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== 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 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, 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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|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>
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* [[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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== 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#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, 1977.
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==2000s==
*In computational group theory, [[Optimal solutions for Rubik's Cube|God's Number]] for the [[Rubik's
*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? —
</ref>
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