21. Euler Og Runge-Kutta Metoder Biografi af carle runge (18561927) Biografi af Martin Kutta (1867-1944) Metodener også kendt som Heun's metode og er simpelthen en forbedring af Eulers http://www.frhavn-gym.dk/matematik/mrunge.html

Dette er et interaktivt kursus i numerisk løsning af differentialligninger ud fra Euler's metode og Runge-Kutta metoder. Klassetrin:Højt niveau, Gymnasiet. Tag allerførst en kopi af denne side Introduktion Ved praktisk løsning af differentialligninger stilles man ofte over for et Begyndelsesværdiproblem Vi har differentialligningen med et startpunkt og skal undersøge, hvordan udviklingen er som tiden går. Det viser sig nu, at der under generelle betingelser er en entydig løsning. Augustin Cauchy (1789-1857) har som den første studeret på dette eksistens- og entydighedsproblem for differentialligninger. Linieelementer Den første angrebsvinkel til vort arbejde med problemet skal være Linieelementer Betragt f.eks. differentialligningen (¤) dy/dt = -2*t*y Vælges (t,y)=(2,½) bliver ifølge (¤) dy/dt= -2. Vi tegner nu igennem (2,½) en lille liniestykke med hældning -2 og snakker om Linieelementet (2,½,-2) Hvis vi tegner linieelementer for et utal af punkter i planen, antydes der en serie af løsningskurver. På SOS-math finder du linieelementer (slope field) og gode eksempler.

22. ADIFOR - Generating Derivative Codes From Fortran Programs - Bischof, Carle, Cor bischof92adifor, author = Christian H. Bischof and Alan carle and George F multistepmethods (context) Dahlquist - 1963 19 Implicit runge-Kutta processes http://citeseer.nj.nec.com/bischof91adifor.html

Alternate document: Details ADIFOR Exception Handling ( Christian Bischof, George Corliss, Andreas Griewank ADIFOR Generating Derivative Codes from Fortran Programs (1991) (Make Corrections) (100 citations) Christian Bischof, Alan Carle, George Corliss, Andreas Griewank, Paul Hovland Scientific Programming Home/Search Context Related View or download: rice.edu/pub/CRPC PCTR91185S.ps.gz anl.gov/adifor/CRPCTR91185.ps Cached: PS.gz PS PDF DjVu ... Help From: rice.edu/CRPC/softli (more) Homepages: C.Bischof A.Carle G.Corliss A.Griewank ... (Update Links) Rate this article: (best) Comment on this article (Enter summary) Abstract: . The numerical methods employed in the solution of many scientific computing problems require the computation of derivatives of a function f : R n !R m . Both the accuracy and the computationalrequirements of the derivative computation are usually of critical importance for the robustness and speed of the numerical solution. ADIFOR (Automatic Differentiation In FORtran) is a source transformation tool that accepts Fortran 77 code for the computation of a function and writes portable... (Update) Context of citations to this paper: More coe#cients using the chain rule [8] Although still in its formative stages, AD is now a very powerful tool (e.g. ADOL C [9] and ADIFOR

23. R Index Roth, Klaus (706*) Roth, Leonard (97*) Routh, Edward (152) Rudio, Ferdinand (268*)Rudolff, Christoff (172) Ruffini, Paolo (2196*) runge, carle (332*) Russell http://www.math.hcmuns.edu.vn/~algebra/history/history/Indexes/R.html

24. CAAM 453 - Rice University - Fall 2002 Lecture 18 rungeKutta methods - Biography of carle runge - Biographyof Martin Kutta. Lecture 17 Error analysis for one-step ODE solvers. http://www.caam.rice.edu/~caam453/notes.html

Notes, Links, and Supplementary Material Course outline [for those preparing for the qualifying exam]: pdf postscript Lecture 40: Gaussian elimination: conditioning and stability Lecture 39: Gaussian elimination; pivoting Lecture 38: Practical modifications of the QR algorithm; Gaussian elimination Lecture 37: Convergence of the QR algorithm (power method, inverse iteration) Lecture 36: QR algorithms for eigenvalue problems Lecture 35: Solving discrete least squares problems via the QR factorization Lecture 34: QR factorization Lecture 33: Unitary invariance of the 2-norm; Householder reflectors Lecture 32: More on the singular value decomposition Lecture 31: Review of matrix theory, introduction to the singular value decomposition Lecture 30: Axioms of floating point arithmetic Lecture 29: Floating point number systems Lecture 28: Finite precision arithmetic Lecture 27: Method of lines for PDEs Lecture 26: Finite differences methods for elliptic PDEs Lecture 25: Boundary value problems (collocation, finite differences) Lecture 24: Absolute stability; 2-point boundary value problems (shooting)

