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         Approximations Expansions:     more books (94)
  1. Asymptotology: Ideas, Methods, and Applications (Mathematics and Its Applications) by Igor V. Andrianov, Leonid I. Manevitch, 2002-11-30
  2. Multiscale Problems and Methods in Numerical Simulations: Lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, September 9-15, 2001 ... Mathematics / Fondazione C.I.M.E., Firenze) by James H. Bramble, Albert Cohen, et all 2004-01-12
  3. Exponential Sums and their Applications (Mathematics and its Applications) by N.M Korobov, 2010-11-02
  4. Symbolic Asymptotics (Algorithms and Computation in Mathematics) by John R. Shackell, 2010-11-02
  5. Singular Integral Operators, Factorization and Applications: International Workshop on Operator Theory and Applications IWOTA 2000, Portugal (Operator Theory: Advances and Applications)
  6. Haar Series and Linear Operators (Mathematics and Its Applications) by I. Novikov, E. Semenov, 2010-11-02
  7. Nonsmooth Equations in Optimization: Regularity, Calculus, Methods and Applications (Nonconvex Optimization and Its Applications) by Diethard Klatte, B. Kummer, 2010-11-02
  8. Spline Functions and Multivariate Interpolations (Mathematics and Its Applications) by Borislav D. Bojanov, H. Hakopian, et all 2010-11-02
  9. Total Positivity and its Applications (Mathematics and Its Applications)
  10. The FitzHugh-Nagumo Model: Bifurcation and Dynamics (Mathematical Modelling: Theory and Applications) by C. Rocsoreanu, A. Georgescu, et all 2010-11-02
  11. Global Analysis in Linear Differential Equations (Mathematics and Its Applications) by M. Kohno, 1999-04-30
  12. The Theory of Cubature Formulas (Mathematics and Its Applications) by S.L. Sobolev, Vladimir L. Vaskevich, 2010-11-02
  13. Multivariate Spline Functions and Their Applications (Mathematics and Its Applications) by Ren-Hong Wang, 2010-11-02
  14. Functional Equations, Inequalities and Applications

41. 1 Introduction
Included in Stage (ii) are expansions in series of Chebyshev polynomials (``Chebyshev series ), minimax polynomial and rational approximations,
http://math.nist.gov/mcsd/Reports/2001/nesf/node2.html
Next: 2 Mathematical Developments Up: Numerical Evaluation of Special Previous: Contents
1 Introduction
continues to be one of the best-selling mathematical books of all time The purpose of the present paper is to provide some assistance to those mathematicians, engineers, scientists, and statisticians who discover that they need to generate numerical values of the special functions in the course of solving their problems. ``Generate'' is the operative word here: we are thinking primarily of either software or numerical approximations that can be programmed fairly easily. Numerical tables are not covered in this survey. Furthermore, for the most part we shall concentrate on the functions themselves; only in certain cases do we include, for example, zeros, inverse functions or indefinite integrals. Elementary functions, also, are excluded . Lastly, we believe that the majority of readers would prefer us to emphasize the more useful algorithms rather than make an attempt to be encyclopedic: algorithms or approximations that have clearly been superseded are omitted. We identify three stages in the development of computational procedures for the special functions:
  • Derivation of relevant mathematical properties.
  • 42. EconPapers: Stochastic Expansions And Asymptotic Approximations
    By Michael A Magdalinos; Stochastic expansions and Asymptotic approximations.
    http://econpapers.repec.org/article/cupetheor/v_3A8_3Ay_3A1992_3Ai_3A3_3Ap_3A343
    EconPapers Home
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    Stochastic Expansions and Asymptotic Approximations
    Michael A Magdalinos Econometric Theory , 1992, vol. 8, issue 3, pages 343-67 There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it. Related works:
    This item may be available elsewhere in EconPapers: Search for items with the same title. Access Statistics for this article More articles in Econometric Theory from Cambridge University Press
    Address: The Edinburgh Building, Shaftesbury Road, Cambridge CB2 2RU UK
    Series data maintained by Mike Eden ( This site is part of RePEc and all the data displayed here is part of the RePEc data set. Is your work missing from RePEc? Here is how to contribute Questions or problems? Check the EconPapers FAQ or send mail to Econpapers is hosted by the Department of Business, Economics, Statistics and Informatics at –rebro University. Page updated 2007-10-22 Handle: RePEc:cup:etheor:v:8:y:1992:i:3:p:343-67

    43. Higher Order Large Deviation Approximations Applied To CDO Pricing
    Keywords Edgeworth expansions, Large Deviation approximations, GramCharlier expansions, Saddle-point, base correlation, CDO, copula, local correlation.
    http://www.defaultrisk.com/pp_cdo_18.htm
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    by Philipp J. Schönbucher
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    Default Risk .com Higher Order Large Deviation Approximations Applied to CDO Pricing by Laurent Veilex of Credit Suisse February 2007 Abstract: We propose a Large Deviation approximation for the loss distribution of a credit portfolio and compare it as well as higher order Saddle-point and Edgeworth expansions with the standard recursion method for the pricing of CDO tranches. Keywords: Edgeworth expansions, Large Deviation Approximations, Gram-Charlier expansions, Saddle-point, base correlation, CDO, copula, local correlation. Books Referenced in this Paper: what is this?

