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List Of Numerical Analysis Topics
This is a list of numerical analysis topics. General *Validated numerics * Iterative method *Rate of convergence — the speed at which a convergent sequence approaches its limit **Order of accuracy — rate at which numerical solution of differential equation converges to exact solution * Series acceleration — methods to accelerate the speed of convergence of a series **Aitken's delta-squared process — most useful for linearly converging sequences **Minimum polynomial extrapolation — for vector sequences **Richardson extrapolation **Shanks transformation — similar to Aitken's delta-squared process, but applied to the partial sums **Van Wijngaarden transformation — for accelerating the convergence of an alternating series *Abramowitz and Stegun — book containing formulas and tables of many special functions **Digital Library of Mathematical Functions — successor of book by Abramowitz and Stegun *Curse of dimensionality *Local convergence and global convergence — whet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Difference Quotient
In single-variable calculus, the difference quotient is usually the name for the expression : \frac which when taken to the limit as ''h'' approaches 0 gives the derivative of the function ''f''. The name of the expression stems from the fact that it is the quotient of the difference of values of the function by the difference of the corresponding values of its argument (the latter is (''x'' + ''h'') - ''x'' = ''h'' in this case). The difference quotient is a measure of the average rate of change of the function over an interval (in this case, an interval of length ''h''). The limit of the difference quotient (i.e., the derivative) is thus the instantaneous rate of change. By a slight change in notation (and viewpoint), for an interval 'a'', ''b'' the difference quotient : \frac is called the mean (or average) value of the derivative of ''f'' over the interval 'a'', ''b'' This name is justified by the mean value theorem, which states that for a differentiable function ''f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ABS Methods
ABS methods, where the acronym contains the initials of Jozsef Abaffy, Charles G. Broyden and Emilio Spedicato, have been developed since 1981 to generate a large class of algorithms for the following applications: * solution of general linear algebraic systems, determined or underdetermined, * full or deficient rank; * solution of linear Diophantine systems, i.e. equation systems where the coefficient matrix and the right hand side are integer valued and an integer solution is sought; this is a special but important case of Hilbert's tenth problem, the only one in practice soluble; * solution of nonlinear algebraic equations; * solution of continuous unconstrained or constrained optimization. At the beginning of 2007 ABS literature consisted of over 400 papers and reports and two monographs, one due to Abaffy and Spedicato and published in 1989, one due to Xia and Zhang and published, in Chinese, in 1998. Moreover three conferences had been organized in China. Research on ABS ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sinc Numerical Methods
In numerical analysis and applied mathematics, sinc numerical methods are numerical techniques for finding approximate solutions of partial differential equations and integral equations based on the translates of sinc function and Cardinal function C(f,h) which is an expansion of f defined by :C(f,h)(x)=\sum_^\infty f(kh) \, \textrm \left(\dfrac-k \right) where the step size h>0 and where the sinc function is defined by :\textrm(x)=\frac Sinc approximation methods excel for problems whose solutions may have singularities, or infinite domains, or boundary layers. The truncated Sinc expansion of f is defined by the following series: : C_(f,h)(x) = \displaystyle \sum_^ f(kh) \, \textrm \left(\dfrac-k \right) . Sinc numerical methods cover *function approximation, *approximation of derivatives, *approximate definite and indefinite integration, *approximate solution of initial and boundary value ordinary differential equation (ODE) problems, *approximation and inversion of Fourier an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Level Set (data Structures)
In computer science a level set data structure is designed to represent discretely sampled dynamic level sets functions. A common use of this form of data structure is in efficient image rendering. The underlying method constructs a signed distance field that extends from the boundary, and can be used to solve the motion of the boundary in this field. Chronological developments The powerful level-set method is due to Osher and Sethian 1988.Osher, S. & Sethian, J. A. 1988. "Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations". ''Journal of Computation Physics'' 79:12–49. However, the straightforward implementation via a dense d-dimensional array of values, results in both time and storage complexity of O(n^d), where n is the cross sectional resolution of the spatial extents of the domain and d is the number of spatial dimensions of the domain. Narrow band The narrow band level set method, introduced in 1995 by Adalsteinsson and Se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Level-set Method
Level-set methods (LSM) are a conceptual framework for using level sets as a tool for numerical analysis of surfaces and shapes. The advantage of the level-set model is that one can perform numerical computations involving curves and surfaces on a fixed Cartesian grid without having to parameterize these objects (this is called the ''Eulerian approach''). Also, the level-set method makes it very easy to follow shapes that change topology, for example, when a shape splits in two, develops holes, or the reverse of these operations. All these make the level-set method a great tool for modeling time-varying objects, like inflation of an airbag, or a drop of oil floating in water. The figure on the right illustrates several important ideas about the level-set method. In the upper-left corner we see a shape; that is, a bounded region with a well-behaved boundary. Below it, the red surface is the graph of a level set function \varphi determining this shape, and the flat blue region r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Collocation Method
