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Line 1: {{short description\|None}} This is a timeline of key developments in [[computational mathematics]]. == 1940s == * Monte Carlo simulation (voted one of the top 10 [[algorithm]]s of the 20th century) invented at Los Alamos by von Neumann, Ulam and Metropolis.<ref>{{cite journal\|last=Metropolis\|first=N.\|title=The Beginning of the Monte Carlo method\|journal=Los Alamos Science\|year=1987\|volume=~~No.~~ 15~~, Page 125~~\|page=125–130\|url=http://library.lanl.gov/cgi-bin/getfile?15-12.pdf~~}}. Accessed~~ \|access-date=5 ~~may~~May 2012.}}</ref><ref>S. Ulam, R. D. Richtmyer, and J. von Neumann(1947). [http://library.lanl.gov/cgi-bin/getfile?00329286.pdf Statistical methods in neutron diffusion]. Los Alamos Scientific Laboratory report LAMS–551.</ref><ref>N. Metropolis and S. Ulam (1949). The Monte Carlo method. Journal of the American Statistical Association 44:~~335-341~~335–341.</ref> * Dantzig introduces the [[simplex method\|simplex algorithm]] (voted one of the top 10 algorithms of the 20th century).<ref>{{cite web\|title=SIAM News, November 1994.\|url=http://www.stanford.edu/group/SOL/dantzig.html\|accessdate=6 June 2012}} Systems Optimization Laboratory, Stanford University Huang Engineering Center (site host/mirror).</ref> * First [[Computational Fluid Dynamics\|hydro simulations]] at Los Alamos occurred.<ref>Richtmyer, R. D. (1948). Proposed Numerical Method for Calculation of Shocks. Los Alamos, NM: Los Alamos Scientific Laboratory LA-671.</ref><ref>A Method for the Numerical Calculation of Hydrodynamic Shocks. Von Neumann, J.; Richtmyer, R. D. Journal of Applied Physics, Vol. 21, ppp.~~232-237~~ 232–237</ref> * Ulam and von Neumann introduce the notion of cellular automata.<ref>Von Neumann, J., Theory of Self-~~Reproduiing~~Reproducing Automata, Univ. of Illinois Press, Urbana, 1966.</ref> * [[Manchester Small-Scale Experimental Machine#First programs\|A routine for the Manchester Baby]] written to factor a large number (2^18), one of the first in [[computational number theory]].<ref>[http://curation.cs.manchester.ac.uk/digital60/www.digital60.org/birth/manchestercomputers/mark1/manchester.html The Manchester Mark 1.]</ref> The Manchester group would make several other breakthroughs in [[Mersenne primes\|this area]].<ref>[http://news.bbc.co.uk/2/hi/technology/7465115.stm One tonne 'Baby' marks its birth: Dashing times.] By Jonathan Fildes, Science and technology reporter, BBC News.</ref> * LU decomposition technique first discovered. == 1950s == * [[Magnus Hestenes\|Hestenes]], [[Eduard Stiefel\|Stiefel]], and [[Cornelius Lanczos\|Lanczos]], all from the Institute for Numerical Analysis at the [[NIST\|National Bureau of Standards]], initiate the development of [[Iterative method\|Krylov subspace iteration method]]s.<ref>Magnus R. Hestenes and Eduard Stiefel, Methods of Conjugate Gradients for Solving Linear Systems, J. Res. Natl. Bur. Stand. 49, 409–436 (1952).</ref><ref>Eduard Stiefel, U¨ ber einige Methoden der Relaxationsrechnung (in German), Z. Angew. Math. Phys. 3, 1–33 (1952).</ref><ref>Cornelius Lanczos, Solution of Systems of Linear Equations by Minimized Iterations, J. Res. Natl. Bur. Stand. 49, 33–53 (1952).</ref><ref>Cornelius Lanczos, An Iteration Method for the Solution of the Eigenvalue Problem of Linear Differential and Integral Operators, J. Res. Natl. Bur. Stand. 45, 255–282 (1950).