In computer science and operations research , approximation algorithms are algorithm s used to find approximate solutions to optimization problem s. Approximation algorithms are often associated with NP ... expensive due to the input size. A typical example for an approximationalgorithm is the one ... twice as large as the optimal one. This is a constant factor approximationalgorithm with a factor ... of the optimum result. For example, in the case of a approximationalgorithm A it has been ... the relative performance guarantee . An approximationalgorithm has an absolute performance guarantee ... algorithm and has an approximation ratio of r n . Likewise, a problem with an r n approximationalgorithm is said to be r n approximable or have an approximation ratio of r n . ref name ausiello99complexity ... performance guarantee math Rho A math of some approximationalgorithm A , where x refers ... problem of c min c means that the algorithm has an approximation ratio of c for arbitrary ... scheme . An term may appear when an approximationalgorithm introduces a multiplicative error ... wants provable solution quality and provable run time bounds. Ideally, the approximation is optimal up to a small constant factor for instance within 5 of the optimal solution . Approximation algorithms ... be approximated within any factor greater than 1 such a family of approximation algorithms is often called a polynomial time approximation scheme or PTAS . Others are impossible to approximate within ... approximation algorithms emerge from the linear programming relaxation of the integer program. Not all approximation algorithms are suitable for all practical applications. They often use IP LP semidefinite ... which lead to difficult implementation problems. Also, some approximation algorithms have ... the ideas into a linear time algorithm. Such algorithms are also worthwhile in some applications ..., it has now been shown that Johnson s 1974 approximation algorithms for Max SAT, Set Cover, Independent ... more details
Context date October 2009 Polynomial expansions such as the Taylor series expansion are often convenient for theoretical work but less useful for practical applications. For practical work it is often desirable to minimize the maximum absolute or relative error of a polynomial fit for any given number of terms in an effort to reduce computational expense of repeated evaluation. Algorithms that minimize the maximum error are known as Minimax approximation algorithms . One popular approach is the Remez algorithm . External links http mathworld.wolfram.com MinimaxApproximation.html Minimax approximation algorithm at MathWorld http www.liftminds.com lesson 74 Polynomial approximations and series polynomial approximation Polynomial approximations and series polynomial approximation Category Numerical analysis math stub ... more details
Certainty Unreferenced date September 2007 An approximation is an Accuracy and precision inexact representation of something that is still close enough to be useful. Although approximation is most often ... representation is known, an approximation may yield a sufficiently accurate solution while reducing ... the convergence rate of evolutionary algorithms, fitness approximation that leads to build model of the fitness function to choose smart search steps is a good solution. The type of approximation ... to this data, and the savings usually in time and effort that can be achieved by approximation ... laws theory and empirical measurement s, which are constantly compared to one another. The approximation ... . Mathematics class wikitable align right Symbols representing approximation style font size xx large & 8784 general approximation style font size xx large asymptotic analysis Approximation usually occurs ... using a small number of Significant figures significant digits . Approximation theory is a branch of mathematics, a quantitative part of functional analysis . Diophantine approximation deals with approximations of real number s by rational number s. Related to approximation of functions is the Asymptotic ... key and typing X. See also col begin col break Approximation error Congruence relation Estimation Fitness approximation Fermi estimate Linear approximation Newton s method col break Numerical analysis Orders of approximation Runge Kutta methods Successive Approximation ADC Taylor series Least squares col break Wiktionarypar approximation col end Category Approximations Category Numerical analysis br Tostadur bg cs Aproximace da Approksimation de Approximation es Aproximaci n eo Proksimuma kalkulado fr Approximation ko hi is N mundun it Approssimazione he nl Benadering ja pl Aproksymacja pt Aproxima o ru simple Approximation fi Approksimaatio sv Approximation uk ... more details
