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
Unreferenced date September 2007 Certainty An approximation is a representation of something that is not exact, but still close enough to be useful. Although approximation is most often applied to number s, it is also frequently applied to such things as Function mathematics mathematical functions , shape ..., an approximation may yield a sufficiently accurate solution while reducing the complexity of the problem ... 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 used depends ..., and the savings usually in time and effort that can be achieved by approximation. Science ... measurement s, which are constantly compared to one another. The approximation also refers ... wikitable align right Symbols representing approximation style font size xx large & 8784 general approximation style font size xx large asymptotic analysis Approximation usually occurs when an exact ... 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 analysis .... See also col begin col break Approximately equals sign Approximation error Congruence relation Estimation Fermi estimate Fitness approximation Least squares col break Linear approximation Newton s method Numerical analysis Orders of approximation Runge Kutta methods Successive approximation ADC Taylor series col break Wiktionarypar approximation col end Category Approximations Category Numerical analysis bg br Tostadur cs Aproximace da Approksimation de Approximation es Aproximaci n eo Proksimuma kalkulado fr Approximation ko hi is N mundun it Approssimazione he ... pt Aproxima o ru simple Approximation fi Approksimaatio sv Approximation uk ... 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
In computer science , hardness of approximation is a field that studies the algorithmic complexity of finding near optimal solutions to optimization problem s. 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 NP P . 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
reach an exact solution, approximationalgorithm s seek an approximation that is close to the true ... that the random element will be decreased as the algorithm settles down to a solution. Approximation ...Lead rewrite date April 2011 Image Euclid flowchart 1.png 200px thumb lright 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 in two loops IF the test B ... a in location A THEN the algorithm specifies B B A meaning the number b a replaces the old b . Similarly IF A B THEN A A B. The process terminates when the contents of B is 0, yielding the g.c.d. in A. Algorithm ... and computer science , an algorithm IPAc en audio en us algorithm.ogg l r m from Algoritmi ... are used for calculation , data processing , and automated reasoning . More precisely, an algorithm is an effective method expressed as a finite list ref Any classical mathematical algorithm, for example ... instructions ref Well defined with respect to the agent that executes the algorithm There is a computing ... 1987 2 . ref for calculating a Function mathematics function . ref an algorithm is a procedure for computing ... and initial input perhaps null string empty , 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 ... ref A procedure which has all the characteristics of an algorithm except that it possibly lacks ... states, eventually producing output ref An algorithm has one or more outputs, i.e. quantities which ... processes not including the input is an algorithm is debatable. Rogers opines that a computation ... Moschovakis chapter What is an algorithm? title Mathematics Unlimited &mdash 2001 and beyond editor1 ... definition For a detailed presentation of the various points of view around the definition of algorithm see Algorithm characterizations . For examples of simple addition algorithms specified in the detailed ... more details
context date June 2011 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 ... and Kiefer Wolfowitz algorithms. Robbins Monro algorithm The Robbins Monro algorithm, introduced in 1951 by Herbert Robbins and Sutton Monro, ref name rm A Stochastic Approximation Method, Herbert Robbins ... variable math N x math where math mathbb E N x M x math . The structure of the algorithm is to then generate ... of math f x math belongs to the interior of math Theta math , then the Robbins Monro algorithm ... math x in Theta math . ref name jsacks Asymptotic distribution of Stochastic Approximation, J. Sacks ... Approximation Approach to Stochastic Programming, A. Nemirovski, A. Juditsky, G. Lan and A. Shapiro ... the Robbins Monro algorithm is theoretically able to achieve math O 1 n math under the assumption of twice .... This is primarily due to the fact that the algorithm is very sensitive to the choice of the step ..., Polyak and Juditsky, ref name pj Acceleration of Stochastic Approximation by Averaging, B.T. ... of the iterates. The algorithm would have the following structure math x n 1 x n b n alpha N x n ... not, hence the longer steps. Under the assumptions outlined in the Robbins Monro algorithm, the resulting ... the nonasymptotic rate math O 1 sqrt n math . Kiefer Wolfowitz algorithm The Kiefer Wolfowitz algorithm ... 466. ref was introduced in 1952, and was motivated by the publication of the Robbins Monro algorithm. However, the algorithm was presented as a method which would stochastically estimate the maximum ... mathbb E N x M x math , can be made at any point math x math . The structure of the algorithm follows ... approximation, while the sequence math a n math specifies a sequence of positive step sizes ... math f x math has a unique point of maximum minimum and is strong concave convex The algorithm was first ... and math c n n 1 3 math . Subsequent Developments and Important Issues The Kiefer Wolfowitz algorithm ... 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. e.g., the accurate reading of a piece of paper is 4.5cm but since the ruler does not use decimals, you round it to 5cm. 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 0.003 and in the second it is only 0.000003. Another example would be if you measured a beaker and read, 5mL. The correct reading would have been 6mL. This means that your error Approximate error would be 16.66666.. error. 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 ... Erreur d approximation ko it Errore relativo he kk nl Benaderingsfout ... 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 ... function is shown in blue to the right is another N degree polynomial that is a better approximation ... of degree N . Chebyshev approximation One can obtain polynomials very close to the optimal one ... rapid converging power series, than for the log function. Chebyshev approximation is the basis for Clenshaw Curtis quadrature , a numerical integration technique. Remez algorithm The Remez algorithm sometimes ... x over a given interval. It is an iterative 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 ... the end points of the interval of approximation , these equations need to be solved math P x 1 f ... algorithm, approximating e sup x sup over the interval 1, 1 . Vertical divisions are 10 sup 4 sup .... 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 ... 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 . 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. Formal description Empty section date July 2010 See also Eikonal equation 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 Math stub Quantum stub ... more details
Boussinesq approximation may refer to several modelling concepts as introduced by Joseph Valentin Boussinesq 1842 1929 , a French mathematician and physicist known for advances in fluid dynamics 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 in modelling the turbulence Reynolds stresses , the Boussinesq approximation results in the use of an eddy viscosity concept disambig ... more details
Sparse approximation also referred to as sparse decomposition is the problem of estimating a Sparsity sparse multi dimensional vector, satisfying a System of linear equations linear system of equations given high dimensional observed data and a design matrix . Sparse approximation techniques have found wide use in applications such as image processing, audio processing, biology, and document analysis. Sparse decomposition Noiseless observations Consider a System of linear equations linear system of equations math x D alpha math , where math D math is an math m times p math matrix mathematics matrix math m ll p math and math x in mathbb R m, alpha in mathbb R p math . math D math , called as the dictionary or the design matrix, is given. The problem is to estimate the signal math alpha math , subject to the constraint that it is sparse. The underlying motivation for sparse decomposition problems is that even though the observed values are in high dimensional math m math space, the actual signal is organized in some lower dimensional subspace math k ll m math . Sparsity implies that only a few components of math x math are non zero and the rest are zero. This implies that math x math can be decomposed as a linear combination of only a few math m times 1 math vectors in math D math , called atoms. math D math itself is over complete math m ll p . math Such vectors are called as the Basis vectors basis of math x math . However, unlike other Dimensionality reduction dimensionality reducing decomposition techniques such as Principal component analysis Principal Component Analysis ..., and employing a sparse solution. Variations There are several variations to the basic sparse approximation ... There are several algorithms that have been developed for solving sparse approximation problem. Matching pursuit Matching Pursuit is a greedy iterative algorithm for approximatively solving the original .... The algorithm maintains an active set of atoms already picked, and adds a new atom at each iteration ... 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 0th order is the term scientist s use for a first estimation educated guess at an answer. Many simplifying assumptions ... 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 ... math y 3,3,5 , math math y sim f x 3.67 , math is an approximate fit to the data. First order approximation ... is often given the town has 4 10 sup 3 sup or four thousand residents . A first order approximation ... approximation , straight line with a slope a polynomial of degree 1. For example, math x 0,1,2 ... by simply averaging the y values. Other methods for selecting a constant approximation can be used. Second order approximation also 2nd order is the term scientists use for a decent quality answer ... approximation of a function that is, mathematically determining a formula to fit multiple data ... referred to by number. A third order approximation would be required to fit four data ... at the annual calibration. In this usage, the ordinality of the approximation is not exact, but is used ... 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 accurate within 0.5 for angles under about 10 Citation needed date May 2011 but its inaccuracy grows significantly 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 ... 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
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
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 range or target set 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 Least squares function approximation Radial basis function network Fitness approximation DEFAULTSORT Function Approximation Category Statistical models 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 ... of transcendental numbers. This link between diophantine approximation and transcendental number ... M. Schmidt generalised this to the case of simultaneous approximation. The proofs were difficult ... ref R. J. Duffin and A. C. Schaeffer, Khintchine s problem in metric Diophantine approximation , Duke .... This showed that Diophantine approximation results were closely related to the general problem ... problems remaining in Diophantine approximation, for example the Littlewood conjecture . Recent ... approach to classical problems in Diophantine approximation. Among its notable successes are the proof ... Khinchin in metric Diophantine approximation have also been obtained within this framework ... J.W.S. Cassels authorlink J. W. S. Cassels title An introduction to Diophantine approximation series ... title Flows on homogeneous spaces and Diophantine approximation on manifolds journal Ann. Math. volume ... year 1995 isbn 0 387 94456 7 Grigory Margulis , Diophantine approximation, lattices and flows on homogeneous ... approximation . Lecture Notes in Mathematics 785. Springer. 1980 1996 with minor corrections ... http people.math.jussieu.fr miw articles pdf HCMUNS10.pdf Diophantine Approximation historical ... Approximation Category Number theory Category Diophantine approximation de Diophantische Approximation es Aproximaci n diof ntica fr Approximation diophantienne it Approssimazione diofantea he ... 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
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
Image MazurGes.jpg thumb right The construction of a Banach space without the approximation property earned Per Enflo a live goose in 1972, which had been promised by Stanislaw Mazur in 1936. In mathematics , a Banach space is said to have the approximation property AP , 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 Per Enflo published the first counterexample in a 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 ... 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 of finite ... G. Bartle title MR0402468 53 6288 Review of Per Enflo s A counterexample to the approximation problem ... 402468 Per Enflo Enflo, P. A counterexample to the approximation property in Banach spaces. Acta Math ... Association of America. Kwapie , S. On Enflo s example of a Banach space without the approximation ... more details
In mathematics , the Lanczos approximation is a method for computing the Gamma function numerically, published by Cornelius Lanczos in 1964. It is a practical alternative to the more popular Stirling s approximation for calculating the Gamma function with fixed precision. Introduction The Lanczos approximation consists of the formula math Gamma z 1 sqrt 2 pi left z g frac 1 2 right z frac 1 2 e left z g frac 1 2 right A g z math for the Gamma function, with math A g z frac 1 2 p 0 g p 1 g frac z z 1 p 2 g frac z z 1 z 1 z 2 cdots. math Here g is a Constant mathematics constant that may be chosen arbitrarily subject to the restriction that Re z g 1 2 0. The coefficients p , which depend on g , are slightly more difficult to calculate see below . Although the formula as stated here is only valid for arguments in the right complex half plane , it can be extended to the entire complex plane by the reflection formula , math Gamma 1 z Gamma z pi over sin pi z . math The series A is convergent series convergent , and may be truncated to obtain an approximation with the desired precision. By choosing an appropriate g typically a small integer , only some 5 10 terms of the series are needed to compute the Gamma function with typical single precision single or double precision double floating point floating point precision. If a fixed g is chosen, the coefficients can be calculated in advance ... s and multiplying by stored constants. The Lanczos approximation was popularized by Numerical Recipes ... return sqrt 2 pi t z 0.5 exp t x source See also Stirling s approximation Spouge s approximation References ... authorlink Cornelius Lanczos title A Precision Approximation of the Gamma Function jstor 2949767 journal ... Gamma approximation cite web last1 Toth first1 Viktor year 2005 url http www.rskey.org lanczos.htm title Programmable Calculators The Lanczos Approximation MathWorld urlname LanczosApproximation title Lanczos Approximation Category Gamma and related functions Category Numerical analysis ... more details
distinguish2 the Born Oppenheimer approximation In scattering theory and, in particular in quantum mechanics , the Born approximation consists of taking the incident field in place of the total field as the driving field at each point in the scatterer. Born approximation is named after Max Born , winner of the 1954 Nobel Prize for physics. It is the perturbation theory perturbation method applied to scattering by an extended body. It is accurate if the scattered field is small, compared to the incident field, in the scatterer. For example, the radar scattering of radio waves by a light styrofoam column can be approximated by assuming that each part of the plastic is polarized by the same electric field that would be present at that point without the column, and then calculating the scattering as a radiation integral over that polarization distribution. Born approximation to the Lippmann Schwinger equation The Lippmann Schwinger equation for the scattering state math vert Psi mathbf p pm rangle math with a momentum p and out going or in going &minus boundary condition s is math vert Psi mathbf p pm rangle vert Psi mathbf p circ rangle G circ E p pm i0 V vert Psi mathbf p pm rangle math where math G circ math is the free particle Green s function , 0 is a positive infinitesimal quantity ... approximation math vert Psi mathbf p pm rangle vert Psi mathbf p circ rangle G circ E p pm i0 V vert ... point of the Born series. Applications The Born approximation is used in quite different physical contexts. In neutron scattering , the first order Born approximation is almost always adequate ... , or grazing incidence small angle scattering . Distorted wave Born approximation DWBA The Born approximation is simplest when the incident waves math vert Psi mathbf p circ rangle math are plane .... In the distorted wave Born approximation DWBA , the incident waves are solutions math vert Psi mathbf ... i0 V 1 vert Psi mathbf p 1 pm rangle math and the Born approximation math vert Psi mathbf p pm rangle ... more details
File 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 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 function math f x, y math with real values, one can approximate math f x, y math for math x, y math close to math a, b math by the formula math f left x,y right approx f left a,b right frac partial f partial x left a,b right left x a right frac partial f partial y left a,b right left y b right . math The right hand side is the equation of the plane tangent to the graph of math z f x, y math at math a, b . math In the more general case of Banach space s, one has math f x approx f a Df a x a math where math Df a math is the Fr chet derivative of math f math at math a math . See also Euler s method Finite differences Finite difference ... fr Approximation affine he it Approssimazione lineare nl Lineaire benadering ja ... more details
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