41. 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 Christopher F. Baum ( 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 2005-06-17 Handle: RePEc:cup:etheor:v:8:y:1992:i:3:p:343-67

42. Asymptotic Expansions For The Laplace Approximations Of Sums Of Banach Space-val Asymptotic expansions for the Laplace approximations of sums of Banach spacevaluedrandom variables. Source Ann. Probab. 33, no. 1 (2005), 300–336 http://dx.doi.org/10.1214/009117904000001017

Current Issue Past Issues Search this Journal Editorial Board ... Other IMS and IMS Related Journals Sergio Albeverio and Song Liang Asymptotic expansions for the Laplace approximations of sums of Banach space-valued random variables Source: Ann. Probab. Abstract: Let X i i N , be i.i.d. B -valued random variables, where B B into R . An asymptotic evaluation of Z n E (exp( n i n X i n ))), up to a factor (1+ o (1)), has been gotten in Bolthausen [ Probab. Theory Related Fields Probab. Theory Related Fields Z n as n Primary Subjects: Keywords: Laplace approximation; asymptotic expansions; i.i.d. random vectors; Banach space-valued random variables; Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text Alternatively, the document is available for a cost of $15. Select the "buy article" button below to purchase this document from a secured VeriSign, Inc. site. Euclid Identifier: euclid.aop/1108141728

43. Cookies Required Scalar multipole expansions have been widely used in many areas of physics asgeneral approximations and expansions . Mathematical methods in physics . http://dx.doi.org/10.1063/1.333885

What is Scitation? News Contact Us Help Welcome to Scitation. Sign In Register EXIT Home ... Usage Reports Welcome! Sign In Sign up for free Send Feedback ... Learn more about our new features! Cookies Required ALERT! This service requires a web browser and/or firewall/network configuration that supports and accepts cookies. You may have been redirected to this page for one or more of the following reasons: You are using a browser that supports cookies, but cookie acceptance is disabled. You are using an older browser that does not support cookies. You are utilizing a personal firewall that is configured to override your browser settings and reject cookies. You are accessing Scitation from a network or proxy that is configured to reject cookies. (1) Browsers that support cookies: You may be using a browser that does support cookies, but you have cookies disabled. Setting up your browser to accept cookies is straightforward in most current versions of Scitation's recommended browsers . Once you have enabled cookies, you may continue to the URL you requested or return to the previous page (2) Browsers that do not support cookies: If you are using a browser that does not support cookies, you must upgrade to a cookie-capable browser. Consult the Scitation

44. Chebyshev Expansions - Curves - Math Library For C# And VB.NET: BLAS, LAPACK, Mo NET supports Chebyshev approximations through the the ChebyshevSeries class. Using Chebyshev expansions outside of this interval is usually not http://www.extremeoptimization.com/Mathematics/UsersGuide/Curves/ChebyshevExpans

Extreme Optimization Mathematics Library for .NET User's Guide Curves Extreme Optimization Mathematics Library for .NET User's Guide Up Curves Next Piecewise Curves and Cubic Splines Previous Polynomials Table of Contents Chebyshev Expansions Chebyshev polynomials form a special class of polynomials whose properties make them especially suited for approximating other functions. As such, they form an essential part of any numerical library. The Extreme Optimization Mathematics Library for .NET supports Chebyshev approximations through the the ChebyshevSeries class. The ChebyshevSeries class A Chebyshev expansion is a linear combination of Chebyshev polynomials. The Chebyshev polynomials are never formed explicitly. All calculations can be performed using only the coefficients. The Chebyshev polynomials provide an alternate basis for representating general polynomials. Two characteristics make Chebyshev polynomials especially attractive. They are mutually orthogonal, and there exists a simple recurrence relation between consecutive polynomials. Chebyshev polynomials are defined over the interval [-1, 1]. Using Chebyshev expansions outside of this interval is usually not meaningful and is to be avoided. To allow expansions over any finite interval, transformations are applied wherever necessary.