algorithm s seek an approximation that is close to the true solution. Approximation may use either ... element will be decreased as the algorithm settles down to a solution. Approximation algorithms are those ... Flow chart of an algorithm Euclid s algorithm for calculating the greatest common divisor g.c.d. of two numbers a and b in locations named A and B. The algorithm proceeds by successive subtractions ... than or equal to the number a in location A THEN the algorithm specifies B B A meaning the number b ..., yielding the g.c.d. in A. Algorithm derived from Scott 2009 13 symbols and drawing style from Tausworthe 1977 . In mathematics and computer science , an algorithm IPAc en confirm icon en us algorithm.ogg ... algorithm, for example, can be described in a finite number of English words Rogers 1987 2 . ref of well defined instructions ref Well defined with respect to the agent that executes the algorithm ... Rogers 1987 2 . ref for calculating a Function mathematics function . ref an algorithm ... and initial input perhaps null , ref An algorithm has zero or more inputs, i.e., quantity quantities which are given to it initially before the algorithm begins Knuth 1973 5 . ref the instructions ... all the characteristics of an algorithm except that it possibly lacks finiteness may be called a computational ... output ref An algorithm has one or more outputs, i.e. quantities which have a specified relation to the inputs ... is an algorithm is debatable. Rogers opines that a computation is carried out in a discrete stepwise ... of the various points of view around the definition of algorithm see Algorithm characterizations . For examples of simple addition algorithms specified in the detailed manner described in Algorithm characterizations , see Algorithm examples . While there is no generally accepted formal definition of algorithm .... ref Stone 1973 4 ref For some people, a program is only an algorithm if it stops eventually for others, a program is only an algorithm if it stops before a given number of calculation steps. ref ... more details
In computer science , hardness of approximation is a field that studies the complexity of finding near optimal solutions to optimization problems. It complements the study of approximationalgorithm s by proving, for certain problems, a limit on the factors with which their solution can be efficiently approximated. Typically such limits show a factor of approximation beyond which a problem becomes NP hard, implying that finding a polynomial time approximation for the problem is impossible unless P NP . Some hardness of approximation results, however, are based on other hypotheses, a notable one among which is the unique games conjecture . Since the early 1970s it was known that many optimization problems could not be solved in polynomial time unless NP P , but in many of these problems the optimal solution could be efficiently approximated to a certain degree. In the early 1990s, with the development of PCP complexity PCP theory, it became clear that there is a limit to the approximability of many of these optimization problems for many optimization problems there is a threshold beyond which they are NP hard to approximate. Hardness of approximation theory deals with studying the approximation threshold of such problems. Examples For an example of an NP hard optimization problem that is hard to approximate, see set cover . See also PCP theorem Further reading http www.cs.berkeley.edu luca pubs inapprox.ps Survey, a good starter by Luca Trevisan External links http www.cs.washington.edu education courses 533 05au CSE 533 The PCP Theorem and Hardness of Approximation, Autumn 2005 , syllabus from the University of Washington , Venkatesan Guruswami and Ryan O Donnell Category Computational complexity theory Category Mathematical optimization comp sci theory stub ... more details
Rational approximation may refer to Diophantine approximation , the approximation of real numbers by rational numbers. Pad approximation , the approximation of functions by rational functions. See also Simple rational approximation Mathdab ... more details
Stochastic approximation methods are a family of iterative stochastic optimization algorithm s that attempt to find zeroes or extrema of functions which cannot be computed directly, but only estimated via noisy observations. The first, and prototypical, algorithms of this kind were the Robbins Monro and Kiefer Wolfowitz algorithms. NOTOC Robbins Monro algorithm In the Herbert Robbins Robbins Monro algorithm, introduced in 1951 ref name rm A Stochastic Approximation Method, Herbert Robbins and Sutton Monro, Annals of Mathematical Statistics 22 , 3 September 1951 , pp. 400 407. ref , one has a function math M x math for which one wishes to find the value of math x math , math x 0 math , satisfying math M x 0 alpha math . However, what is observable is not math M x math , but rather a random variable math N x math such