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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=
* Dantzig introduces the [[simplex method|simplex
* 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, pp. 232–237</ref>
* Ulam and von Neumann introduce the notion of cellular automata.<ref>Von Neumann, J., Theory of Self-
* [[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]].
* 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
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|first4=A.H. |last4=Teller
|first5=E. |last5=Teller |authorlink5=Edward Teller
|title=
|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.
* [[Enrico Fermi]], [[Stanislaw Ulam]], [[John Pasta]], and [[Mary Tsingou]], discover the [[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.
*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>
* 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, II" ''The Computer Journal'', vol. 4, no. 4, pages 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]].
* [[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 |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|bibcode = 1963JAtS...20..130L }}</ref>
* [[Martin Kruskal|Kruskal]] and [[Norman Zabusky|Zabusky]] follow up the [[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
* [[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
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| year = 1967
| volume = 159
| issue=1| pages = 98–103
| doi=10.1103/PhysRev.159.98
|bibcode = 1967PhRv..159...98V | doi-access=free
</ref> hence the alternative names Störmer's method or the Verlet-Störmer method) for dynamics.<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.
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== 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#
</ref><ref>Appel, K. and Haken, W. "The Solution of the Four-Color Map Problem." Sci. Amer. 237, 108–121, 1977.</ref>
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==2000s==
*In computational group theory, [[Optimal solutions for Rubik's Cube|God's
*Mathematicians completely map the E8-group.<ref>[
==
* Hales completes the proof of Kepler's conjecture.<ref>[http://blog.kleinproject.org/?p=742 What is the way of packing oranges? —
</ref>
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* [[Timeline of mathematics#20th century|Timeline of mathematics from the 20th century onwards]]
* [[Timeline of numerical analysis after 1945]]
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== References ==
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