In mathematics, a collocation method is a method for the numerical solution of ordinary differential equations, partial differential equations and integral equations. The idea is to choose a finite-dimensional space of candidate solutions (usually polynomials up to a certain degree) and a number of points in the domain (called ''collocation points''), and to select that solution which satisfies the given equation at the collocation points. Ordinary differential equations Suppose that the ordinary differential equation : y'(t) = f(t,y(t)), \quad y(t_0)=y_0, is to be solved over the interval _0,t_0+c_k h/math>. Choose c_k from 0 ≤ ''c''1< ''c''2< … < ''c''''n'' ≤ 1. The corresponding (polynomial) collocation method approximates the solution ''y'' by the polynomial ''p'' of degree ''n'' which satisfies the initial condition , and the differential equation at all ''collocation points' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Timeline Of Numerical Analysis After 1945
The following is a timeline of numerical analysis after 1945, and deals with developments after the invention of the modern electronic computer, which began during Second World War. For a fuller history of the subject before this period, see timeline and history of mathematics. 1940s * Monte Carlo simulation (voted one of the top 10 algorithms of the 20th century) invented at Los Alamos by von Neumann, Ulam and Metropolis. * Crank–Nicolson method was developed by Crank and Nicolson. * Dantzig introduces the simplex method (voted one of the top 10 algorithms of the 20th century) in 1947. * Turing formulated the LU decomposition method. 1950s * Successive over-relaxation was devised simultaneously by D.M. Young, Jr. and by H. Frankel in 1950. * Hestenes, Stiefel, and Lanczos, all from the Institute for Numerical Analysis at the National Bureau of Standards, initiate the development of Krylov subspace iteration methods. Voted one of the top 10 algorithms of the 20th century. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
International Workshops On Lattice QCD And Numerical Analysis
International is an adjective (also used as a noun) meaning "between nations". International may also refer to: Music Albums * ''International'' (Kevin Michael album), 2011 * ''International'' (New Order album), 2002 * ''International'' (The Three Degrees album), 1975 *''International'', 2018 album by L'Algérino Songs * The Internationale, the left-wing anthem * "International" (Chase & Status song), 2014 * "International", by Adventures in Stereo from ''Monomania'', 2000 * "International", by Brass Construction from ''Renegades'', 1984 * "International", by Thomas Leer from ''The Scale of Ten'', 1985 * "International", by Kevin Michael from ''International'' (Kevin Michael album), 2011 * "International", by McGuinness Flint from ''McGuinness Flint'', 1970 * "International", by Orchestral Manoeuvres in the Dark from '' Dazzle Ships'', 1983 * "International (Serious)", by Estelle from '' All of Me'', 2012 Politics * Political international, any transnational organization of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nick Trefethen
Lloyd Nicholas Trefethen (born 30 August 1955) is an American mathematician, professor of numerical analysis and head of the Numerical Analysis Group at the Mathematical Institute, University of Oxford. Education Trefethen was born 30 August 1955 in Boston, Massachusetts, the son of mechanical engineer Lloyd M. Trefethen and codebreaker, poet, teacher and editor Florence Newman Trefethen. He obtained his bachelor's degree from Harvard University in 1977 and his master's from Stanford University in 1980. His PhD was on ''Wave Propagation and Stability for Finite Difference Schemes'' supervised by Joseph E. Oliger at Stanford University. Career and research Following his PhD, Trefethen went on to work at the Courant Institute of Mathematical Sciences in New York, Massachusetts Institute of Technology, and Cornell University, before being appointed to a chair at the University of Oxford and a Fellowship of Balliol College, Oxford. , he has published around 150 journal papers sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hundred-dollar, Hundred-digit Challenge Problems
The Hundred-dollar, Hundred-digit Challenge problems are 10 problems in numerical mathematics published in 2002 by . A $100 prize was offered to whoever produced the most accurate solutions, measured up to 10 significant digits. The deadline for the contest was May 20, 2002. In the end, 20 teams solved all of the problems perfectly within the required precision, and an anonymous donor aided in producing the required prize monies. The challenge and its solutions were described in detail in the book . The problems From : # \lim_\int_\varepsilon^1 x^ \cos\left(x^ \log x\right)\,dx # A photon moving at speed 1 in the ''xy''-plane starts at ''t'' = 0 at (''x'', ''y'') = (0.5, 0.1) heading due east. Around every integer lattice point (''i'', ''j'') in the plane, a circular mirror of radius 1/3 has been erected. How far from the origin is the photon at ''t'' = 10? # The infinite matrix ''A'' with entries a_=1, a_=1/2, a_=1/3, a_=1/4, a_=1/5, a_=1/6, \dots is a bounded operator on \ell^2. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
History Of Numerical Solution Of Differential Equations Using Computers
Differential equations, in particular Euler equations, rose in prominence during World War II in calculating the accurate trajectory of ballistics, both rocket-propelled and gun or cannon type projectiles. Originally, mathematicians used the simpler calculus of earlier centuries to determine velocity, thrust, elevation, curve, distance, and other parameters. New weapons, however, such as Germany's giant cannons, the " Paris Gun" (Encyclopedia Astronautica) and " Big Bertha," and the V-2 rocket, meant that projectiles would travel hundreds of miles in distance and dozens of miles in height, in all weathers. As a result, variables such as diminished wind resistance in thin atmospheres and changes in gravitational pull reduced accuracy using the historic methodology. There was the additional problem of planes that could now fly hundreds of miles an hour. Differential equations were applied to stochastic processes. Developing machines that could speed up human calculation of differenti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |