions are again elementary functions.
Hermite (1822-1901) found an algorithm for integrating rational functions.
In the 1940s Ostrowski extended this algorithm to rational expressions involving the logarithm.

In the 20th century before computers, mathematicians developed the theory of integration and applied it to write tables of integrals and integral transforms. Among these mathematicians were Watson, Titchmarsh, Barnes, Mellin, Meijer, Grobner, Hofreiter, Erdelyi, Lewin, Luke, Magnus, Apelblat, Oberhettinger, Gradshteyn, Ryzhik, Exton, Srivastava, Prudnikov, Brychkov, and Marichev.

In 1969 Risch made the major breakthrough in algorithmic indefinite integration when he published his work on the general theory and practice of integrating elementary functions. His algorithm does not automatically apply to all classes of elementary functions because at the heart of it there is a hard differential equation that needs to be solved. Efforts since then have been directed at handling this equation algorithmically for various sets of elementary functions. These efforts have led to an increasingly complete algorithmization of the Risch scheme. In the 1980s some progress was also made in extending his method to certain classes of special functions.
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K-Means Clustering Algorithm (W3)

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Knuth-Morris-Pratt algorithm (W3)

The Knuth-Morris-Pratt string searching algorithm (or KMP algorithm) searches for occurrences of a "word" W within a main "text string" S by employing the observation that when a mismatch occurs, the word itself embodies sufficient information to determine where the next match could begin, thus bypassing re-examination of previously matched characters.

The algorithm was conceived by Donald Knuth and Vaughan Pratt and independently by James H. Morris in 1977, but the three published it jointly.

Contents

• 1 KMP algorithm
• 1.1 Worked example of the search algorithm
• 1.2 Description of and pseudocode for the search algorithm
• 1.3 Efficiency of the search algorithm
• 2 "Partial match" table (also known as "failure function")
• 2.1 Worked example of the table-building algorithm
• 2.2 Description of and pseudocode for the table-building algorithm
• 2.3 Efficiency of the table-building algorithm
• 3 Efficiency of the KMP algorithm
• 4 External links
• 5 References

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SIGACT (W3)

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Kraus, Johannes (Autor)
Margenow, Svetozar (Autor)
Robust Algebraic Multilevel Methods and Algorithms

Kurzbeschreibung
This book deals with algorithms for the solution of linear systems of algebraic equations with large-scale sparse matrices, with a focus on problems that are obtained after discretization of partial differential equations using finite element methods. Provides a systematic presentation of the recent advances in robust algebraic multilevel methods. Can be used for advanced courses on the topic.

Über den Autor
Johannes Kraus, Radon Institute for Computational and Applied Mathematics, Linz, Austria;Svetozar Margenov, Institute for Parallel Processing, Sofia, Bulgaria.

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Meinhardt, Hans (Autor)
Prusinkiewicz, P. (Illustrator)
Fowler, D. R. (Illustrator)
The Algorithmic Beauty of Sea Shells
The Virtual Laboratory

Kurzbeschreibung
For centuries scientists have tried to understand the growth and development of multicellular organisms. More recently, with the help of mathematical models and computerized simulations, they have discovered algorithmic patterns and models that seem to describe the dynamic processes in which organisms grow, reproduce, and respond to external factors. In this fascinating and beautifully illustrated book, Hans Meinhardt explains and illustrates these structural growth patterns in the case of sea shells. The book delightfully conveys the intuitive appeal and the "touch of magic" in this research. A diskette packaged with the book contains a program that allows the reader to run the simulations on a PC. New patterns can be generated interactively to provide an insight into the process of biological pattern formation.

4th ed., 2009, XIV, 269 p. With CD-ROM.
Hardcover, ISBN 978-3-540-92141-7
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A fascinating reconstruction of the processes forming the patterns on the shells of tropical sea snails.
Corresponding algorithms are described in detail.
An accompanying CD-ROM gives all the algorithms for simulating the natural pattern formation and growth processes on a PC.

The patterns on the shells of tropical sea snails are not only compellingly beautiful but also tell a tale of biological development. The decorative patterns are records of their own genesis, which follows laws like those of dune formation or the spread of a flu epidemic. Hans Meinhardt has analyzed the dynamical processes that form these patterns and retraced them faithfully in computer simulations. His book is exciting not only for the astonishing scientific knowledge it reveals but also for its fascinating pictures.

Many of the simulations are supplied on the accompanying CD-ROM in HTML format. Thus, the highly dynamic behavior of the interactions on which shell patterning is based can be inspected like conventional websites with any browser, also on a Macintosh or Linux system. The animations are given as separate files that also can be integrated, e.g., into a PowerPoint presentation if desired. Also included are animated simulations of models that describe more general problems in the development of higher organisms, e.g., formation of organizing regions, pattern formation in hydra, gene activation, formation of net-like structures, and chemotactic orientation.

Table of contents

Shell patterns as dynamic systems.- Pattern formation.- Oscillation and travelling waves.- Superposition of stable and periodic patterns.- Meshwork of oblique lines and staggered dots.- Branch initiation by global control.- The big problem: two or more time-dependent patterns.- Triangles.- Parallel lines with tongues.- Shell models in three dimensions.- The computer program.- Appendix: Pattern formation in the development of higher-level organisms.