that math E N x x M x math . The algorithm is then to construct a sequence math x 1, x 2, dots math which satisfies math x n 1 x n a n alpha N x n math . Here, math a 1, a 2, dots math is a sequence of positive step sizes. Herbert Robbins Robbins and Monro proved ref name ... , math a n math must converge slowly. Kiefer Wolfowitz algorithm In the Kiefer Wolfowitz algorithm ... a year after the Robbins Monro algorithm, one wishes to find the maximum, math x 0 math ... step sizes which serve the same function as in the Robbins Monro algorithm, and math c 1, c 2, dots ... models, and so on. ref name kushneryin Stochastic Approximation Algorithms and Applications , Harold ... Approximation and Recursive Algorithms and Applications , 2003, ISBN 0387008942. ref ref Stochastic Approximation and Recursive Estimation , Mikhail Borisovich Nevel son and Rafail Zalmanovich Has ... and R. Douglas Martin were the first to use stochastc approximation in 1975 when dealing with Robust ... approximation . IEEE Trans. Inform. Theory, 21 pp.263 271 1975 . ref See also Stochastic gradient descent Stochastic optimization Simultaneous perturbation stochastic approximation References reflist ... more details
The approximation error in some data is the discrepancy between an exact value and some approximation to it. An approximation error can occur because the measurement of the data is not precise due to the instruments , or approximations are used instead of the real data e.g., 3.14 instead of pi . In the mathematics mathematical field of numerical analysis , the numerical stability of an algorithm in numerical analysis indicates how the error is propagated by the algorithm. Overview One commonly distinguishes between the relative error and the absolute error . The absolute error is the absolute value magnitude of the difference between the exact value and the approximation. The relative error is the absolute error divided by the magnitude of the exact value. The percent error is the relative error expressed in terms of per 100. As an example, if the exact value is 50 and the approximation is 49.9, then the absolute error is 0.1 and the relative error is 0.1 50 0.002. The relative error is often used to compare approximations of numbers of widely differing size for example, approximating the number 1,000 with an absolute error of 3 is, in most applications, much worse than approximating the number 1,000,000 with an absolute error of 3 in the first case the relative error is .003 and in the second it is only .000003. Definitions Given some value v and its approximation v sub approx sub , the absolute error is math epsilon v v text approx , math where the vertical bars denote the absolute value . If math v ne 0, math the relative error is math eta frac v v text approx v left frac v v text approx v right , math and the percent error is math delta frac v v text approx v times 100 left frac v v text approx v right times 100. math These definitions can be extended to the case when math v math and math v text approx math are Euclidean vector n dimensional vectors , by replacing ... de Fehlerschranke es Error de aproximaci n eo Ekarto fr Erreur d approximation ko it Errore relativo ... more details
In mathematics , approximation theory is concerned with how function mathematics function s can best be approximation approximated with simpler function mathematics functions , and with Quantitative property quantitative ly characterization mathematics characterizing the approximation error errors introduced ... related topic is the approximation of functions by generalized Fourier series , that is, approximations ... or Rational function rational ratio of polynomials approximations. The objective is to make the approximation ... Image Logerror.png thumb 300px Error between optimal polynomial and log x red , and Chebyshev approximation ... optimal polynomial and exp x red , and Chebyshev approximation and exp x blue over the interval 1 ... approximation to f than P . In particular, Q is closer to f than P for each value x sub i sub where ... for a polynomial of degree N . Chebyshev approximation One can obtain polynomials very close to the optimal ... for the log function. Chebyshev approximation is the basis for Clenshaw Curtis quadrature , a numerical integration technique. Remez algorithm The Remez algorithm sometimes spelled Remes is used to produce ... algorithm that converges to a polynomial that has an error function with N 2 level extrema. By the theorem above, that polynomial is optimal. Remez algorithm uses the fact that one can construct ... math and math x n 2 math are presumably the end points of the interval of approximation , these equations ... of Remez algorithm, approximating e sup x sup over the interval 1, 1 . Vertical divisions are 10 sup ... be optimal. The second step of Remez algorithm consists of moving the test points to the approximate ... from looking at the graph that the point at 0.1 should have been at about 0.28. The way to do this in the algorithm ... must also be able to calculate the first and second derivatives of f x . Remez algorithm requires an ability .... The entire algorithm must be carried out to higher precision than the desired precision of the result ... the result converges to the desired accuracy. The algorithm converges very rapidly. Convergence is quadratic ... more details
