This is a timeline of key developments in computational mathematics.

- Monte Carlo simulation (voted one of the top 10 algorithms of the 20th century) invented at Los Alamos by von Neumann, Ulam and Metropolis.
^{ [1] }^{ [2] }^{ [3] } - Dantzig introduces the simplex algorithm (voted one of the top 10 algorithms of the 20th century).
^{ [4] } - First hydro simulations at Los Alamos occurred.
^{ [5] }^{ [6] } - Ulam and von Neumann introduce the notion of cellular automata.
^{ [7] } - A routine for the Manchester Baby written to factor a large number (2^18), one of the first in computational number theory.
^{ [8] }The Manchester group would make several other breakthroughs in this area.^{ [9] }^{ [10] } - LU decomposition technique first discovered.

- Hestenes, Stiefel, and Lanczos, all from the Institute for Numerical Analysis at the National Bureau of Standards, initiate the development of Krylov subspace iteration methods.
^{ [11] }^{ [12] }^{ [13] }^{ [14] }Voted one of the top 10 algorithms of the 20th century. *Equations of State Calculations by Fast Computing Machines*introduces the Metropolis–Hastings algorithm.^{ [15] }Also, important earlier independent work by Alder and S. Frankel.^{ [16] }^{ [17] }- Enrico Fermi, Stanislaw Ulam, John Pasta, and Mary Tsingou, discover the Fermi–Pasta–Ulam–Tsingou problem.
^{ [18] } - In network theory, Ford & Fulkerson compute a solution to the maximum flow problem.
^{ [19] } - Householder invents his eponymous matrices and transformation method (voted one of the top 10 algorithms of the 20th century).
^{ [20] } - Molecular dynamics invented by Alder and Wainwright
^{ [21] } - John G.F. Francis
^{ [22] }and Vera Kublanovskaya^{ [23] }invent QR factorization (voted one of the top 10 algorithms of the 20th century).

- First recorded use of the term "finite element method" by Ray Clough,
^{ [24] }to describe the methods of Courant, Hrenikoff and Zienkiewicz, among others. See also here. - Using computational investigations of the 3-body problem, Minovitch formulates the gravity assist method.
^{ [25] }^{ [26] } - Molecular dynamics was invented independently by Aneesur Rahman.
^{ [27] } - 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.
^{ [28] } - Kruskal and Zabusky follow up the Fermi–Pasta–Ulam–Tsingou problem with further numerical experiments, and coin the term "soliton".
^{ [29] }^{ [30] } - Birch and Swinnerton-Dyer conjecture formulated through investigations on a computer.
^{ [31] } - Grobner bases and Buchberger's algorithm invented for algebra
^{ [32] } - Frenchman Verlet (re)discovers a numerical integration algorithm,
^{ [33] }(first used in 1791 by Delambre, by Cowell and Crommelin in 1909, and by Carl Fredrik Störmer in 1907,^{ [34] }hence the alternative names Störmer's method or the Verlet-Störmer method) for dynamics.^{ [33] } - Risch invents algorithm for symbolic integration.
^{ [35] }

- Computer algebra replicates and extends the work of Delaunay in lunar theory.
^{ [36] } - Mandelbrot, from studies of the Fatou, Julia and Mandelbrot sets, coined and popularized the term 'fractal' to describe these structures' self-similarity.
^{ [37] }^{ [38] } - Kenneth Appel and Wolfgang Haken prove the four colour theorem, the first theorem to be proved by computer.
^{ [39] }^{ [40] }^{ [41] }

- Fast multipole method invented by Rokhlin and Greengard (voted one of the top 10 algorithms of the 20th century).
^{ [42] }^{ [43] }^{ [44] }

- The appearance of the first research grids using volunteer computing – GIMPS (1996) and distributed.net (1997).
- Kepler conjecture is almost all but certainly proved algorithmically by Thomas Hales in 1998.

- In computational group theory, God's Number for the Rubik's cube is shown to be 20.
^{ [45] }^{ [46] } - Mathematicians completely map the E8-group.
^{ [47] }^{ [48] }^{ [49] }

- Hales completes the proof of Kepler's conjecture.
^{ [50] }^{ [51] }^{ [52] }

**John von Neumann** was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest coverage of any mathematician of his time and was said to have been "the last representative of the great mathematicians who were equally at home in both pure and applied mathematics". He integrated pure and applied sciences.