45. Browse MSC2000 approximations and expansions not classified at a more specific level Asymptotic approximations, asymptotic expansions steepest descent, etc. http://www.zblmath.fiz-karlsruhe.de/MATH/msc/zbl/msc/2000/41-XX/dir

Contact Search Browse Instructions ... Main Changes 70th anniversary Zentralblatt MATH Home Facts and Figures Partners and Projects Subscription Service Database Gateway Database Mirrors Reviewer Service Classification ... Serials and Journals database Miscellanea Links to the Mathematical World Display Text version Printer friendly page Internal Browse MSC2000 - by section and classification TOP MSC2000 - Mathematics Subject Classification Scheme 41-XX Approximations and expansions [For all approximation theory in the complex domain, see and ; for all trigonometric approximation and interpolation, see and ; for numerical approximation, see Classification Topic X-ref General reference works handbooks, dictionaries, bibliographies, etc. Instructional exposition textbooks, tutorial papers, etc. Research exposition monographs, survey articles Historical must also be assigned at least one classification number from Section 01 Explicit machine computation and programs not the theory of computation or programming Proceedings, conferences, collections, etc.

46. Browse MSC2000 Section 41XX approximations and expansions For all approximation theory 41A60 Asymptotic approximations, asymptotic expansions steepest descent, etc. http://www.zblmath.fiz-karlsruhe.de/MATH/text/msc/zbl/msc/2000/41-XX/dir

ZENTRALBLATT MATH Home Facts and Figures Partners and Projects Subscription ... Graphical Version of this page. Browse MSC2000 - by section and classification Search Browse the MSC2000 by Classification]-[ Instructions for using the classification]-[ Main Changes from MSC1991 to MSC2000] TOP MSC2000 - Mathematics Subject Classification Scheme Section: 41-XX Approximations and expansions [For all approximation theory in the complex domain, see and ; for all trigonometric approximation and interpolation, see and ; for numerical approximation, see 41-00 General reference works handbooks, dictionaries, bibliographies, etc. 41-01 Instructional exposition textbooks, tutorial papers, etc. 41-02 Research exposition monographs, survey articles

47. [41Xxx] --  Approximations And Expansions 41Xxx approximations and expansions. 41A17 . Inequalities inapproximation (Bernstein, Jackson, Nikolprime skiitype inequalities). 41A21 http://jipam-old.vu.edu.au/subj_classf/41Xxx.htm

Approximations and Expansions Inequalities in approximation (Bernstein, Jackson, Nikolprime skii-type inequalities) Padé approximation Rate of convergence, degree of approximation Approximation by other special function classes Best approximation, Chebyshev systems Approximate quadratures Miscellaneous topics Editors R.P. Agarwal G. Anastassiou T. Ando H. Araki A.G. Babenko D. Bainov N.S. Barnett H. Bor J. Borwein P.S. Bullen P. Cerone S.H. Cheng L. Debnath S.S. Dragomir N. Elezovic A.M. Fink A. Fiorenza T. Furuta L. Gajek H. Gauchman C. Giordano F. Hansen D. Hinton A. Laforgia L. Leindler C.-K. Li L. Losonczi A. Lupas R. Mathias T. Mills G.V. Milovanovic R.N. Mohapatra B. Mond M.Z. Nashed C.P. Niculescu I. Olkin B. Opic B. Pachpatte Z. Pales C.E.M. Pearce J. Pecaric L.-E. Persson L. Pick I. Pressman S. Puntanen F. Qi A.G. Ramm T.M. Rassias A. Rubinov S. Saitoh J. Sandor S.P. Singh A. Sofo H.M. Srivastava K.B. Stolarsky G.P.H. Styan L. Toth R. Verma F. Zhang School of Communications and Informatics Victoria University of Technology JIPAM is published by the School of Communications and Informatics which is part of the Faculty of Engineering and Science , located in Melbourne, Australia. All correspondence should be directed to the