25. Full Alphabetical Index Translate this page Johann (146) Roth, Leonard (97*) Routh, Edward (152) Rudio, Ferdinand (268*)Rudolff, Christoff (172) Ruffini, Paolo (167) runge, carle (332*) Russell http://www.geocities.com/Heartland/Plains/4142/matematici.html

Completo Indice Alfabetico Cliccare su una lettera sottostante per andare a quel file. A B C D ... XYZ Cliccare sotto per andare agli indici alfabetici separati A B C D ... XYZ Il numero di parole nella biografia e' dato in parentesi. Un * indica che c'e' un ritratto. A Abbe , Ernst (602*) Abel , Niels Henrik (286*) Abraham bar Hiyya (240) Abraham, Max Abu Kamil Shuja (59) Abu'l-Wafa al'Buzjani (243) Ackermann , Wilhelm (196) Adams, John Couch Adams, Frank Adelard of Bath (89) Adler , August (114) Adrain , Robert (79) Aepinus , Franz (124) Agnesi , Maria (196*) Ahlfors , Lars (725*) Ahmed ibn Yusuf (60) Ahmes Aida Yasuaki (114) Aiken , Howard (94) Airy , George (313*) Aitken , Alexander (825*) Ajima , Chokuyen (144) Akhiezer , Naum Il'ich (248*) al'Battani , Abu Allah (194) al'Biruni , Abu Arrayhan (306*) al'Haitam , Abu Ali (269*) al'Kashi , Ghiyath (73) al'Khwarizmi , Abu (123*) Albanese , Giacomo (282) Albert of Saxony Albert, Abraham Adrian (121*) (158*) Alberti , Leone (181*) Alberto Magno, San (109*) Alcuin di York (237*) Aleksandrov , Pave (160*) Alembert , Jean d' (291*) Alexander , James (163) Amringe , Howard van (354*) Amsler , Jacob (82) Anassagora di Clazomenae (169) Anderson , Oskar (67) Andreev , Konstantin (117) Angeli , Stefano degli (234) Anstice , Robert (209) Antemio of Tralles (55) Antifone il Sofista (125) Apollonio di Perga (276) Appell , Paul (1377) Arago , Dominique (345*) Arbogasto , Louis (87) Arbuthnot , John (251*) Archimede di Siracusa (467*) Archita of Tarentum (103) Argand , Jean (81) Aristeo il Vecchio (44) Aristarco di Samo (183) Aristotele Arnauld , Antoine (179)

26. R Index Roth, Leonard (97*) Routh, Edward (152) Rudin, Mary (1857*) Rudio, Ferdinand (268*)Rudolff, Christoff (172) Ruffini, Paolo (2196*) runge, carle (332*) Russell http://math.ichb.ro/History/Indexes/R.html

27. Courant He married Nerina runge, carle runge's daughter, on 22 January 1919 and acouple of months later began teaching as a privatdozent at Göttingen. http://math.ichb.ro/History/Mathematicians/Courant.html

28. History Of Mathematicians Used In Wi2023 The method of Heun; The rungeKutta method ( Martin Wilhelm Kutta (1867-1944),carle David Tolmé runge (1856-1927)). As an application http://ta.twi.tudelft.nl/nw/users/vuik/wi212tn/hist.html