The eikonal approximation is a method of approximation useful in wave scattering equations within the realms of quantum mechanics , optics , quantum electrodynamics , and partial wave expansion Informal description The main advantage the eikonal approximation offers is that the equations reduce to a differential equation in a single variable. This reduction into a single variable is the result of the straight line approximation or the eikonal approximation which allows us to choose the straight line as a special direction. Relation to the WKB approximation The early steps involved in the eikonal approximation in quantum mechanics are very closely related to the WKB approximation . The WKB approximation involves an expansion in terms of Planck s constant . It, like the eikonal approximation, reduces the equations into a differential equation in a single variable. But the difficulty with the WKB approximation is that this variable is described by the trajectory of the particle which, in general, is complicated. The advantage of the eikonal approximation is that the classical trajectory is a straight line. Formal description Empty section date July 2010 References http www.nhn.ou.edu shajesh eikonal sp.pdf Eikonal Approximation K. V. Shajesh Department of Physics and Astronomy, University of Oklahoma Category Mathematical analysis ... more details
Boussinesq approximation may refer to Boussinesq approximation buoyancy for buoyancy driven flows for small density differences in the fluid Boussinesq approximation water waves for long waves propagating on the surface of a fluid layer under the action of gravity Turbulence modeling and Viscosity Eddy viscosity eddy viscosity for the Boussinesq approximation resulting in the use of an eddy viscosity to model the turbulence Reynolds stresses disambig ... more details
unreferenced date March 2008 In science , engineering , and other quantitative disciplines, orders of approximation refer to formal or informal terms for how precise an approximation is, and to indicate progressively more refined approximations in increasing order of precision, a zeroth order approximation, a first order approximation, a second order approximation, and so forth. Formally, an n th order approximation is one where the order of magnitude of the error is at most math x n math , or in terms of big O notation , the error is math O x n . math In suitable circumstances, approximating a function by a Taylor polynomial of degree n yields an n th order approximation, by Taylor s theorem a first order approximation is a linear approximation , and so forth. The term is also used more loosely, as detailed below. Usage in science and engineering Zeroth order approximation also 0 sup th ... it has 3,914 people in actuality. This is also sometimes referred to as an order of magnitude approximation. A zeroth order approximation of a function mathematics function that is, mathematics mathematically ... approximation also 1 sup st sup order is the term scientists use for a further educated guess at an answer ... order approximation of a function that is, mathematically determining a formula to fit multiple data points will be a linear approximation , straight line with a slope a polynomial of degree 1. For example ... to the data, obtained by simply averaging the y values. Other methods for selecting a constant approximation can be used. Second order approximation also 2 sup nd sup order is the term scientists use ... is generally given. A second order approximation of a function that is, mathematically determining ... of reality, they are not typically referred to by number. A third order approximation ... about at the annual calibration. In this usage, the ordinality of the approximation is not exact, but is used .... See also Linearization Taylor approximation Category Numerical analysis ... more details
Refimprove date December 2009 In particle physics , the quenched approximation is an approximation often used in lattice gauge theory in which the quantum loops of fermion s in Feynman diagram s are neglected. Equivalently, the corresponding one loop determinants are set to one. This approximation is often forced upon the physicists because the calculation with the Grassmann number s is computationally very difficult in lattice gauge theory. Recent calculations ref name urlRev. Mod. Phys. 82, 1349 2010 Nonperturbative QCD simulations with 2 1 flavors of improved staggered quarks cite web url http link.aps.org doi 10.1103 RevModPhys.82.1349 author A. Bazavov et. al title Rev. Mod. Phys. 82, 1349 2010 Nonperturbative QCD simulations with 2 1 flavors of improved staggered quarks format work accessdate doi 10.1103 RevModPhys.82.1349 ref typically avoid the quenched approximation. References Reflist DEFAULTSORT Quenched Approximation Category Lattice models Quantum stub pt Aproxima o temperada ... more details
In geometric optics , the paraxial approximation is a small angle approximation used in Gaussian optics and Ray tracing physics ray tracing of light through an optical system such as a lens optics lens . ref name Greivenkamp cite book first John E. last Greivenkamp year 2004 title Field Guide to Geometrical Optics publisher SPIE others SPIE Field Guides vol. FG01 isbn 0 8194 5294 7 pages 19 20 ref A paraxial ray is a Ray optics ray which makes a small angle to the optical axis of the system, and lies close to the axis throughout the system. ref name Greivenkamp Generally, this allows three important approximations for in radian s for calculation of the ray s path ref name Greivenkamp math begin align sin theta & approx theta tan theta & approx theta end align math and math cos theta approx 1 math The paraxial approximation is used in Gaussian optics and first order raytracing. ref name Greivenkamp Ray transfer matrix analysis is one method that uses the approximation. In some cases, the second order approximation is also called paraxial . To second order, the approximations above for sine and tangent do not change the next term in their Taylor series expansion is zero , while for cosine the second order approximation is math cos theta approx 1 theta 2 over 2 . math The paraxial approximation is fairly accurate for angles under about 10 , but is inaccurate for larger angles. For larger angles it is often necessary to distinguish between meridional ray s, which lie in a plane containing the optical axis , and sagittal ray s, which do not. References reflist External links http demonstrations.wolfram.com ParaxialApproximationAndTheMirror Paraxial Approximation and the Mirror by David Schurig, The Wolfram Demonstrations Project . Category Geometrical optics ca Aproximaci paraxial de Paraxiale Optik es Aproximaci n paraxial fr Approximation de Gauss it Approssimazione parassiale hu Paraxi lis k zel t s nl Paraxiale benadering ja ru uk ... more details
In mathematics , an approximation to the identity refers to a sequence or net that converges to the identity in some algebra. Specifically, it can mean Nascent delta function , most commonly Mollifier , more narrowly Approximate identity , more abstractly disambig ... more details
Infobox Journal cover discipline mathematics abbreviation Constr. Approx. publisher Springer Science Business Media Springer country United States U.S. ISSN 0176 4276 eISSN 1432 0940 CODEN CONSA? history 1985 to present website http www.math.vanderbilt.edu ca Constructive Approximation is an international mathematics journal dedicated to Approximations and Expansions and related research in computation, function theory, functional analysis, interpolation spaces and interpolation of operators, numerical analysis, space of functions, special functions, and applications. ref cite web url http www.springer.com west home?SGWID 4 102 70 1056700 detailsPage journal description&changeHeader true&SHORTCUT www.springer.com journal 00365 about title Constructive Approximation Numerical Analysis Journals, Books & Online Media ¦ Springer accessdate 2007 04 30 ref References references External links http www.math.vanderbilt.edu ca Constructive Approximation web site Category Mathematics journals Category Approximation theory Category English language journals Category Publications established in 1985 Category Springer academic journals Category Bimonthly journals math journal stub sci journal stub ... more details
Orphan date December 2008 Cleanup rewrite date May 2009 A fundamental principle in the field of acoustics , the acoustic approximation states that an acoustic wave is created by a small, adiabatic, pressure ripple riding on a comparatively large equilibrium bias pressure . Typically, the acoustic pressure is on the order of a few ppm of the equilibrium pressure. By extension, the acoustic approximation also guarantees that an acoustic wave travels at a speed exactly equal to the local speed of sound , such that the Mach number math M 1 math See also Sound References reflist External links Category Acoustics physics stub ... more details