48. Numerical Approximations Using Chebyshev Polynomial Expansions Numerical approximations using Chebyshev polynomial expansions Elgendi s methodrevisited. Bogdan Mihaila1 and Ioana Mihaila2 http://www.iop.org/EJ/S/UNREG/abstract/0305-4470/35/3/317

@import url(http://ej.iop.org/style/nu/EJ.css); Quick guide Site map Athens login IOP login: Password: Create account Alerts Contact us Journals Home ... This issue B Mihaila et al J. Phys. A: Math. Gen. Numerical approximations using Chebyshev polynomial expansions: El-gendi's method revisited Bogdan Mihaila and Ioana Mihaila Physics Division, Argonne National Laboratory, Argonne, IL 60439, USA Department of Mathematics and Statistics, Coastal Carolina University, Conway, SC 29528-6054, USA E-mail: bogdan@theory.phy.anl.gov Received 12 July 2001, in final form 6 November 2001 Published 11 January 2002 Print publication: Issue 3 (25 January 2002) Abstract. We present numerical solutions for differential equations by expanding the unknown function in terms of Chebyshev polynomials and solving a system of linear equations directly for the values of the function at the extrema (or zeros) of the Chebyshev polynomial of order N (El-gendi's method). The solutions are exact at these points, apart from round-off computer errors and the convergence of other numerical methods used in solving the linear system of equations. Applications to initial value problems in time-dependent quantum field theory, and second-order boundary value problems in fluid dynamics are presented. PACS numbers: 02.70.-c, 02.30.Mv, 02.60.Jh, 02.60.Nm, 02.60.Lj, 02.70.Bf

49. Laplace Approximations For Hypergeometric Functions With Matrix Argument, Roland In this paper we present Laplace approximations for two functions of Asy mptotic expansions for distributions of latent roots in multivariate analysis. http://projecteuclid.org/Dienst/UI/1.0/Display/euclid.aos/1031689021

Current Issue Past Issues Search this Journal Editorial Board ... Other IMS and IMS Related Journals Roland W. Butler and Andrew T. A. Wood Laplace approximations for hypergeometric functions with matrix argument Source: Ann. Statist. Abstract: References Primary Subjects: Seconday Subjects: Keywords: Full-text: Open access Download the full-text in the following format: PDF (194 KB) Euclid Identifier: euclid.aos/1031689021 Digital Object Identifier (DOI): 10.1214/aos/1031689021 Mathmatical Reviews number (MathSciNet): Zentralblatt Math Identifier: To Table of Contents for this Issue References ABRAMOWITZ, M. and STEGUN, I. A. (1972). Handbook of Mathematical Functions, 9th ed. Dover, New York. BAGCHI, P. and KADANE, J. B. (1991). Laplace approximations to posterior moments and marginal distributions on circles, spheres, and cy linders. Canad. J. Statist. 19 67-77. Mathematical Reviews: BARNDORFF-NIELSEN, O. E. and WOOD, A. T. A. (1998). On large deviations and choice of ancillary for p and r . Bernoulli 4 35-63. Mathematical Reviews: BINGHAM, C., CHANG, T. and RICHARDS, D. (1992). Approximating the matrix Fisher and Bingham distributions: applications to spherical regression and Procrustes analysis. J. Multivariate Anal. 41 314-337.

50. [math/0503601] Asymptotic Expansions For The Laplace Approximations Of Sums Of B Asymptotic expansions for the Laplace approximations of sums of Banach spacevaluedrandom variables. Authors Sergio Albeverio, Song Liang http://arxiv.org/abs/math/0503601

Mathematics, abstract math.PR/0503601 From: Sergio Albeverio [ view email ] Date: Fri, 25 Mar 2005 14:37:45 GMT (125kb,S) Asymptotic expansions for the Laplace approximations of sums of Banach space-valued random variables Authors: Sergio Albeverio Song Liang Comments: Published at this http URL in the Annals of Probability ( this http URL ) by the Institute of Mathematical Statistics ( this http URL Report-no: IMS-AOP-AOP299 Subj-class: Probability MSC-class: 62E20, 60F10, 60B12. (Primary) Journal-ref: Annals of Probability 2005, Vol. 33, No. 1, 300-336 DOI: Full-text: PostScript PDF , or Other formats References and citations for this submission: CiteBase (autonomous citation navigation and analysis) Which authors of this paper are endorsers? Links to: arXiv math find abs