History of mathematicians In this document we give some information of mathematicians which work or names are used in the course wi2023 "Numerieke Wiskunde voor technici". November 1947 can be seen as the birthday of modern numerical analysis. 1. Ordinary differential equations Ordinary differential equations are splitted into two classes: initial value problems and boundary value problems. In Chapter 1 initial value problems are considered. Several numerical integration methods are given and analysed as there are Euler methods ( Leonhard Euler (1707-1783) The method of Heun The Runge-Kutta method ( Martin Wilhelm Kutta (1867-1944) Carle David Tolmé Runge (1856-1927) As an application of the theory given in Chapter 1 of this course a simulation (using a Java-applet) of a double pendulum is possible. The integration is done by a Runge-Kutta method. To analyse the discretisation error of these methods the Taylor ( Brook Taylor (1685-1731) ) polynomial is used together with numerical integration methods: midpoint rule, trapezium rule and the integration method of Simpson ( Thomas Simpson (1710-1761) ). The order symbol of Landau (

29. History Of Mathematicians Used In Wi3097 The rungeKutta method Martin Wilhelm Kutta (1867-1944) and carle David Tolmérunge (1856-1927). 5. Finite differences for boundary value problems. http://ta.twi.tudelft.nl/nw/users/vuik/wi3097/hist.html

History of mathematicians In this document we give some information of mathematicians which work or names are used in the course wi3097 "Numerieke methoden voor differentiaalvergelijkingen". 1. Mathematical preliminaries Modern applied mathematics started in the 17 and 18 century with people like Simon Stevin (1548-1620) René Descartes (1596-1650) Isaac Newton (1642-1727) and Leonhard Euler (1707-1783) . Numerical aspects were used in analysis in a natural way; the name numerical mathematics was unknown. Numerical methods developed by Newton, Euler and later Carl Friedrich Gauss (1777-1855) play an important role in present day numerical mathematics. Additional information about Simon Stevin (in Dutch). November 1947 can be seen as the birthday of modern numerical analysis. The Taylor polynomial ( Brook Taylor (1685-1731) ) is used to analyse the error in various numerical approximations. The order symbol of Landau ( Edmund Georg Hermann Landau (1877-1938) ) is used to give a short notation of the approximation errors. 2. Interpolation

30. [HETHZ] Developed 1901 by Martin Kutta and later published by carle runge. 2) = runge-KUTTAAAA!!! =- Combat cry of the HETHZ clan. Don't ask why http://www.rungekutta.com/main_e.html

Hethz Homepage , [HETHZ], [HETHZ]Clan, [HETHZ] Modifikation, [HETHZ]Weasel, [HETHZ]LagChicken, [HETHZ]spastic, [HETHZ]SneakyViper, [HETHZ]FatalBranch, [HETHZ]Ghostface, [HETHZ]Lynx, [HETHZ]DeepBlue, [HETHZ]voege, [HETHZ]deathman, [HETHZ]ElekJoker, rungekutta Rungekutta: 1) Runge-Kutta Algorithm. A numerical Method for an approximate solution for ordinary differential equations. Developed 1901 by Martin Kutta and later published by Carle Runge. 2) "-= RUNGE-KUTTAAAA !!! =-". Combat cry of the [HETHZ] clan. Don't ask why... Runge-Kutta just rocks as a battle cry :-)

31. Numerical Solutions Of Differential Equations Links carle David Tolmé runge (18561927); Home Lectures Assignments Maple Quizzes Links Books Contact Page maintained by GW Delius. http://www.york.ac.uk/depts/maths/teaching/gwd/numerical/links.html

Numerical Solutions of Differential Equations 2003 Home Lectures Assignments Maple ... Contact Links On this page I am planning to list a few links to pages which I found interesting, entertaining or useful. If you find pages which you think might be interesting to others in the course, please let me know and I will post them here. Some disasters attributable to bad numerical computing Numerical Recipes on-line Entries in MathWorld on Differential Equations Solving Entry for Numerical Analysis in The Mathematical Atlas Scientific Computing and Associated Fields Resource Guide Biographies of some Mathematicians whose work we have met in the course: John Couch Adams Leonhard Euler Martin Wilhelm Kutta Carle David Tolmé Runge ... Contact Page maintained by G W Delius The Department of Mathematics

32. A 1907?), WDB; runge, carle David Tolme (1856-1927), Maths Archive;Russell, Bertrand Arthur William (1872-1970), Bjorn's Guide; Russell http://members.aol.com/jayKplanr/images.htm

return home An Alphabetical A-Z List of Famous Scientists and Mathematicians Indicates a portrait photograph or illustration is included. browse a section: A B C D ... Z A Abel, Niels Henrik Maths Archive Adams, John Couch Maths Archive Adams, Walter S. BM Agassiz, Louis UCMP Agnesi, Maria Gaetana Maths Archive Agnesi, Maria Gaetana ASC Aitken, Robert G. BM Alexander, Albert Ernest AAS Alfred Day Hershey BDB Ambartsumian, Viktor A. BM Ampere, Andre Marie 17th and 18th C Mathematicians Antoine, Albert C. Faces Apollonius of Perga (200 BC-100 BC), Maths Archive Arago, Francois Jean Dominique 17th and 18th C Mathematicians Arbogast, Antoine 17th and 18th C Mathematicians Arbuthnot, John Maths Archive Archimedes of Syracuse (287 BC - 212 BC), Maths Archive Aristarchus of Samos (310 BC-230 BC), Maths Archive Aristotle (384 BC-322 BC), Maths Archive Aristotle (384-322 BC), Bjorn's Guide Arrhenius, Svante August 1992 Institute Artin, Emil Maths Archive Artzt, Karen WDB Atanasoff, John Vincent

33. IndexArts-Sample Indexes Rousseau, 48, 67 Rubenistes, 67, 75 Rubens, 61 runge, Philip Otto, 118 S Sand, George,125 U UltraClassiques, 37 V Van Gogh, Vincent, 136 Vernet, carle, 94 http://members.aol.com/indexarts/delacroix.htm

IndexArts Sample Index David to Delacroix, 18th to 19th Century French Painting author Walter Friedlaender (Painting reproductions indicated by fig. Works of art in italics) A abstraction, 49 anticlassical, 49, 50 B Barbus: See Les Primitifs baroque, 54, 104, 125, 136 Bartolini, Lorenzo, 72 Bourbon Restoration, 58 C Caravaggio, 14 Carracci, Annibale, 58 Cézanne, Paul, 136 Chénier, André, 19 classicism, 37, 51, 72 coloring, 46, 136 Correggio, 58 D D'Angiviller, 15 David, Jacques-Louis, 12–35 antiquity, 13, 26 Bara , 25, fig.13 Belisarius Asking Alms , 14, fig.3 Brutus Bringing Home of the Bodies of His Sons , 18–19, fig.7 Distribution of the Eagles , 28-30, fig.15 Leonidas at Thermopylae , 31 Les Primitifs, 46–50 M. de Sériziat , 33, fig.19 Marat , 24, 25, 35, fig.14 Napoleon, 20–21 neoclassicism, 17 The Oath of the Horatii , 14-17, 35, fig. 1 Paris and Helen , 18, 33, fig.6 Pope Pius VII , 34, fig.21 portraits, 32–34 The Rape of the Sabines , 25–27, fig.8; realism, 24 Revolution, 20, 35 romantic features, 17 Sappho and Phaon Socrates Drinking the Hemlock , 17, fig.4

34. Matlab Links John Couch Adams; carle David Tolmé runge; Martin Wilhelm Kutta.Finite Difference TimeDomain Resources A free PostScript viewer http://implicit.che.utah.edu/Group/LinksFrm.htm

Matlab Links MathWorks Introduction to Matlab (postscript) Matlab Primer (postscript) Control Tutorial for Matlab Matlab Toolboxes Free Toolboxes Control Toolbox (pdf) Robust Control Toolbox On-line Reference pdf Manual Matlab tutorial from University of Utah Matlab tutorial from University of Florida People Here are links to some of the mathematicians, scientists, and engineers, whose contribution will be studied during this course. Leonhard Euler Brook Taylor Simon Denis Poisson Pierre-Simon Laplace ... Finite Difference Time-Domain Resources A free PostScript viewer is available at http://www.cs.wisc.edu/~ghost/aladdin/get550.html

35. History Of Mathematicians Used In Wi2091, Wi2092 The rungeKutta method Martin Wilhelm Kutta (1867-1944) and carle DavidTolmé runge (1856-1927); The Adams-Bashforth method. 5. Finite http://dutita0.twi.tudelft.nl/users/vuik/wi211/hist.html

History of mathematicians In this document we give some information of mathematicians which work or names are used in the course wi2091, wi2092 "Numerieke methoden voor differentiaal vergelijkingen". 1. Introduction Modern applied mathematics started in the 17 and 18 century with people like Simon Stevin (1548-1620) René Descartes (1596-1650) Isaac Newton (1642-1727) and Leonhard Euler (1707-1783) . Numerical aspects were used in analysis in a natural way; the name numerical mathematics was unknown. Numerical methods developed by Newton, Euler and later Carl Friedrich Gauss (1777-1855) play an important role in present day numerical mathematics. Additional information about Simon Stevin (in Dutch). The order symbol of Landau ( Edmund Georg Hermann Landau (1877-1938) ) is used to give a short notation of the approximation errors. 2. Interpolation In the error estimate of linear interpolation we use 'Rolle's Theorem' ( Michel Rolle (1652-1719) ). Thereafter linear interpolation is generalized to Lagrange interpolation ( Joseph-Louis Lagrange (1736-1813) ). In Hermite polynomials (

36. Bremen General Guide Address. (In German.). SCIENCE runge, carle David Tolmé Mathematicianborn in 1856. Short biography, poster, references. (In English.). http://www.gates96.sk/cam/Europe/Germany/Bremen/general.html

37. EOSSAA L'HERITAGE E 11.94Q 3 2 Rice, Matt ADHS 11.95Q 2 3 runge, Alec RENFREW Stephan THOUSANDISL 12.44 3 12 Scanlon, Kurt KINGSTON CVI 12.47 3 13 carle, Justin ST. http://wecssatf.tripod.com/2001-2002/results_EOSSAA.htm

38. Mathematical Remarks applicability to the empirical sciences. carle runge Doctoral Dissertation,Berlin, April 23, 1880. The value of a mathematical http://www.math.hmc.edu/~jacobsen/quotes.html

Each progress in mathematics is based on the discovery of stronger tools and easier methods, which at the time makes it easier to understand earlier methods. By making these stronger tools and easier methods his own, it is possible for the individual researcher to orientate himself in the different branches of mathematics. The organic unity of mathematics is inherent in the nature of this science, for mathematics is the foundation of all exact knowledge of natural phenomena. David Hilbert (1900 Paris Lecture) Technical skill is mastery of complexity while creativity is mastery of simplicity. E. C. Zeeman (note that this quote applies equally well to music and mathematics). Although it may be fashionable to acknowledge that everything is connected to everything else in principle, some things are more tightly connected to each other than to all the rest. Such a little knot of causal interactions goes by the name of a system. Art Winfree Mathematics is not a deductive science - that's a cliché. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation and guesswork. Paul Halmos Cauchy is mad and there is nothing that can be done about him, although, right now, he is the only one who knows how mathematics should be done.

39. Nature Publishing Group There is a chapter on the computation of tables, and another on the numerical solutionof differential equations in which carle runge takes pride of place. http://www.nature.com/cgi-taf/DynaPage.taf?file=/nature/journal/v403/n6771/full/

40. Current Institutional Reports - Oct 95 GD Pusch, C. Bischof, and A. carle, On Automatic Differentiation of Codes with Baker A. Tang, Stability analysis of continuous implicit rungeKutta methods http://www.netlib.org/signum-reports/archive/30_4_oct95.html

FROM THE INSTITUTIONAL REPORTS EDITOR With this issue the SIGNUM Newsletter begins providing an online version of its Institutional Reports column. Point your Web browser to http://www.netlib.org/signum-reports/ to see the new online offerings. Going online should offer advantages both to readers and to submitters of report listings. Exactly what direction the online service takes, however, will depend on feedback we get from you. The easiest way to send your ideas is to visit the Web page and click a button. If you don't have Web access, see my address on the inside front cover for other ways to reach me. Tom Rowan CURRENT INSTITUTIONAL REPORTS Please note: Reports can be obtained by writing to the person or address given for the publishing institution. We recommend requesting reports by author, title and number, since the information listed in this column has been transcribed at least once from the original sources. URLs included in the contact's address point to reports that are available online. ARGONNE NATIONAL LABORATORY Math and Computer Science Division Argonne National Laboratory 9700 Cass Avenue Argonne, IL 60439 Attn: Dr. Gail Pieper URL: http://www.mcs.anl.gov/Divisional/publications.html

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