    44. 2008-01-19T111904Z Http//locutus.cs.dal.ca8088/perl/oai2
    of successive approximations 7375626A656374733D31312D58583131415858 41xx approximations and expansions 41Axx approximations and expansions
    http://locutus.cs.dal.ca:8088/perl/oai2?verb=ListSets

    45. Mutual Information And Kullback-Leibler Divergence
    Some authors have used approximations of mutual information based on polynomial density expansions 36,1, which lead to the use of higherorder cumulants
    http://www.cis.hut.fi/aapo/papers/NCS99web/node26.html
    Next: Non-linear cross-correlations Up: Multi-unit contrast functions Previous: Likelihood and network entropy
    Mutual information and Kullback-Leibler divergence
    Theoretically the most satisfying contrast function in the multi-unit case is, in our view, mutual information. Using the concept of differential entropy defined in Eq. ( ), one defines the mutual information I between m (scalar) random variables y i i m , as follows
    where H denotes differential entropy. The mutual information is a natural measure of the dependence between random variables. It is always non-negative, and zero if and only if the variables are statistically independent. Thus the mutual information takes into account the whole dependence structure of the variables. Finding a transform that minimizes the mutual information between the components s i is a very natural way of estimating the ICA model [ ]. This approach gives at the same time a method of performing ICA according to the general Definition 1 in Section . For future use, note that by the properties of mutual information, we have for an invertible linear transformation
    The use of mutual information can also be motivated using the Kullback-Leibler divergence, defined for two probability densities

    46. Approximation: Theory And Applications
    Approximation Theory and its Applications Series in approximations and 2000 Mathematics Subject Classification approximations and expansions
    http://www.ift.uib.no/~antonych/Approx.html

    GAMS (Generalized Algebraic Modeling System) Development Corporation

    47. Eitan Tadmor Course, Homepage For AMSC666 Fall 2003
    Least Squares approximations II. (Generalized) Fourier expansions. Bessel, Parseval, . Least Squares approximations III. Discrete expansions
    http://www.cscamm.umd.edu/people/faculty/tadmor/courses/666_fall03/
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    Numerical Analysis I
    AMSC 666, Fall 2003
    Basic Information
    Lecture CSIC Bldg. 3118; TuTh 11-12:15pm Instructor Professor Eitan Tadmor Contact tel.: x5-0648 Email: eo("tadmor","math.umd.edu",""); Eitan tadmor Office Hours By appointment ( eo("cgray","cscamm.umd.edu",""); eitan Tadmor
    CSCAMM 4119 CSIC Bldg. #406
    Teaching Assistant eo("bincheng","cscamm.umd.edu",""); eitan Tadmor TA Office Hours 4117 CSIC Bldg. #404 MW 10-11am Midterm Tu Nov 4 11-12:15pm 4117 CSIC Bldg. #404 Final Mo Dec 15 8-10am 4117 CSIC Bldg. #404 Grading 25% Homework, 25% Mid-Term, 50% Final
    Course Description
  • Ten steps of Approximation Theory
  • General overview
      On the choice of norm: L vs. L
    • Weierstrass' density theorem Bernstein polynomials

    Least Squares Approximations I. A general overview
      Gramm mass matrix ill-conditioning of monomials in L

    Least Squares Approximations II. (Generalized) Fourier expansions
      Bessel, Parseval, ... Orthogonal polynomials: Legendre, Chebyshev, Sturm's sequence
    Assignment #1 [ pdf file
  • Least Squares Approximations III. Discrete expansions
      Examples of discrete least squares.
  • 48. Higher-Order Approximations To The Distributions Of Fit Indexes Under Fixed Alte
    The approximations are given by the Edgeworth expansions for the distributions of the fit indexes under arbitrary distributions.
    http://www.eric.ed.gov/ERICWebPortal/recordDetail?accno=EJ766743

    49. Series Expansions - Wolfram Mathematica
    Series expansions. Power series are in many ways the algebraic analog of limitedprecision numbers. Mathematica can generate series approximations to
    http://reference.wolfram.com/mathematica/guide/SeriesExpansions.html
    baselang='SeriesExpansions.en'; PreloadImages('/common/images2003/link_products_on.gif','/common/images2003/link_purchasing_on.gif','/common/images2003/link_forusers_on.gif','/common/images2003/link_aboutus_on.gif','/common/images2003/link_oursites_on.gif'); DOCUMENTATION CENTER SEARCH Mathematica Mathematics and Algorithms Calculus Series Expansions Power series are in many ways the algebraic analog of limited-precision numbers. Mathematica can generate series approximations to virtually any combination of built-in mathematical functions. It will then automatically combine series truncating to the correct order. Mathematica supports not only ordinary power series, but also Laurent series and Puiseux series, as well as complex asymptotic expansions for special functions with elaborate branch cut structures. Many of the formulas used are original to Mathematica Series construct a series expansion in one or more variables Normal convert from a series expansion to an ordinary expression O symbolic representation of a higher-order series term Assumptions Assuming give assumptions about parameters Coefficient coefficient of a particular term in an ordinary power series CoefficientList coefficients in an ordinary power series SeriesCoefficient coefficient of a term in a general series InverseSeries find the functional inverse of a series ComposeSeries find the functional composition of series Limit find the limit of a series at its expansion point Integrate integrate a series D differentiate a series LogicalExpand expand out equations for series

    50. Cumulant Approximations And Renormalized Wigner-Kirkwood Expansion For Quantum B
    Cumulant approximations and renormalized WignerKirkwood expansion for quantum Boltzmann densities. Authors, Royer, Antoine. Affiliation
    http://adsabs.harvard.edu/abs/1985PhRvA..32.1729R
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    Title:
    Cumulant approximations and renormalized Wigner-Kirkwood expansion for quantum Boltzmann densities Authors:
    Affiliation:
    Publication:

    Physical Review A (General Physics), Volume 32, Issue 3, September 1985, pp.1729-1743 ( PhRvA Homepage Publication Date:
    Origin:

    AIP,
    (c) 1985: The American Physical Society DOI:
    10.1103/PhysRevA.32.1729
    Bibliographic Code:
    Abstract
    V =p /2m+V, is expressed as a cumulant expansion in powers of v=V-W, where W(x)=V(X)+V'(X)(x-X)+(1/2)V''(X )(x-X) V ). By Taylor expanding v(x) about X in the cumulant expansion, we obtain an expansion which is a resummation over powers of V''(X) of the Wigner-Kirkwood (WK) expansion of lnrho V V ). In lowest order, it yields an approximation initially proposed by Miller. Bibtex entry for this abstract Preferred format for this abstract (see Preferences
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    51. Biorthogonal Wavelets
    This paper looks at the effect of the number of vanishing moments on the approximation power of wavelet expansions. The StrangFix conditions imply that the
    http://bigwww.epfl.ch/publications/unser9604.html
    Biomedical Imaging Group Publications English only BIG Publications
    CONTENTS Home Page News Events People ... Download Algorithms
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    Approximation Power of Biorthogonal Wavelet Expansions
    M. Unser
    IEEE Transactions on Signal Processing, vol. 44, no. 3, pp. 519-527, March 1996.
    This paper looks at the effect of the number of vanishing moments on the approximation power of wavelet expansions. The Strang-Fix conditions imply that the error for an orthogonal wavelet approximation at scale a = 2 -i globally decays as a N , where N is the order of the transform. This is why, for a given number of scales, higher order wavelet transforms usually result in better signal approximations. We prove that this result carries over for the general biorthogonal case and that the rate of decay of the error is determined by the order properties of the synthesis scaling function alone. We also derive asymptotic error formulas and show that biorthogonal wavelet transforms are equivalent to their corresponding orthogonal projector as the scale goes to zero. These results strengthen Sweldens earlier analysis and confirm that the approximation power of biorthogonal and (semi-)orthogonal wavelet expansions is essentially the same. Finally, we compare the asymptotic performance of various wavelet transforms and briefly discuss the advantages of splines. We also indicate how the smoothness of the basis functions is beneficial in reducing the approximation error.
    webmaster.big@epfl.ch

    52. Calculus And Mathematica At UIUC
    Centering expansions for good approximation. Newton s method for root finding. Successes and failures of Newton s method. nbsp; Using the complex
    http://cm.math.uiuc.edu/231syl.php
    The Courses
    Students
    Staff
    Handin Systems
    Links
    Math 231
    Expansions and techiques of integration
    Authors: Bill Davis, Horacio Porta and Jerry Uhl
    Producer: Bruce Carpenter
    Publisher: Distr ibutor:
    3.Approximation
    3.01 Splines
    Mathematics. Remarkable plots explained by order of contact. Splining for smoothness at the knots. Science and math experience.
    3.02 Expansions in Powers of x
    Mathematics. The expansion of a function in powers of as a file of polynomials with higher and higher orders of contact with at . The expansions every literate calculus person knows: , and
    Converting known expansions to others via change of variable.
    Expansions for approximations. Science and math experience. . Expansions by substitution. Expansions by differentiation. ;Expansions by integration. Recognition of expansions. Expansions that satisfy a priori error bounds.
    3.03 Using Expansions