51. [physics/9901005] Numerical Approximations Using Chebyshev Polynomial Expansions Numerical approximations Using Chebyshev Polynomial expansions. Authors BogdanMihaila, Ioana Mihaila Comments minor wording changes, some typos have been http://arxiv.org/abs/physics/9901005

Physics, abstract physics/9901005 From: Bogdan Mihaila [ view email ] Date ( ): Thu, 7 Jan 1999 17:19:43 GMT (27kb) Date (revised ): Wed, 11 Jul 2001 22:30:03 GMT (124kb) Date (revised v3): Wed, 31 Oct 2001 21:16:05 GMT (253kb) Numerical Approximations Using Chebyshev Polynomial Expansions Authors: Bogdan Mihaila Ioana Mihaila Comments: minor wording changes, some typos have been eliminated Subj-class: Computational Physics Journal-ref: J. Phys. A: Math. Gen. 35, 731 (2002) We present numerical solutions for differential equations by expanding the unknown function in terms of Chebyshev polynomials and solving a system of linear equations directly for the values of the function at the extrema (or zeros) of the Chebyshev polynomial of order N (El-gendi's method). The solutions are exact at these points, apart from round-off computer errors and the convergence of other numerical methods used in connection to solving the linear system of equations. Applications to initial value problems in time-dependent quantum field theory, and second order boundary value problems in fluid dynamics are presented. Full-text: PostScript PDF , or Other formats References and citations for this submission: CiteBase (autonomous citation navigation and analysis) Which authors of this paper are endorsers?

52. Warning Main(../prefs.php) Failed To Open Stream No Such File expansions for approximations. The expansion of a function fx in powers of (x b) as a file of polynomials with higher and higher orders of contact with http://www.distancecalculus.com/courses/calcIII.php

Warning : main(../prefs.php): failed to open stream: No such file or directory in /usr/www/users/webprim4/rcurtis2/suffolk/courses/calcIII.php on line Warning /usr/www/users/webprim4/rcurtis2/suffolk/courses/calcIII.php on line Warning /usr/www/users/webprim4/rcurtis2/suffolk/courses/calcIII.php on line Calculus III (Series) Detailed Syllabus Mathematics Remarkable plots explained by order of contact. Splining for smoothness at the knots. The expansion of a function f[x] in powers of x as a file of polynomials with higher and higher orders of contact with f[x] at x = 0. The expansions every literate calculus person knows (1/(1 - x), ex, sin[x] and cos[x]). Expansions for approximations. The expansion of a function f[x] in powers of (x - b) as a file of polynomials with higher and higher orders of contact with f[x] at x = b. Newton's method. Multiplying and dividing expansions. Using expansions to help to calculate limits at a point. Expansions and the complex exponential function. Using expansions to help to get precise estimates of some integrals. Science and Math Experience Experiments geared at discovering that the smoother the transition from one curve to another at a knot, the better both curves approximate each other near the knot.

53. Citebase - Numerical Approximations Using Chebyshev Polynomial Expansions Numerical approximations Using Chebyshev Polynomial expansions. Authors Mihaila,Bogdan; Mihaila, Ioana. We present numerical solutions for differential http://citebase.eprints.org/cgi-bin/citations?id=oai:arXiv.org:physics/9901005

54. Entrez PubMed The subject of this paper is the construction of the exponential asymptoticexpansions of the unstab http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=1

55. MSC91 41XX approximations and expansions, {For all approximation theory in the 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc. http://bieson.ub.uni-bielefeld.de/opus/msc_ebene2.php?zahl=41&anzahl=0

56. 1 Introduction Included in Stage (ii) are expansions in series of Chebyshev polynomials or rational approximations, or truncated Chebyshevseries expansions; see, 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.

57. The Mathematica Book Online: Mathematica Reference Guide | Some Notes On Interna FilledSmallCircle Bessel functions use series and asymptotic expansions. method starting from rational approximations and asymptotic expansions. http://documents.wolfram.com/mathematica/book/section-A.9.4

PreloadImages('/common/images2003/btn_products_over.gif','/common/images2003/btn_purchasing_over.gif','/common/images2003/btn_services_over.gif','/common/images2003/btn_new_over.gif','/common/images2003/btn_company_over.gif','/common/images2003/btn_webresource_over.gif'); Mathematica Documentation All Documentation All of wolfram.com The Mathematica Book Online Mathematica Reference Guide ... Some Notes on Internal Implementation A.9.4 Numerical and Related Functions Number representation and numerical evaluation Large integers and high-precision approximate numbers are stored as arrays of base or digits, depending on the lengths of machine integers. Precision is internally maintained as a floating-point number. IntegerDigits RealDigits and related base conversion functions use recursive divide-and-conquer algorithms. Similar algorithms are used for number input and output. N uses an adaptive procedure to increase its internal working precision in order to achieve whatever overall precision is requested. Floor Ceiling and related functions use an adaptive procedure similar to N to generate exact results from exact input.

58. Numerical And Related Functions FilledSmallCircle PolyLog uses EulerMaclaurin summation, expansions in terms of method starting from rational approximations and asymptotic expansions. http://documents.wolfram.com/v5/TheMathematicaBook/MathematicaReferenceGuide/Som

PreloadImages('/common/images2003/btn_products_over.gif','/common/images2003/btn_purchasing_over.gif','/common/images2003/btn_services_over.gif','/common/images2003/btn_new_over.gif','/common/images2003/btn_company_over.gif','/common/images2003/btn_webresource_over.gif'); Mathematica 5 Documentation All of Documentation Center All of wolfram.com Documentation Mathematica The Mathematica Book Mathematica ... Some Notes on Internal Implementation A.9.4 Numerical and Related Functions Number representation and numerical evaluation Large integers and high-precision approximate numbers are stored as arrays of base or digits, depending on the lengths of machine integers. Precision is internally maintained as a floating-point number. IntegerDigits RealDigits and related base conversion functions use recursive divide-and-conquer algorithms. Similar algorithms are used for number input and output. N uses an adaptive procedure to increase its internal working precision in order to achieve whatever overall precision is requested. Floor Ceiling and related functions use an adaptive procedure similar to N to generate exact results from exact input.

59. Personal Asymptotic approximations of integrals The term by term integration method. Asymptotic expansions of the Whittaker functions for large order parameter. http://www.unavarra.es/personal/jl_lopez/

Curriculum Professor of Mathematics Education Employment History Research Interests Recent Publications (since 1998) (4) Recent Reprints Recent Lectures Teaching Contact Information Education Degree in Physics, 1990 . University of Zaragoza. Ph. Degree in Physics, 1995. University of Zaragoza. Degree in Mathematics, 1997 . University of Zaragoza. Employment History University of Zaragoza, 1995-1999. Associate Professor. State University of Navarra, 1999-present. Full Professor. Research Interests Asymptotic Approximation of Integrals. Analytical Aspects of Special Functions. Singular Perturbation Problems: Asymptotic Approximation. Limit Cycles of Dynamical Systems. Recent Publications (since 1998) 1. Several Series containing Gamma and Polygamma Functions. J. Comp. Appl. Math. 90 (1998) 15-23. 3. A family of multiple integrals analytically solvable. Appl. Math. Lett. 12 (1999) 119- 125. 4. The Whittaker function M as a function of k. Const. Approx. 15 (1998) 83-95. With J. Sesma

60. Handbook Of Mathematical Functions With Formulas, Graphs, And 9.7 Asymptotic expansions 9.8 Polynomial approximations Kelvin Functions 9.10 Asymptotic expansions 9.11 Polynomial approximations Numerical Methods http://www.knovel.com/knovel2/Toc.jsp?BookID=528

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