The **Mandelbrot set** is the set of complex numbers for which the function does not diverge to infinity when iterated from , i.e., for which the sequence , , etc., remains bounded in absolute value.

**Stanisław Marcin Ulam** was a Polish-American scientist in the fields of mathematics and nuclear physics. He participated in the Manhattan Project, originated the Teller–Ulam design of thermonuclear weapons, discovered the concept of the cellular automaton, invented the Monte Carlo method of computation, and suggested nuclear pulse propulsion. In pure and applied mathematics, he proved some theorems and proposed several conjectures.

**Monte Carlo methods**, or **Monte Carlo experiments**, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deterministic in principle. They are often used in physical and mathematical problems and are most useful when it is difficult or impossible to use other approaches. Monte Carlo methods are mainly used in three problem classes: optimization, numerical integration, and generating draws from a probability distribution.

The **MANIAC I** was an early computer built under the direction of Nicholas Metropolis at the Los Alamos Scientific Laboratory. It was based on the von Neumann architecture of the IAS, developed by John von Neumann. As with all computers of its era, it was a one-of-a-kind machine that could not exchange programs with other computers. Metropolis chose the name MANIAC in the hope of stopping the rash of silly acronyms for machine names, although von Neumann may have suggested the name to him.

**Nicholas Constantine Metropolis** was a Greek-American physicist.

**Ralph William Gosper Jr.**, known as **Bill Gosper**, is an American mathematician and programmer. Along with Richard Greenblatt, he may be considered to have founded the hacker community, and he holds a place of pride in the Lisp community. The Gosper curve and the Gosper's algorithm are named after him.

**Monte Carlo N-Particle Transport** (**MCNP**) is a general-purpose, continuous-energy, generalized-geometry, time-dependent, Monte Carlo radiation transport code designed to track many particle types over broad ranges of energies and is developed by Los Alamos National Laboratory. Specific areas of application include, but are not limited to, radiation protection and dosimetry, radiation shielding, radiography, medical physics, nuclear criticality safety, detector design and analysis, nuclear oil well logging, accelerator target design, fission and fusion reactor design, decontamination and decommissioning. The code treats an arbitrary three-dimensional configuration of materials in geometric cells bounded by first- and second-degree surfaces and fourth-degree elliptical tori.

**John R. Pasta** was an American computational physicist and computer scientist who is remembered today for the Fermi–Pasta–Ulam–Tsingou experiment, the result of which was much discussed among physicists and researchers in the fields of dynamical systems and chaos theory, and as the head of the department of Computer Science at the University of Illinois at Urbana-Champaign from 1964 to 1970.

In physics, the **Fermi–Pasta–Ulam–Tsingou problem** or formerly the **Fermi–Pasta–Ulam problem** was the apparent paradox in chaos theory that many complicated enough physical systems exhibited almost exactly periodic behavior – called **Fermi–Pasta–Ulam–Tsingou recurrence** – instead of the expected ergodic behavior. This came as a surprise, as Fermi, certainly, expected the system to thermalize in a fairly short time. That is, it was expected for all vibrational modes to eventually appear with equal strength, as per the equipartition theorem, or, more generally, the ergodic hypothesis. Yet here was a system that appeared to evade the ergodic hypothesis. Although the recurrence is easily observed, it eventually became apparent that over much, much longer time periods, the system does eventually thermalize. Multiple competing theories have been proposed to explain the behavior of the system, and it remains a topic of active research.

**Computational statistics**, or statistical computing, is the bond between statistics and computer science. It means statistical methods that are enabled by using computational methods. It is the area of computational science specific to the mathematical science of statistics. This area is also developing rapidly, leading to calls that a broader concept of computing should be taught as part of general statistical education.

**"Equation of State Calculations by Fast Computing Machines"** is a scholarly article published by Nicholas Metropolis, Arianna W. Rosenbluth, Marshall N. Rosenbluth, Augusta H. Teller, and Edward Teller in the Journal of Chemical Physics in 1953. This paper proposed what became known as the Metropolis Monte Carlo algorithm, which forms the basis for Monte Carlo statistical mechanics simulations of atomic and molecular systems.

This is a timeline of pure and applied mathematics history. It is divided here into three stages, corresponding to stages in the development of mathematical notation: a "rhetorical" stage in which calculations are described purely by words, a "syncopated" stage in which quantities and common algebraic operations are beginning to be represented by symbolic abbreviations, and finally a "symbolic" stage, in which comprehensive notational systems for formulas are the norm.

**Aneesur Rahman** pioneered the application of computational methods to physical systems. His 1964 paper on liquid argon studied a system of 864 argon atoms on a CDC 3600 computer, using a Lennard-Jones potential. His algorithms still form the basis for many codes written today. Moreover, he worked on a wide variety of problems, such as the microcanonical ensemble approach to lattice gauge theory, which he invented with David J E Callaway.

**Lateral computing** is a lateral thinking approach to solving computing problems. Lateral thinking has been made popular by Edward de Bono. This thinking technique is applied to generate creative ideas and solve problems. Similarly, by applying lateral-computing techniques to a problem, it can become much easier to arrive at a computationally inexpensive, easy to implement, efficient, innovative or unconventional solution.

**Robert Davis Richtmyer** was an American physicist, mathematician, educator, author, and musician.

The following timeline starts with the invention of the modern computer in the late interwar period.

The following is a **timeline of scientific computing**, also known as **computational science**.

**Mary Tsingou** is an American physicist and mathematician of Greek descent. She was one of the first programmers on the MANIAC computer at Los Alamos National Laboratory and is best known for having coded the celebrated computer experiment with Enrico Fermi, John Pasta, and Stanislaw Ulam which became an inspiration for the fields of chaos theory and scientific computing and was a turning point in soliton theory.

The following is a **timeline of numerical analysis after 1945**, and deals with developments after the invention of the modern electronic computer, which began during Second World War. For a fuller history of the subject before this period, see timeline and history of mathematics.

- ↑ Metropolis, N. (1987). "The Beginning of the Monte Carlo method" (PDF).
*Los Alamos Science*. No. 15, Page 125.. Accessed 5 may 2012. - ↑ S. Ulam, R. D. Richtmyer, and J. von Neumann(1947). Statistical methods in neutron diffusion. Los Alamos Scientific Laboratory report LAMS–551.
- ↑ N. Metropolis and S. Ulam (1949). The Monte Carlo method. Journal of the American Statistical Association 44:335–341.
- ↑ "SIAM News, November 1994" . Retrieved 6 June 2012. Systems Optimization Laboratory, Stanford University Huang Engineering Center (site host/mirror).
- ↑ Richtmyer, R. D. (1948). Proposed Numerical Method for Calculation of Shocks. Los Alamos, NM: Los Alamos Scientific Laboratory LA-671.
- ↑ A Method for the Numerical Calculation of Hydrodynamic Shocks. Von Neumann, J.; Richtmyer, R. D. Journal of Applied Physics, Vol. 21, pp. 232–237
- ↑ Von Neumann, J., Theory of Self-Reproduiing Automata, Univ. of Illinois Press, Urbana, 1966.
- ↑ The Manchester Mark 1.
- ↑ Miscellaneous Notes: Mersenne Primes. 60 Manchester - 60 years of the Modern Computer
^{[ permanent dead link ]}, Manchester Uni. CS Curation website. - ↑ One tonne 'Baby' marks its birth: Dashing times. By Jonathan Fildes, Science and technology reporter, BBC News.
- ↑ Magnus R. Hestenes and Eduard Stiefel, Methods of Conjugate Gradients for Solving Linear Systems, J. Res. Natl. Bur. Stand. 49, 409–436 (1952).
- ↑ Eduard Stiefel,U¨ ber einige Methoden der Relaxationsrechnung (in German), Z. Angew. Math. Phys. 3, 1–33 (1952).
- ↑ Cornelius Lanczos, Solution of Systems of Linear Equations by Minimized Iterations, J. Res. Natl. Bur. Stand. 49, 33–53 (1952).
- ↑ 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).
- ↑ Metropolis, N.; Rosenbluth, A.W.; Rosenbluth, M.N.; Teller, A.H.; Teller, E. (1953). "Equations of State Calculations by Fast Computing Machines".
*Journal of Chemical Physics*.**21**(6): 1087–1092. Bibcode:1953JChPh..21.1087M. doi:10.1063/1.1699114. OSTI 4390578. - ↑ Unfortunately, Alder's thesis advisor was unimpressed, so Alder and Frankel delayed publication of their results until much later. Alder, B. J. , Frankel, S. P. , and Lewinson, B. A. , J. Chem. