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Gauss–Jordan Elimination
In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used to compute the rank of a matrix, the determinant of a square matrix, and the inverse of an invertible matrix. The method is named after Carl Friedrich Gauss (1777–1855) although some special cases of the method—albeit presented without proof—were known to Chinese mathematicians as early as circa 179 AD. To perform row reduction on a matrix, one uses a sequence of elementary row operations to modify the matrix until the lower left-hand corner of the matrix is filled with zeros, as much as possible. There are three types of elementary row operations: * Swapping two rows, * Multiplying a row by a nonzero number, * Adding a multiple of one row to another row. (subtraction can be achieved by multiplying one row with -1 and adding ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can perform automated deductions (referred to as automated reasoning) and use mathematical and logical tests to divert the code execution through various routes (referred to as automated decision-making). Using human characteristics as descriptors of machines in metaphorical ways was already practiced by Alan Turing with terms such as "memory", "search" and "stimulus". In contrast, a Heuristic (computer science), 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. As an effective method, an algorithm ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scalar (mathematics)
A scalar is an element of a field which is used to define a ''vector space''. In linear algebra, real numbers or generally elements of a field are called scalars and relate to vectors in an associated vector space through the operation of scalar multiplication (defined in the vector space), in which a vector can be multiplied by a scalar in the defined way to produce another vector. Generally speaking, a vector space may be defined by using any field instead of real numbers (such as complex numbers). Then scalars of that vector space will be elements of the associated field (such as complex numbers). A scalar product operation – not to be confused with scalar multiplication – may be defined on a vector space, allowing two vectors to be multiplied in the defined way to produce a scalar. A vector space equipped with a scalar product is called an inner product space. A quantity described by multiple scalars, such as having both direction and magnitude, is called a '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
O Notation
Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Landau, and others, collectively called Bachmann–Landau notation or asymptotic notation. The letter O was chosen by Bachmann to stand for ''Ordnung'', meaning the order of approximation. In computer science, big O notation is used to classify algorithms according to how their run time or space requirements grow as the input size grows. In analytic number theory, big O notation is often used to express a bound on the difference between an arithmetical function and a better understood approximation; a famous example of such a difference is the remainder term in the prime number theorem. Big O notation is also used in many other fields to provide similar estimates. Big O notation characterizes functions according to their growth rates: diff ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Monthly
''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America. The ''American Mathematical Monthly'' is an expository journal intended for a wide audience of mathematicians, from undergraduate students to research professionals. Articles are chosen on the basis of their broad interest and reviewed and edited for quality of exposition as well as content. In this the ''American Mathematical Monthly'' fulfills a different role from that of typical mathematical research journals. The ''American Mathematical Monthly'' is the most widely read mathematics journal in the world according to records on JSTOR. Tables of contents with article abstracts from 1997–2010 are availablonline The MAA gives the Lester R. Ford Awards annually to "authors of articles of expository excellence" published in the ''American Mathematical Monthly''. Editors *2022– ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wilhelm Jordan (geodesist)
Wilhelm Jordan ( 1 March 1842, Ellwangen, Württemberg – 17 April 1899, Hanover) was a German geodesist who conducted surveys in Germany and Africa and founded the German geodesy journal. Biography Jordan was born in Ellwangen, a small town in southern Germany. He studied at the polytechnic institute in Stuttgart and after working for two years as an engineering assistant on the preliminary stages of railway construction he returned there as an assistant in geodesy. In 1868, when he was 26 years old, he was appointed a full professor at Karlsruhe. In 1874 Jordan took part in the expedition of Friedrich Gerhard Rohlfs to Libya. From 1881 until his death he was professor of geodesy and practical geometry at the Technical University of Hannover. He was a prolific writer and his best known work was his ''Handbuch der Vermessungskunde'' (''Handbook of Geodesy''). He is remembered among mathematicians for the Gauss–Jordan elimination algorithm, with Jordan improving the stability ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Human Computer