The need for function approximations arises in many branches of applied mathematics , and computer science in particular. In general, a function approximation problem asks us to select a function mathematics function among a well defined class that closely matches approximates a target function in a task specific way. One can distinguish two major classes of function approximation problems First, for known target functions approximation theory is the branch of numerical analysis that investigates how certain known functions for example, special function s can be approximated by a specific class of functions for example, polynomial s or rational function s that often have desirable properties inexpensive computation, continuity, integral and limit values, etc. . Second, the target function, call it g , may be unknown instead of an explicit formula, only a set of points of the form x , g x is provided. Depending on the structure of the domain mathematics domain and codomain of g , several techniques for approximating g may be applicable. For example, if g is an operation on the real number s, techniques of interpolation , extrapolation , regression analysis , and curve fitting can be used. If the codomain clarify date December 2010 of g is a finite set, one is dealing with a statistical classification classification problem instead. To some extent the different problems regression, classification, fitness approximation have received a unified treatment in statistical learning theory , where they are viewed as supervised learning problems. See also Radial basis function network Fitness approximation DEFAULTSORT Function Approximation Category Statistical models Category Machine learning Category Statistical approximations mathanalysis stub statistics stub hr Aproksimacija funkcija sr ... more details
In number theory , the field of Diophantine approximation , named after Diophantus of Alexandria , deals with the approximation of real number s by rational number s. The absolute value of the difference ... measure of how good the approximation is. However, since the rational numbers are dense in the real .... So this measure tells us nothing about the quality of the approximation. A better measure of the quality of the approximation is by comparison of the difference to the size of the denominator ... approximation because its denominator is smaller. Approximation to algebraic numbers The theory ... numbers. This link between diophantine approximation and transcendental number transcendence ... generalised this to the case of simultaneous approximation. The proofs were difficult, and not effective .... Duffin and A. C. Schaeffer, Khintchine s problem in metric Diophantine approximation , Duke Mathematical ... that Diophantine approximation results were closely related to the general problem of cancellation ... remaining in Diophantine approximation, for example the Littlewood conjecture . Recent developments ... to classical problems in Diophantine approximation. Among its notable successes are the proof of the decades ... Khinchin in metric Diophantine approximation have also been obtained within this framework. See also ... Cassels authorlink J. W. S. Cassels title An introduction to Diophantine approximation series Cambridge ... on homogeneous spaces and Diophantine approximation on manifolds url http jstor.org stable 120997 ... publisher Springer Verlag year 1995 isbn 0 387 94456 7 Grigory Margulis , Diophantine approximation ... 521 80799 9. Wolfgang M. Schmidt . Diophantine approximation . Lecture Notes in Mathematics 785. Springer ... HCMUNS10.pdf Diophantine Approximation historical survey . From Introduction to Diophantine methods course by Michel Waldschmidt. DEFAULTSORT Diophantine Approximation Category Number theory Category Diophantine approximation de Diophantische Approximation fr Approximation diophantienne it Approssimazione ... more details
Unreferenced stub auto yes date December 2009 Orphan date December 2009 The Coopmans Approximation is a method for approximating a fractional order integrator in a continuous process with constant space complexity . The most correct and accurate methods for calculating the fractional integral require a record of all previous history, and therefore would require a constant space complexity solution O n , where n is the number of samples measured for the complete history. The fractor is an analog component useful in control systems . In order to model the components behavior in a digital simulation, or replace the fractor in a digital controller, a linear solution is untenable. In order to reduce the space complexity however, it is necessary to lose information in some way. The Coopmans Approximation is a robust, simple method that uses a simple convolution to compute the fractional integral, then recycles old data back through the convolution. The convolution sets up a weighting table as described by the fractional calculus , which varies based on the size of the table, the sampling rate of the system, and the order of the integral. Once computed the weighting table remains static. The