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useful algorithms rather than make an attempt to be encyclopedic: algorithms or approximations that have clearly been superseded are omitted.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 About EconPapers Working Papers Journal Articles ... Format for printing EconPapers has moved to http://econpapers.repec.org! Please update your bookmarks. 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

Default Risk .com the web's biggest credit risk modeling resource. Submit Your Paper Fitch Ratings Jobs Worldwide Post Your Résumé For Recruiters Featured Book The Analytics of Risk Model Validation by George A. Christodoulakis (Editor), Stephen Satchell (Editor) , Academic Press, November 11, 2007 , Hardcover, 216 pages Training Discounted for Default Risk .com visitors only The Mathematics of Credit Derivatives: The Essential Credit Modelling and Pricing Companion by Philipp J. Schönbucher WBS Training, August 2003, DVD / Interactive CD-ROM Sponsor: Shop at Amazon.com and support 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

Constructive Approximation Links to CA Problems Approximation Theory and its Applications ... Learning Evaluation Functions GAMS (Generalized Algebraic Modeling System) Development Corporation Model Library Subject Index: Agricultural Economics Applied General Equilibrium Modeling Chemistry and Chemical Engineering ... MobiDK: Ministry of Business and Industry, Model for Denmark Mathematical and Computational Science Division, Information Technology Laboratory, National Institute of Standards and Technology Guide to Available Mathematical Software Catalog of Algorithms for Approximation Problem Decision Tree Linear Algebra ... Virtual Pro, Inc.: Programming resources of Russia Can an approximation in multidimensional spaces be constructive indeed And what's about approximation of gradients/Hessians/etc., when a smooth function is given by a finite set of values corrupted with uncorrelated noise The Brachistochrone Challenge "I, Johann Bernoulli, greet the most clever mathematicians in the world. Nothing is more attractive to intelligent people than an honest, challenging problem whose possible solution will bestow fame and remain as a lasting monument. Following the example set by Pascal, Fermat, etc., I hope to earn the gratitude of the entire scientific community by placing before the finest mathematicians of our time a problem which will test their methods and the strength of their intellect. If someone communicates to me the solution of the proposed problem, I shall then publicly declare him worthy of praise."

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/

Your browser does not support scripts 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

Sign on SAO/NASA ADS Physics Abstract Service Find Similar Abstracts (with default settings below Electronic Refereed Journal Article (HTML) References in the article Citations to the Article (1) ... Translate This Page 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 Find Similar Abstracts: Use: Authors Title Abstract Text Return: Query Results Return items starting with number Query Form Database: Astronomy Physics arXiv e-prints SAO/NASA ADS Homepage ADS Sitemap Query Form Basic Search ... FAQ

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 DOWNLOAD PDF Postscript All BibTeX References 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

Home People Student Comments F.A.Q. The Courses Syllabi Students Homework Tips Homework Examples Download Lessons Help! Staff Lab Schedule Grading Tips Instructor Tips Tech Help Handin Systems ClassComm Moodle Links Netmath OSU Program Wolfram MathEverywhere 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 Home Digital Library Podcasts Site Map ... Table of Contents Abstract 1997 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP'97) - Volume 1 p. 455 Analytical Approximations of Fractional Delays: Lagrange Interpolators and Allpass Filters S. Tassart P. Depalle Full Article Text: DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/ICASSP.1997.599673 Send link to a friend Back to Top Additional Information Citation: S. Tassart, P. Depalle, "Analytical Approximations of Fractional Delays: Lagrange Interpolators and Allpass Filters," icassp , p. 455, 1997 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP'97) - Volume 1, 1997

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

Page 3     41-60 of 71    Back | 1  | 2  | 3  | 4  | Next 20