The term "computer", in use from the early 17th century (the first known written reference dates from 1613), meant "one who computes": a person performing mathematical calculations, before electronic computers became commercially available. Alan Turing described the "human computer" as someone who is "supposed to be following fixed rules; he has no authority to deviate from them in any detail." Teams of people, often women from the late nineteenth century onwards, were used to undertake long and often tedious calculations; the work was divided so that this could be done in parallel. The same calculations were frequently performed independently by separate teams to check the correctness of the results. Since the end of the 20th century, the term "human computer" has also been applied to individuals with prodigious powers of mental arithmetic, also known as mental calculators. Origins in sciences Astronomers in Renaissance times used that term about as often as they called themse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a "natural philosopher"), widely recognised as one of the greatest mathematicians and physicists and among the most influential scientists of all time. He was a key figure in the philosophical revolution known as the Enlightenment. His book (''Mathematical Principles of Natural Philosophy''), first published in 1687, established classical mechanics. Newton also made seminal contributions to optics, and shares credit with German mathematician Gottfried Wilhelm Leibniz for developing infinitesimal calculus. In the , Newton formulated the laws of motion and universal gravitation that formed the dominant scientific viewpoint for centuries until it was superseded by the theory of relativity. Newton used his mathematical description of gravity to derive Kepler's laws of planetary motion, account for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Liu Hui
Liu Hui () was a Chinese mathematician who published a commentary in 263 CE on ''Jiu Zhang Suan Shu (The Nine Chapters on the Mathematical Art).'' He was a descendant of the Marquis of Zixiang of the Eastern Han dynasty and lived in the state of Cao Wei during the Three Kingdoms period (220-280 CE) of China. His major contributions as recorded in his commentary on ''The Nine Chapters on the Mathematical Art'' include a proof of the Pythagorean theorem, theorems in solid geometry, an improvement on Archimedes's approximation of , and a systematic method of solving linear equations in several unknowns. In his other work, ''Haidao Suanjing (The Sea Island Mathematical Manual)'', he wrote about geometrical problems and their application to surveying. He probably visited Luoyang, where he measured the sun's shadow. Mathematical work Liu Hui expressed mathematical results in the form of decimal fractions that utilized metrological units (i.e., related units of length with base 10 s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Nine Chapters On The Mathematical Art
''The Nine Chapters on the Mathematical Art'' () is a Chinese mathematics book, composed by several generations of scholars from the 10th–2nd century BCE, its latest stage being from the 2nd century CE. This book is one of the earliest surviving mathematical texts from China, the first being '' Suan shu shu'' (202 BCE – 186 BCE) and ''Zhoubi Suanjing'' (compiled throughout the Han until the late 2nd century CE). It lays out an approach to mathematics that centres on finding the most general methods of solving problems, which may be contrasted with the approach common to ancient Greek mathematicians, who tended to deduce propositions from an initial set of axioms. Entries in the book usually take the form of a statement of a problem, followed by the statement of the solution and an explanation of the procedure that led to the solution. These were commented on by Liu Hui in the 3rd century. History Original book The full title of ''The Nine Chapters on the Mathemat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangular Form
In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called if all the entries ''above'' the main diagonal are zero. Similarly, a square matrix is called if all the entries ''below'' the main diagonal are zero. Because matrix equations with triangular matrices are easier to solve, they are very important in numerical analysis. By the LU decomposition algorithm, an invertible matrix may be written as the product of a lower triangular matrix ''L'' and an upper triangular matrix ''U'' if and only if all its leading principal minors are non-zero. Description A matrix of the form :L = \begin \ell_ & & & & 0 \\ \ell_ & \ell_ & & & \\ \ell_ & \ell_ & \ddots & & \\ \vdots & \vdots & \ddots & \ddots & \\ \ell_ & \ell_ & \ldots & \ell_ & \ell_ \end is called a lower triangular matrix or left triangular matrix, and a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Augmented Matrix
In linear algebra, an augmented matrix is a matrix obtained by appending the columns of two given matrices, usually for the purpose of performing the same elementary row operations on each of the given matrices. Given the matrices and , where A = \begin 1 & 3 & 2 \\ 2 & 0 & 1 \\ 5 & 2 & 2 \end , \quad B = \begin 4 \\ 3 \\ 1 \end, the augmented matrix (''A'', ''B'') is written as (A, B) = \left begin 1 & 3 & 2 & 4 \\ 2 & 0 & 1 & 3 \\ 5 & 2 & 2 & 1 \end\right This is useful when solving systems of linear equations. For a given number of unknowns, the number of solutions to a system of linear equations depends only on the rank of the matrix representing the system and the rank of the corresponding augmented matrix. Specifically, according to the Rouché–Capelli theorem, any system of linear equations is inconsistent (has no solutions) if the rank of the augmented matrix is greater than the rank of the coefficient matrix; if, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