data table is initialized as all zeros, which represents a lack of activity for all previous time. New data is added to the data buffer in the fashion of a ring buffer, so that the newest point is written over the oldest data point. The convolution is solved by multiplying corresponding elements from the weight and data tables, and summing the resulting products. As described, the loss of the old data by overwriting with new data will cause echoes in a continuous system as disturbances that were ... Approximation, where the old data point, multiplied by its corresponding weight term, as added to the newest ... history. This approximation has the desirable effect of removing the echo, while preserving the space complexity of the solution. The negative effect of the approximation is that the phase ... more details
In mathematics , Spouge s approximation is a formula for the gamma function due to John L. Spouge. The formula is a modification of Stirling s approximation , and has the form math Gamma z 1 z a z 1 2 e z a left c 0 sum k 1 a 1 frac c k z k varepsilon a z right math where a is an arbitrary positive integer and the coefficients are given by math c 0 sqrt 2 pi , math math c k frac 1 k 1 k 1 k a k 1 2 e k a quad k in 1,2, dots, a 1 . math Spouge has proved that, if Re z 0 and a 2, the relative error in discarding &epsilon sub a sub z is bounded by math ,a 1 2 2 pi a 1 2 . math The formula is similar to the Lanczos approximation , but has some distinct features. Whereas the Lanczos formula exhibits faster convergence, Spouge s coefficients are much easier to calculate and the error can be set arbitrarily low. The formula is therefore feasible for arbitrary precision arithmetic arbitrary precision evaluation of the gamma function. However, special care must be taken to use sufficient precision when computing the sum due to the large size of the coefficients c k, as well as their alternating sign. For example, for a 49, you must compute the sum using about 65 decimal digits of precision in order to obtain the promised 40 decimal digits of accuracy. See also Stirling s approximation Lanczos approximation References Citation last Spouge first John L. year 1994 title Computation of the gamma, digamma, and trigamma functions journal SIAM Journal on Numerical Analysis volume 31 issue 3 pages 931 944 doi 10.1137 0731050 External links http en.literateprograms.org Gamma function with Spouge s formula Mathematica Gamma function with Spouge s formula Mathematica implementation at LiteratePrograms Category Gamma and related functions Category Arbitrary precision algorithms hu Spouge formula numtheory stub ... more details
In mathematics , a Banach space is said to have the approximation property AP in short , if every compact operator is a limit of finite rank operator s. The converse is always true. Every Hilbert space has this property. There are, however, Banach space s which do not Enflo published the first counterexample in an 1973 article. However, a lot of work in this area was done by Grothendieck 1955 . Later many other counterexamples were found. The space of bounded operator s on math ell 2 math does not have the approximation property Szankowski . The spaces math ell p math for math p neq 2 math and math c 0 math see Sequence space have closed subspaces that do not have the approximation property. Definition A Banach space math X math is said to have the approximation property, if, for every compact set math K subset X math and every math varepsilon 0 math , there is an operator mathematics operator math T colon X to X math of finite rank so that math Tx x leq varepsilon math , for every math x in K math . Some other flavours of the AP are studied Let math X math be a Banach space and let math 1 leq lambda infty math . We say that math X math has the math lambda math approximation property math lambda math AP , if, for every compact set math K subset X math and every math varepsilon 0 math , there is an operator mathematics operator math T colon X to X math of finite rank so that math ... is said to have bounded approximation property BAP , if it has the math lambda math AP for some math lambda math . A Banach space is said to have metric approximation property MAP , if it is 1 AP. A Banach space is said to have compact approximation property CAP , if in the definition of AP an operator ... Robert G. Bartle title MR0402468 53 6288 Review of Per Enflo s A counterexample to the approximation ... year 1977 MR 402468 Per Enflo Enflo, P. A counterexample to the approximation property in Banach spaces ... s example of a Banach space without the approximation property . S minaire Goulaouic Schwartz 1972 ... more details