    53. Miscellanea Interpolation Rules Suggested By Asymptotic Expansions
    Your browser may not have a PDF reader available. Google recommends visiting our text version of this document.
    http://biomet.oxfordjournals.org/cgi/reprint/79/3/651.pdf

    54. Analytical Approximations Of Fractional Delays: Lagrange Interpolators And Allpa
    Your browser may not have a PDF reader available. Google recommends visiting our text version of this document.
    http://doi.ieeecomputersociety.org/10.1109/ICASSP.1997.599673
    var sc_project=2763585; var sc_invisible=0; var sc_partition=28; var sc_security="3c678009"; var addtoLayout=0; var addtoMethod=0; var AddURL = escape(document.location.href); // this is the page's URL var AddTitle = escape(document.title); // this is the page title Advanced Search CS Search Google Search

    55. Adaptive Greedy Approximations
    When using a suitable dictionary, the expansion of a function into its coherent structures yields a compact approximation. We demonstrate a denoising
    http://research.microsoft.com/research/pubs/view.aspx?type=Publication&id=780

    56. Explaining The Saddlepoint Approximation
    We give an elementary motivation and explanation of approximation techniques, starting with Taylor series expansions and progressing to the Laplace
    http://www.questia.com/PM.qst?a=o&se=gglsc&d=5001894379

    57. Keith Price Bibliography Wavelet Representations
    MathAnal(25), No 5, 1994, pp. 14121433. BibRef 9400 Unser, M.Michael, Vanishing moments and the approximation power of wavelet expansions,
    http://www.visionbib.com/bibliography/compute79.html
    4.10.1 Wavelet Representations
    Chapter Contents (Back) Wavelets Representation, Wavelets . Wavelet transformations are generated by a smoothing filter and a wavelet filter. Different wavelet filters can be applied. One interpretation is that the wavelet transform is the same as applying quadrature mirror filters. The wavelet provides both time and frequency information. (Or in images space and frequency.) For general wavelet information see: Wavelet.Org:
    WWW Version
    Grosssmann, A. , and Morlet, J.
    Decomposition of Hardy Functions into Square Integrable Wavelets of Constant Shape
    , 1984, pp. 723-736. The mathmatical introduction of wavelets. Later adopted by computer vision. BibRef Pentland, A.P.[Alex P.]
    Interpolation Using Wavelet Bases
    PAMI(16)
    , No. 4, April 1994, pp. 410-414.
    IEEE Abstract
    IEEE Top Reference
    WWW Version
    BibRef
    Earlier:
    Surface Interpolation Using Wavelets
    WWW Version
    BibRef And: Spatial and Temporal Surface Interpolation Using Wavelet Bases SPIE(1570) , 1991, pp. 43-62. Surface Reconstruction Regularization BibRef Mann, S.[Steve] Wavelets and 'Chirplets': Time-Frequence 'Perspectives' with Applications AMV Strategies92 1992, pp. 99-128. Both space and time domain sampling.

    58. The Map Expansion Obtained With Recombinant Inbred Strains And Intermated Recomb
    Since the recursive formulas turned out to be tedious, approximation formulas were added, giving a useful tool to determine the map expansions for each
    http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=1450404

    59. Approximation-theoretic Analysis Of Translation Invariant Wavelet
    Your browser may not have a PDF reader available. Google recommends visiting our text version of this document.
    http://ieeexplore.ieee.org/iel5/7594/20726/00959122.pdf

    60. First Steps In Numerical Analysis
    Note that this expansion has only oddpowered terms, although the polynomial approximation is of degree (2k - 1 ) - it has only k terms.
    http://kr.cs.ait.ac.th/~radok/math/mat7/step5.htm
    STEP 5
    ERRORS
    Approximation to functions
    An important procedure in Analysis is to represent a given function as an infinite serie s of terms involving simpler or otherwise more appropriate functions. Thus, if f is the given function, it may be represented as the series expansion involving the set of functions ( f i ). Mathematicians have spent a lot of effort in discussing the convergence of series , i.e., on defining conditions for which the partial sum approximates the function value f (x) ever more closely as n increases. In Numerical Analysis , we are primarily concerned with such convergent series; computation of the sequence of partial sums is an approximation process in which the truncation error may be made as small as we please by taking a sufficient number of terms into account.
  • Taylor series
    The most important expansion to represent a function is the Taylor series . If f is suitably smooth in the neighbourhood of some chosen point x where here h = x - x denotes the displacement from x to the point x in the neighbourhood, and the
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