In mathematics and computer science, an algorithm is an unambiguous specification of how to solve a class of problems. Algorithms can perform calculation, data processing, automated reasoning, other tasks; as an effective method, an algorithm can be expressed within a finite amount of space and time and in a well-defined formal language for calculating a function. Starting from an initial state and initial input, the instructions describe a computation that, when executed, proceeds through a finite number of well-defined successive states producing "output" and terminating at a final ending state; the transition from one state to the next is not deterministic. The concept of algorithm has existed for centuries. Greek mathematicians used algorithms in the sieve of Eratosthenes for finding prime numbers, the Euclidean algorithm for finding the greatest common divisor of two numbers; the word algorithm itself is derived from the 9th century mathematician Muḥammad ibn Mūsā al-Khwārizmī, Latinized Algoritmi.
A partial formalization of what would become the modern concept of algorithm began with attempts to solve the Entscheidungsproblem posed by David Hilbert in 1928. Formalizations were framed as attempts to define "effective calculability" or "effective method"; those formalizations included the Gödel–Herbrand–Kleene recursive functions of 1930, 1934 and 1935, Alonzo Church's lambda calculus of 1936, Emil Post's Formulation 1 of 1936, Alan Turing's Turing machines of 1936–37 and 1939. The word'algorithm' has its roots in Latinizing the name of Muhammad ibn Musa al-Khwarizmi in a first step to algorismus. Al-Khwārizmī was a Persian mathematician, astronomer and scholar in the House of Wisdom in Baghdad, whose name means'the native of Khwarazm', a region, part of Greater Iran and is now in Uzbekistan. About 825, al-Khwarizmi wrote an Arabic language treatise on the Hindu–Arabic numeral system, translated into Latin during the 12th century under the title Algoritmi de numero Indorum; this title means "Algoritmi on the numbers of the Indians", where "Algoritmi" was the translator's Latinization of Al-Khwarizmi's name.
Al-Khwarizmi was the most read mathematician in Europe in the late Middle Ages through another of his books, the Algebra. In late medieval Latin, English'algorism', the corruption of his name meant the "decimal number system". In the 15th century, under the influence of the Greek word ἀριθμός'number', the Latin word was altered to algorithmus, the corresponding English term'algorithm' is first attested in the 17th century. In English, it was first used in about 1230 and by Chaucer in 1391. English adopted the French term, but it wasn't until the late 19th century that "algorithm" took on the meaning that it has in modern English. Another early use of the word is from 1240, in a manual titled Carmen de Algorismo composed by Alexandre de Villedieu, it begins thus: Haec algorismus ars praesens dicitur, in qua / Talibus Indorum fruimur bis quinque figuris. Which translates as: Algorism is the art by which at present we use those Indian figures, which number two times five; the poem is a few hundred lines long and summarizes the art of calculating with the new style of Indian dice, or Talibus Indorum, or Hindu numerals.
An informal definition could be "a set of rules that defines a sequence of operations". Which would include all computer programs, including programs that do not perform numeric calculations. A program is only an algorithm if it stops eventually. A prototypical example of an algorithm is the Euclidean algorithm to determine the maximum common divisor of two integers. Boolos, Jeffrey & 1974, 1999 offer an informal meaning of the word in the following quotation: No human being can write fast enough, or long enough, or small enough† to list all members of an enumerably infinite set by writing out their names, one after another, in some notation, but humans can do something useful, in the case of certain enumerably infinite sets: They can give explicit instructions for determining the nth member of the set, for arbitrary finite n. Such instructions are to be given quite explicitly, in a form in which they could be followed by a computing machine, or by a human, capable of carrying out only elementary operations on symbols.
An "enumerably infinite set" is one whose elements can be put into one-to-one correspondence with the integers. Thus and Jeffrey are saying that an algorithm implies instructions for a process that "creates" output integers from an arbitrary "input" integer or integers that, in theory, can be arbitrarily large, thus an algorithm can be an algebraic equation such as y = m + n – two arbitrary "input variables" m and n that produce an output y. But various authors' attempts to define the notion indicate that the word implies much more than this, something on the order of: Precise instructions for a fast, efficient, "good" process that specifies the "moves" of "the computer" to find and process arbitrary input integers/symbols m and n, symbols + and =... and "effectively" produce, in a "reasonable" time, output-integer y at a specified place and in a specified format
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (. Sometimes referred to as the Princeps mathematicorum and "the greatest mathematician since antiquity", Gauss had an exceptional influence in many fields of mathematics and science, is ranked among history's most influential mathematicians. Johann Carl Friedrich Gauss was born on 30 April 1777 in Brunswick, in the Duchy of Brunswick-Wolfenbüttel, to poor, working-class parents, his mother was illiterate and never recorded the date of his birth, remembering only that he had been born on a Wednesday, eight days before the Feast of the Ascension. Gauss solved this puzzle about his birthdate in the context of finding the date of Easter, deriving methods to compute the date in both past and future years, he was christened and confirmed in a church near the school he attended as a child. Gauss was a child prodigy. In his memorial on Gauss, Wolfgang Sartorius von Waltershausen says that when Gauss was three years old he corrected a math error his father made. Many versions of this story have been retold since that time with various details regarding what the series was – the most frequent being the classical problem of adding all the integers from 1 to 100.
There are many other anecdotes about his precocity while a toddler, he made his first groundbreaking mathematical discoveries while still a teenager. He completed his magnum opus, Disquisitiones Arithmeticae, in 1798, at the age of 21—though it was not published until 1801; this work was fundamental in consolidating number theory as a discipline and has shaped the field to the present day. Gauss's intellectual abilities attracted the attention of the Duke of Brunswick, who sent him to the Collegium Carolinum, which he attended from 1792 to 1795, to the University of Göttingen from 1795 to 1798. While at university, Gauss independently rediscovered several important theorems, his breakthrough occurred in 1796 when he showed that a regular polygon can be constructed by compass and straightedge if the number of its sides is the product of distinct Fermat primes and a power of 2. This was a major discovery in an important field of mathematics. Gauss was so pleased with this result that he requested that a regular heptadecagon be inscribed on his tombstone.
The stonemason declined, stating that the difficult construction would look like a circle. The year 1796 was productive for both Gauss and number theory, he discovered a construction of the heptadecagon on 30 March. He further advanced modular arithmetic simplifying manipulations in number theory. On 8 April he became the first to prove the quadratic reciprocity law; this remarkably general law allows mathematicians to determine the solvability of any quadratic equation in modular arithmetic. The prime number theorem, conjectured on 31 May, gives a good understanding of how the prime numbers are distributed among the integers. Gauss discovered that every positive integer is representable as a sum of at most three triangular numbers on 10 July and jotted down in his diary the note: "ΕΥΡΗΚΑ! num = Δ + Δ' + Δ". On 1 October he published a result on the number of solutions of polynomials with coefficients in finite fields, which 150 years led to the Weil conjectures. Gauss remained mentally active into his old age while suffering from gout and general unhappiness.
For example, at the age of 62, he taught himself Russian. In 1840, Gauss published his influential Dioptrische Untersuchungen, in which he gave the first systematic analysis on the formation of images under a paraxial approximation. Among his results, Gauss showed that under a paraxial approximation an optical system can be characterized by its cardinal points and he derived the Gaussian lens formula. In 1845, he became an associated member of the Royal Institute of the Netherlands. In 1854, Gauss selected the topic for Bernhard Riemann's inaugural lecture "Über die Hypothesen, welche der Geometrie zu Grunde liegen". On the way home from Riemann's lecture, Weber reported that Gauss was full of excitement. On 23 February 1855, Gauss died of a heart attack in Göttingen. Two people gave eulogies at his funeral: Gauss's son-in-law Heinrich Ewald, Wolfgang Sartorius von Waltershausen, Gauss's close friend and biographer. Gauss's brain was preserved and was studied by Rudolf Wagner, who found its mass to be above average, at 1,492 grams, the cerebral area equal to 219,588 square millimeters.
Developed convolutions were found, which in the early 20th century were suggested as the explanation of his genius. Gauss was a Lutheran Protestant, a member of the St. Albans Evangelical Lutheran church in Göttingen. Potential evidence that Gauss believed in God comes from his response after solving a problem that had defeated him: "Finally, two days ago, I succeeded—not on account of my hard efforts, but by th
Magnus Rudolph Hestenes was an American mathematician best known for his contributions to calculus of variations and optimal control. As a pioneer in computer science, he devised the conjugate gradient method, published jointly with Eduard Stiefel. Born in Bricelyn, Hestenes earned his Ph. D. at the University of Chicago in 1932 under Gilbert Bliss. His dissertation was titled "Sufficient Conditions for the General Problem of Mayer with Variable End-Points." After teaching as an associate professor at Chicago, in 1947 he moved to a professorship at UCLA. He continued there until his retirement in 1973, during that time he served as department chair from 1950–58. While a professor, Hestenes supervised the thesis research of 34 students, among them Glen Culler, Richard Tapia and Jesse Wilkins, Jr.. Hestenes received the Guggenheim and Fulbright awards, was a vice president of the American Mathematical Society, was an invited speaker at the 1954 International Congress of Mathematicians in Amsterdam.
He died on May 1991 in Los Angeles, California. Magnus Hestenes at the Mathematics Genealogy Project Obituary in the Numerical Analysis Digest Magnus Hestenes at Find a Grave
Cornelius Lanczos was a Hungarian mathematician and physicist, born on February 2, 1893, died on June 25, 1974. According to György Marx he was one of The Martians, he was born in Székesfehérvár to dr. Károly Lőwy and Adél Hahn. Lanczos' Ph. D. thesis was on relativity theory. He sent his thesis copy to Einstein, Einstein wrote back, saying: "I studied your paper as far as my present overload allowed. I believe I may say this much: this does involve competent and original brainwork, on the basis of which a doctorate should be obtainable... I gladly accept the honorable dedication."In 1924 he discovered an exact solution of the Einstein field equation representing a cylindrically symmetric rigidly rotating configuration of dust particles. This was rediscovered by Willem Jacob van Stockum and is known today as the van Stockum dust, it is one of the simplest known exact solutions in general relativity and is regarded as an important example, in part because it exhibits closed timelike curves. Lanczos served as assistant to Albert Einstein during the period of 1928–29.
In 1927 Lanczos married Maria Rupp. He was offered a one-year visiting professorship from Purdue University. For a dozen years Lanzos split his life between two continents, his wife Maria Rupp stayed with Lanczos' parents in Székesfehérvár year-around while Lanczos went to Purdue for half the year, teaching graduate students matrix mechanics and tensor analysis. In 1933 his son Elmar was born. Maria was too ill to travel and died several weeks from tuberculosis; when the Nazis purged Hungary of Jews in 1944, of Lanczos' family, only his sister and a nephew survived. Elmar moved to Seattle and raised two sons; when Elmar looked at his own firstborn son, he said: "For me, it proves that Hitler did not win." During the McCarthy era, Lanczos came under suspicion for possible communist links. In 1952, he left the U. S. and moved to the School of Theoretical Physics at the Dublin Institute for Advanced Studies in Ireland, where he succeeded Schrödinger and stayed until 1968. In 1956 Lanczos published Applied Analysis.
The topics covered include "algebraic equations and eigenvalue problems, large scale linear systems, harmonic analysis, data analysis and power expansions...illustrated by numerical examples worked out in detail." The contents of the book are stylized "parexic analysis lies between classical analysis and numerical analysis: it is the theory of approximation by finite algorithms." Lanczos did pioneering work along with G. C. Danielson on what is now called the fast Fourier transform, but the significance of his discovery was not appreciated at the time, today the FFT is credited to Cooley and Tukey.. Lanczos was the one, he discovered the diagonalizable matrix. Working in Washington DC at the U. S. National Bureau of Standards after 1949, Lanczos developed a number of techniques for mathematical calculations using digital computers, including: the Lanczos algorithm for finding eigenvalues of large symmetric matrices, the Lanczos approximation for the gamma function, the conjugate gradient method for solving systems of linear equations.