</ref> Voted one of the top 10 algorithms of the 20th century. * ''[[Equations of State Calculations by Fast Computing Machines]]'' introduces the [[Metropolis–Hastings algorithm]].<ref>{{cite journal \|first1=N. \|last1=Metropolis \|authorlink1=Nicholas Metropolis Line 14 ⟶ 20: \|first4=A.H. \|last4=Teller \|first5=E. \|last5=Teller \|authorlink5=Edward Teller \|title=[[Equations of State Calculations by Fast Computing Machines]] \|journal=[[Journal of Chemical Physics]] \|volume=21 \|issue=6 \|pages=1087–1092 \|year=1953 \|doi=10.1063/1.1699114 \|bibcode = 1953JChPh..21.1087M \|title-link=Equations of State Calculations by Fast Computing Machines \|osti=4390578 \|s2cid=1046577 }}</ref> Also, important earlier independent work by Alder and S. Frankel.<ref>Unfortunately, Alder's thesis advisor was unimpressed, so Alder and Frankel delayed publication of their results until much later. [http://scitation.aip.org/content/aip/journal/jcp/23/3/10.1063/1.1742004 Alder, B. J. , Frankel, S. P. , and Lewinson, B. A. , J. Chem. Phys., 23, 3 (1955)].</ref><ref>[http://www.hp9825.com/html/stan_frankel.html Stanley P. Frankel, Unrecognized Genius], HP9825.COM (accessed 29 Aug 2015).</ref> * [[Enrico Fermi]], [[Stanislaw Ulam]], ~~and~~[[John Pasta]], ~~with help from~~and [[Mary Tsingou]], discover the [[~~Fermi-Pasta-Ulam~~Fermi–Pasta–Ulam–Tsingou problem]].<ref>Fermi, E. (posthumously); Pasta, J.; Ulam, S. (1955) : [http://www.osti.gov/accomplishments/documents/fullText/ACC0041.pdf Studies of Nonlinear Problems (accessed 25 Sep 2012)]. Los Alamos Laboratory Document LA-1940. [http://www.cs.princeton.edu/courses/archive/fall09/cos323/papers/fpu55.pdf Also appeared] in 'Collected Works of Enrico Fermi', E. Segre ed. , [[University of Chicago Press]], Vol. II, 978–988, 1965. Recovered 21 Dec 2012</ref> In network theory, Ford & Fulkerson compute [[Ford–Fulkerson algorithm\|a solution to the maximum flow problem]].<ref>Ford, L. R.; Fulkerson, D. R. (1956). [http://www.cs.yale.edu/homes/lans/readings/routing/ford-max_flow-1956.pdf "Maximal flow through a network"] . [[Canadian Journal of Mathematics]]. 8: 399–404.</ref> Molecular dynamics invented by Alder and Wainwright<ref>Alder, B. J.; T. E. Wainwright (1959). "Studies in Molecular Dynamics. I. General Method". J. Chem. Phys. 31 (2): 459. Bibcode 1959JChPh..31..459A. doi:10.1063/1.1730376</ref>▼ * "[[Equations of State Calculations by Fast Computing Machines]]" introduces the [[Metropolis–Hastings algorithm]].<ref>Metropolis, N.; Rosenbluth, A.W.; Rosenbluth, M.N.; Teller, A.H.; Teller, E. (1953): [http://www.aliquote.org/pub/metropolis-et-al-1953.pdf Equations of State Calculations by Fast Computing Machines] (Retrieved 3 May 2012). Journal of Chemical Physics 21 (6): 1087–1092. Bibcode 1953JChPh..21.1087M. [[doi:10.1063/1.1699114]].</ref> * Householder invents his [[Householder matrix\|eponymous matrices]] and [[Householder transformation\|transformation method]] (voted one of the top 10 algorithms of the 20th century).<ref>{{cite journal\|first=A. S. \|last=Householder \|title=Unitary Triangularization of a Nonsymmetric Matrix\|journal=[[Journal of the ACM]] \|volume=5 \|issue=4 \|year=1958 \|pages=339–342\|doi=10.1145/320941.320947 \|mr=0111128\|s2cid=9858625 \|url=https://hal.archives-ouvertes.fr/hal-01316095/file/p339householderb.pdf }}</ref> ▲* Molecular dynamics invented by Alder and Wainwright<ref>Alder, B. J.; T. E. Wainwright (1959). "Studies in Molecular Dynamics. I. General Method". J. Chem. Phys. 31 (2): 459. Bibcode 1959JChPh..31..459A. doi:10.1063/1.1730376</ref> * [[John