Image TangentGraphic2.svg thumb 300px Tangent line at a , f a In mathematics , a linear approximation is an approximation of a general function mathematics function using a linear function more precisely, an affine function . They are widely used in the method of finite differences to produce first order methods for solving or approximating solutions to equations. Definition Given a twice continuously differentiable function f of one real number real variable, Taylor s theorem for the case n 1 states that math f x f a f a x a R 2 math where math R 2 math is the remainder term. The linear approximation is obtained by dropping the remainder math f x approx f a f a x a . math ref Some calculus textbooks write d x for x a the change in x , and then define d f f a x a so as to have a numerical equality d f f a d x . This may be useful as a mnemonic for the fact that f x f a the change in f is approximated by f a x a , but it conflicts with the actual definition of d f as a differential form . ref This is a good approximation for x when it is close enough to a since a curve, when closely observed, will begin to resemble a straight line. Therefore, the expression on the right hand side is just the equation for the tangent line to the graph of f at a , f a . For this reason, this process is also called the tangent line approximation . Linear approximations for vector geometric vector functions of a vector variable are obtained in the same way, with the derivative at a point replaced by the Jacobian matrix and determinant Jacobian matrix. For example, given a differentiable ... a math where math Df a math is the Fr chet derivative of math f math at math a math . To find an approximation ... 2 3 3 frac 1 3 sqrt 3 x 2 math According to linear approximation math f 25 approx f 27 f 27 25 27 3 ... year 2005 isbn 0 7641 2382 3 oclc doi page 118 DEFAULTSORT Linear Approximation Category Differential ... es Aproximaci n lineal fr Approximation affine he nl Lineaire benadering ja pl ... more details
In mathematics , approximation adjusts a Fourier series Fourier summation to eliminate the Gibbs phenomenon which would otherwise occur at discontinuities . A approximated summation for a series of period T can be written as follows math s theta frac 1 2 a 0 sum k 1 m 1 mathrm sinc Bigl frac k m Bigr cdot left a k cos Bigl frac 2 pi k T theta Bigr b k sin Bigl frac 2 pi k T theta Bigr right , math in terms of the normalized sinc function math mathrm sinc , x frac sin pi x pi x . math Here, the term math mathrm sinc Bigl frac k m Bigr math is the Lanczos factor , which is responsible for eliminating most of the Gibbs phenomenon. It does not do so entirely, however, but one can square or even cube the expression to serially attenuate Gibbs Phenomenon in the most extreme cases. See also Lanczos resampling References Unreferenced date January 2007 math stub Category Fourier series Category Numerical analysis bs Sigma aproksimacija fr Approximation sigma ... more details
The Kirkwood superposition approximation was introduced by Matsuda 2000 as a means of representing a discrete probability distribution . The name apparently refers to a 1942 paper by John Gamble Kirkwood John G. Kirkwood . ref Kirkwood, J. G. & Boggs, E. M. 1942 . The radial distribution function in liquids . J. Chem. Phys. 10, 394. ref The Kirkwood approximation for a discrete probability density function math P x 1 ,x 2 , ldots ,x n math is given by math P prime x 1,x 2, ldots ,x n frac frac frac prod mathcal T n 1 subseteq mathcal V p mathcal T n 1 prod mathcal T n 2 subseteq mathcal V p mathcal T n 2 vdots prod mathcal T 1 subseteq mathcal V p mathcal T 1 math where math prod mathcal T i subseteq mathcal V p mathcal T i math is the product of probabilities over all subsets of variables of size i in variable set math scriptstyle mathcal V math . This kind of formula has been considered by Watanabe 1960 and, according to Watanabe, also by Robert Fano. For the three variable case, it reduces to simply math P prime x 1,x 2,x 3 frac p x 1,x 2 p x 2,x 3 p x 1,x 3 p x 1 p x 2 p x 3 math The Kirkwood approximation does not generally produce a valid probability distribution the normalization condition is violated . Watanabe claims that for this reason informational expressions of this type are not meaningful, and indeed there has been very little written about the properties of this measure. The Kirkwood approximation is the probabilistic counterpart of the interaction information . Judea Pearl 1988 3.2.4 indicates that an expression of this type can be exact in the case of a decomposable model, that is, a probability distribution that admits a graph mathematics graph structure whose clique graph theory cliques form a tree graph theory tree . In such cases, the numerator contains the product of the intra clique joint distributions and the denominator contains the product of the clique intersection distributions. References reflist Jakulin, A. & Bratko, I. 2004 , Quantifying ... more details
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