In 1962, Lanczos showed that the Weyl tensor, which plays a fundamental role in general relativity, can be obtained from a tensor potential, now called the Lanczos potential. Lanczos resampling is based on a windowed sinc function as a practical upsampling filter approximating the ideal sinc function. Lanczos resampling is used in video up-sampling for digital zoom applications and image scaling. Books such as The Variational Principles of Mechanics show his explanatory ability and enthusiasm as a physics teacher. 1956: Applied Analysis, Prentice Hall 1961: Linear Differential Operators, Van Nostrand Company, ISBN 048665656X 1966: Albert Einstein and the cosmic world order: six lectures delivered at the University of Michigan in the Spring of 1962, Interscience Publishers 1966: Discourse on Fourier Series, Oliver & Boyd 1968: Numbers without End, Edinburgh: Oliver & Boyd 1970: Judaism and Science, Leeds University Press ISBN 085316021X 1970: The Variational Principles of Mechanics, University of Toronto Press ISBN 0-8020-1743-6, fourth edition 1974 Dover paperback 1974: The Einstein Decade, Granada Publishing ISBN 0236176323 1979: Space through the Ages: the Evolution of the geometric Ideas from Pythagoras to Hilbert and Einstein, Academic Press ISBN 0124358500 1998: Cornelius Lanczos: Collected Published Papers with Commentaries, North Carolina State University ISBN 0-929493-01-X 1924: "Über eine stationäre Kosmologie im Sinne der Einsteinischen Gravitationstheorie", Zeitschrift für Physik 21: 73 doi:10.1007/BF01328251 1962: "The splitting of the Riemann tensor", Reviews of Modern Physics 34: 379 doi:10.1103/RevModPhys.34.379 The Martians Brendan Scaife.
Studies in Numerical Analysis: Papers in Honour of Cornelius Lanczos. Dublin. ISBN 0-12-621150-7. O'Connor, John J.. Cornelius Lanczos at the Mathematics Genealogy Project Cornelius Lanczos, Collected published papers with commentaries, published by North Carolina State University Photo gallery of Lanczos by Nicholas Higham Series of historic video tapes p
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis. Numerical analysis finds application in all fields of engineering and the physical sciences, but in the 21st century the life sciences, social sciences, medicine and the arts have adopted elements of scientific computations; as an aspect of mathematics and computer science that generates and implements algorithms, the growth in power and the revolution in computing has raised the use of realistic mathematical models in science and engineering, complex numerical analysis is required to provide solutions to these more involved models of the world. Ordinary differential equations appear in celestial mechanics. Before the advent of modern computers, numerical methods depended on hand interpolation in large printed tables. Since the mid 20th century, computers calculate the required functions instead; these same interpolation formulas continue to be used as part of the software algorithms for solving differential equations.
One of the earliest mathematical writings is a Babylonian tablet from the Yale Babylonian Collection, which gives a sexagesimal numerical approximation of the square root of 2, the length of the diagonal in a unit square. Being able to compute the sides of a triangle is important, for instance, in astronomy and construction. Numerical analysis continues this long tradition of practical mathematical calculations. Much like the Babylonian approximation of the square root of 2, modern numerical analysis does not seek exact answers, because exact answers are impossible to obtain in practice. Instead, much of numerical analysis is concerned with obtaining approximate solutions while maintaining reasonable bounds on errors; the overall goal of the field of numerical analysis is the design and analysis of techniques to give approximate but accurate solutions to hard problems, the variety of, suggested by the following: Advanced numerical methods are essential in making numerical weather prediction feasible.
Computing the trajectory of a spacecraft requires the accurate numerical solution of a system of ordinary differential equations. Car companies can improve the crash safety of their vehicles by using computer simulations of car crashes; such simulations consist of solving partial differential equations numerically. Hedge funds use tools from all fields of numerical analysis to attempt to calculate the value of stocks and derivatives more than other market participants. Airlines use sophisticated optimization algorithms to decide ticket prices and crew assignments and fuel needs; such algorithms were developed within the overlapping field of operations research. Insurance companies use numerical programs for actuarial analysis; the rest of this section outlines several important themes of numerical analysis. The field of numerical analysis predates the invention of modern computers by many centuries. Linear interpolation was in use more than 2000 years ago. Many great mathematicians of the past were preoccupied by numerical analysis, as is obvious from the names of important algorithms like Newton's method, Lagrange interpolation polynomial, Gaussian elimination, or Euler's method.
To facilitate computations by hand, large books were produced with formulas and tables of data such as interpolation points and function coefficients. Using these tables calculated out to 16 decimal places or more for some functions, one could look up values to plug into the formulas given and achieve good numerical estimates of some functions; the canonical work in the field is the NIST publication edited by Abramowitz and Stegun, a 1000-plus page book of a large number of used formulas and functions and their values at many points. The function values are no longer useful when a computer is available, but the large listing of formulas can still be handy; the mechanical calculator was developed as a tool for hand computation. These calculators evolved into electronic computers in the 1940s, it was found that these computers were useful for administrative purposes, but the invention of the computer influenced the field of numerical analysis, since now longer and more complicated calculations could be done.
Direct methods compute the solution to a problem in a finite number of steps. These methods would give the precise answer. Examples include Gaussian elimination, the QR factorization method for solving systems of linear equations, the simplex method of linear programming. In practice, finite precision is used and the result is an approximation of the true solution. In contrast to direct methods, iterative methods are not expected to terminate in a finite number of steps. Starting from an initial guess, iterative methods form successive approximations that converge to the exact solution only in the limit. A convergence test involving the residual, is specified in order to decide when a sufficiently accurate solution has been found. Using infinite precision arithmetic these methods would not reach the solution within a finite number of steps. Examples include Newton's method, the bisection method, Jacobi iteration. In computational matrix algebra, iterative methods are generall
ArXiv is a repository of electronic preprints approved for posting after moderation, but not full peer review. It consists of scientific papers in the fields of mathematics, astronomy, electrical engineering, computer science, quantitative biology, mathematical finance and economics, which can be accessed online. In many fields of mathematics and physics all scientific papers are self-archived on the arXiv repository. Begun on August 14, 1991, arXiv.org passed the half-million-article milestone on October 3, 2008, had hit a million by the end of 2014. By October 2016 the submission rate had grown to more than 10,000 per month. ArXiv was made possible by the compact TeX file format, which allowed scientific papers to be transmitted over the Internet and rendered client-side. Around 1990, Joanne Cohn began emailing physics preprints to colleagues as TeX files, but the number of papers being sent soon filled mailboxes to capacity. Paul Ginsparg recognized the need for central storage, in August 1991 he created a central repository mailbox stored at the Los Alamos National Laboratory which could be accessed from any computer.
Additional modes of access were soon added: FTP in 1991, Gopher in 1992, the World Wide Web in 1993. The term e-print was adopted to describe the articles, it began as a physics archive, called the LANL preprint archive, but soon expanded to include astronomy, computer science, quantitative biology and, most statistics. Its original domain name was xxx.lanl.gov. Due to LANL's lack of interest in the expanding technology, in 2001 Ginsparg changed institutions to Cornell University and changed the name of the repository to arXiv.org. It is now hosted principally with eight mirrors around the world, its existence was one of the precipitating factors that led to the current movement in scientific publishing known as open access. Mathematicians and scientists upload their papers to arXiv.org for worldwide access and sometimes for reviews before they are published in peer-reviewed journals. Ginsparg was awarded a MacArthur Fellowship in 2002 for his establishment of arXiv; the annual budget for arXiv is $826,000 for 2013 to 2017, funded jointly by Cornell University Library, the Simons Foundation and annual fee income from member institutions.
This model arose in 2010, when Cornell sought to broaden the financial funding of the project by asking institutions to make annual voluntary contributions based on the amount of download usage by each institution. Each member institution pledges a five-year funding commitment to support arXiv. Based on institutional usage ranking, the annual fees are set in four tiers from $1,000 to $4,400. Cornell's goal is to raise at least $504,000 per year through membership fees generated by 220 institutions. In September 2011, Cornell University Library took overall administrative and financial responsibility for arXiv's operation and development. Ginsparg was quoted in the Chronicle of Higher Education as saying it "was supposed to be a three-hour tour, not a life sentence". However, Ginsparg remains on the arXiv Scientific Advisory Board and on the arXiv Physics Advisory Committee. Although arXiv is not peer reviewed, a collection of moderators for each area review the submissions; the lists of moderators for many sections of arXiv are publicly available, but moderators for most of the physics sections remain unlisted.
Additionally, an "endorsement" system was introduced in 2004 as part of an effort to ensure content is relevant and of interest to current research in the specified disciplines. Under the system, for categories that use it, an author must be endorsed by an established arXiv author before being allowed to submit papers to those categories. Endorsers are not asked to review the paper for errors, but to check whether the paper is appropriate for the intended subject area. New authors from recognized academic institutions receive automatic endorsement, which in practice means that they do not need to deal with the endorsement system at all. However, the endorsement system has attracted criticism for restricting scientific inquiry. A majority of the e-prints are submitted to journals for publication, but some work, including some influential papers, remain purely as e-prints and are never published in a peer-reviewed journal. A well-known example of the latter is an outline of a proof of Thurston's geometrization conjecture, including the Poincaré conjecture as a particular case, uploaded by Grigori Perelman in November 2002.
Perelman appears content to forgo the traditional peer-reviewed journal process, stating: "If anybody is interested in my way of solving the problem, it's all there – let them go and read about it". Despite this non-traditional method of publication, other mathematicians recognized this work by offering the Fields Medal and Clay Mathematics Millennium Prizes to Perelman, both of which he refused. Papers can be submitted in any of several formats, including LaTeX, PDF printed from a word processor other than TeX or LaTeX; the submission is rejected by the arXiv software if generating the final PDF file fails, if any image file is too large, or if the total size of the submission is too large. ArXiv now allows one to store and modify an incomplete submission, only finalize the submission when ready; the time stamp on the article is set. The standard access route is through one of several mirrors. Sev
Eduard L. Stiefel was a Swiss mathematician. Together with Cornelius Lanczos and Magnus Hestenes, he invented the conjugate gradient method, gave what is now understood to be a partial construction of the Stiefel–Whitney classes of a real vector bundle, thus co-founding the study of characteristic classes. Stiefel entered the Swiss Federal Institute of Technology in 1928, he received his Ph. D. in 1935 under Heinz Hopf. Stiefel completed his habilitation in 1942. Besides his academic pursuits, Stiefel was active as a military officer, rising to the rank of colonel in the Swiss army during World War II. Stiefel achieved his full professorship at ETH Zurich in 1948, the same year he founded the Institute for Applied Mathematics; the objective of the new institute was to construct an electronic computer. He spent a year in the United States commencing in August, 1951. During this time, he met Magnus Hestenes and many other scientists at the National Bureau of Standards and these professional associations served him well during the remainder of his career at Zurich.
Stiefel manifold Stiefel–Whitney class Hestenes, Magnus R.. "Methods of Conjugate Gradients for Solving Linear Systems". Journal of Research of the National Bureau of Standards. 49: 409. Doi:10.6028/jres.049.044. How Professor Eduard Stiefel Got to NBS-INA-UCLA in August 1951 John Todd's lecture about his association with Eduard Stiefel. Eduard L. Stiefel at the Mathematics Genealogy Project Numerical Analysis in Zurich – 50 Years Ago by Martin H. Gutknecht of ETH Zurich. "Direction fields and teleparallelism in n-dimensional manifolds," translation of thesis by D. H. Delphenich