G.F. Francis]] <ref> J.G.F. Francis, "The QR Transformation, I", ''The Computer Journal'', vol. 4, no. 3, pages 265-271 (1961, received Oct 1959) [http://comjnl.oxfordjournals.org/cgi/content/abstract/4/3/265 online at oxfordjournals.org];<br> J. G. F. Francis, "The QR Transformation, I", ''The Computer Journal'', vol. 4, no. 3, pages 265–271 (1961, received Oct 1959) [http://comjnl.oxfordjournals.org/cgi/content/abstract/4/3/265 online at oxfordjournals.org];<br /> J. G. F. Francis, "The QR Transformation, II" ''The Computer Journal'', vol. 4, no. 4, pages ~~332-345~~332–345 (1962) [http://comjnl.oxfordjournals.org/cgi/content/abstract/4/4/332 online at oxfordjournals.org].<br /></ref> and [[Vera Kublanovskaya]]<ref>Vera N. Kublanovskaya (1961), "On some algorithms for the solution of the complete eigenvalue problem," USSR Computational Mathematics and Mathematical Physics, 1(3), pages 637–657 (1963, received Feb 1961). Also published in: Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki [Journal of Computational Mathematics and Mathematical Physics], 1(4), pages 555–570 (1961).</ref> invent [[QR factorization]] (voted one of the top 10 algorithms of the 20th century). == 1960s == * [[finite element method#History\|First recorded use]] of the term "finite element method" by [[Ray W. Clough\|Ray Clough]],<ref>RW Clough, “The Finite Element Method in Plane Stress Analysis,” Proceedings of 2nd ASCE Conference on Electronic Computation, Pittsburgh, PA, Sept. 8, 9, 1960.</ref> to describe the methods of Courant, Hrenikoff and Zienkiewicz, among others. See also [[Structural analysis#Timeline\|here]]. * Using computational investigations of the [[3 -body problem]], Minovitch formulates the [[gravity assist]] method.<ref>Minovitch, Michael: "A method for determining interplanetary free-fall reconnaissance trajectories," Jet Propulsion Laboratory Technical Memo TM-312-130, pages 38-44 (23 August 1961).</ref><ref>Christopher Riley and Dallas Campbell, Oct 22, 2012. [http://www.bbc.co.uk%2Fnews%2Fscience-environment-20033940&ei=j-29UZ6sNIexPInBgfAG&usg=AFQjCNEj30660hWJWTpfDJohrZek5KxAFA "The maths that made Voyager possible"] {{Webarchive\|url=https://web.archive.org/web/20130730023109/http://www.beyondintractability.org/bi-essay/culture-conflict \|date=2013-07-30 }}. BBC News Science and Environment. Recovered 16 Jun 2013.</ref> * Molecular dynamics was invented independently by [[Aneesur Rahman]].<ref>{{cite journal\|last=Rahman\|first=A\|title=Correlations in the Motion of Atoms in Liquid Argon\|journal=Phys Rev\|year=1964\|volume=136\|issue=2A\|pages=A405–A41\|doi=10.1103/PhysRev.136.A405\|bibcode = 1964PhRv..136..405R }}</ref> * Cooley and Tukey re-invent the [[Fast Fourier transform]] (voted one of the top 10 algorithms of the 20th century), an algorithm first discovered by [[Gauss]]. * [[Martin Kruskal\|Kruskal]] and [[Norman Zabusky\|Zabusky]] follow up the [[Fermi-Pasta-Ulam problem]] with further numerical experiments, and coin the term "soliton".<ref>Zabusky, N. J.; Kruskal, M. D. (1965). "Interaction of 'solitons' in a collisionless plasma and the recurrence of initial states". Phys. Rev. Lett. 15 (6): 240–243. Bibcode 1965PhRvL..15..240Z. doi:10.1103/PhysRevLett.15.240.</ref><ref>http://www.merriam-webster.com/dictionary/soliton ; retrieved 3 nov 2012.</ref>▼ * [[Edward Lorenz]] discovers the [[butterfly effect]] on a computer, attracting interest in [[chaos theory]].<ref>{{cite journal\|last=Lorenz\|first=Edward N.\|title=Deterministic Nonperiodic Flow\|journal=Journal of the Atmospheric Sciences ~~20 (2):~~ \|pages=130–141\|year=1963\|url=http://www.nd.edu/~powers/ame.60611/lorenz.article.pdf\|doi=10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2\|volume=20\|issue=2~~\|pages=130~~\|bibcode = 1963JAtS...20..130L }}</ref> ▲* [[Martin Kruskal\|Kruskal]] and [[Norman Zabusky\|Zabusky]] follow up the [[~~Fermi-Pasta-Ulam~~Fermi–Pasta–Ulam–Tsingou problem]] with further numerical experiments, and coin the term "soliton".<ref>Zabusky, N. J.; Kruskal, M. D. (1965). "Interaction of 'solitons' in a collisionless plasma and the recurrence of initial states". Phys. Rev. Lett. 15 (6): 240–243. Bibcode 1965PhRvL..15..240Z. doi:10.1103/PhysRevLett.15.240.</ref><ref>http://www.merriam-webster.com/dictionary/soliton ; retrieved 3 nov 2012.</ref> * [[Birch and Swinnerton-Dyer conjecture]] formulated through investigations on a computer.<ref>Birch, Bryan; Swinnerton-Dyer, Peter (1965). "Notes on Elliptic Curves (II)". J. Reine Angew. Math. 165 (218): 79–108. doi:10.1515/crll.1965.218.79.</ref> * Grobner bases and Buchberger's algorithm invented for algebra<ref>Bruno Buchberger: Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal (PDF; 1,8 MB). 1965</ref> * Frenchman Verlet (re)discovers [[Verlet integration\|a numerical integration algorithm]],<ref name="Verlet">{{cite journal \| first=Loup \| last=Verlet\| authorlink=Loup Verlet Line 42 ⟶ 51: \| year = 1967 \| volume = 159 \| issue=1\| pages = 98–103 ~~\| url=http://link.aps.org/doi/10.1103/PhysRev.159.98~~ \| doi=10.1103/PhysRev.159.98 \|bibcode = 1967PhRv..159...98V \| doi-access=free ~~\|bibcode = 1967PhRv..159...98V~~ }}</ref> (first used in 1791 by Delambre, by Cowell and Crommelin in 1909, and by [[Fredrik Carl Størmer\|Carl Fredrik Störmer]] in 1907,<ref>{{Cite book \| last1=Press \| first1=WH \| last2=Teukolsky \| first2=SA \| last3=Vetterling \| first3=WT \| last4=Flannery \| first4=BP \| year=2007 \| title=Numerical Recipes: The Art of Scientific Computing \| edition=3rd \| publisher=Cambridge University Press \| ~~publication-place~~___location=New York \| isbn=978-0-521-88068-8 \| chapter=Section 17.4. Second-Order Conservative Equations \| chapter-url=http://apps.nrbook.com/empanel/index.html#pg=928}} </ref> hence the alternative names Störmer's method or the Verlet-Störmer method) for dynamics~~, and the Verlet list~~.<ref name="Verlet" /> * Risch invents algorithm for symbolic integration.<ref>Risch, R. H. (1969). "The problem of integration in finite terms". Transactions of the American Mathematical Society. American Mathematical Society. 139: 167–189. doi:10.2307/1995313. JSTOR 1995313. Risch, R. H. (1970). "The solution of the problem of integration in finite terms". Bulletin of the American Mathematical Society. 76 (3): 605–608. doi:10.1090/S0002-9904-1970-12454-5.</ref> == 1970s == * Mandelbrot, from studies of the [[Fatou set\|Fatou]], [[Julia set\|Julia]] and [[Mandelbrot set]]s, coined and popularized the term 'fractal' to describe these structures' [[self-similarity]].<ref>B. Mandelbrot; ''Les objets fractals, forme, hasard et dimension '' (in French). Publisher: Flammarion (1975), {{ISBN \|9782082106474}}; English translation ''Fractals: Form, Chance and Dimension.'' Publisher: Freeman, W. H & Company. (1977). {{ISBN \|9780716704737}}.</ref><ref>Mandelbrot, Benoît B.; (1983). The Fractal Geometry of Nature. San Francisco: W.H. Freeman. {{ISBN \|0-7167-1186-9}}.</ref> Kenneth Appel and Wolfgang Haken prove the [[four colour theorem]], the [[Computer-assisted proof#~~List of theorems~~Theorems proved with the help of computer programs\|first theorem to be proved by computer]].<ref>Kenneth Appel and Wolfgang Haken, "Every planar map is four colorable, Part I: Discharging," Illinois Journal of Mathematics 21: 429–490, 1977.</ref><ref>Appel, K. and Haken, W. "Every Planar Map is Four-Colorable, II: Reducibility." Illinois J. Math. 21, ~~491-567~~491–567, 1977. </ref><ref>Appel, K. and Haken, W. "The Solution of the Four-Color Map Problem." Sci. Amer. 237, ~~108-121~~108–121, 1977.</ref> == 1980s == [[Fast multipole method]] invented by Rokhlin and ~~Greengaard~~Greengard (voted one of the top 10 algorithms of the 20th century).<ref>L. Greengard, The Rapid Evaluation of Potential Fields in Particle Systems, MIT, Cambridge, (1987).</ref><ref>Rokhlin, Vladimir (1985). "Rapid Solution of Integral Equations of Classic Potential Theory." J. Computational Physics Vol. 60, pp. ~~187-207~~187–207.</ref><ref>L. Greengard and V. Rokhlin, "A fast algorithm for particle simulations," J. Comput. Phys., 73 (1987), no. 2, pp. 325–348.</ref> ==1990s== * The appearance of the first research grids using [[volunteer computing]] -– [[GIMPS]] (1996), and [[distributed.net]] (1997~~) and [[Seti@Home]] (1999~~). * [[Kepler conjecture]] is [[proof by exhaustion\|almost all but certainly proved]] algorithmically by [[Thomas Callister Hales\|Thomas Hales]] in 1998. ==2000s== In computational group theory, [[Optimal solutions for Rubik's Cube\|God's Number]] for the [[Rubik's Cube]] is shown to be 20.<ref>[http://blog.computationalcomplexity.org/2010/09/rubiks-cube-conjecture-proven-do-we.html The Rubik's Cube Conjecture PROVEN! (Do we care?)] Wednesday, September 08, 2010</ref><ref>[http://www.cube20.org God's Number is 20.]</ref> Mathematicians completely map the E8-group.<ref>[https://news.mit.edu/2007/e8 Math research team maps E8: Calculation on paper would cover Manhattan.] MIT News. Elizabeth A. Thomson, News Office; March 18, 2007.</ref><ref>[https://www.math.columbia.edu/~woit/wordpress/?p=534 E8 Media Blitz], [[Peter Woit]].</ref><ref>[https://www.huliq.com/15695/mathematicians-map-e8 Mathematicians Map E8.] {{Webarchive\|url=https://web.archive.org/web/20150924032322/http://www.huliq.com/15695/mathematicians-map-e8 \|date=2015-09-24 }} By Armine Hareyan 2007-03-20 02:21.</ref> ==2010s== * Hales completes the proof of Kepler's conjecture.<ref>[http://blog.kleinproject.org/?p=742 What is the way of packing oranges? — Kepler's conjecture on the packing of spheres.] Posted on May 26, 2015 by Antoine Nectoux. Klein Project Blog: Connecting mathematical worlds.</ref><ref>[https://code.google.com/p/flyspeck/wiki/AnnouncingCompletion Announcement of Completion.] Flyspeck Project, [[Google Code]].</ref><ref>[https://www.newscientist.com/article/dn26041-proof-confirmed-of-400-year-old-fruit-stacking-problem/ Proof confirmed of 400-year-old fruit-stacking problem.] [[New Scientist]], 12 August 2014. </ref> == See also == Line 67 ⟶ 86: * [[Timeline of mathematics#20th century\|Timeline of mathematics from the 20th century onwards]] * [[Timeline of numerical analysis after 1945]] {{subject bar\|portal1=Science~~\|portal2=Computing~~\|portal3=Mathematics}} == References == Line 74 ⟶ 93: == External links == * [https://web.archive.org/web/20140822045448/http://home.gwu.edu/~stroud/mc-classics.html The Monte Carlo Method: Classic Papers] * [http://scienze-como.uninsubria.it/bressanini/montecarlo-history/ Monte Carlo Landmark Papers]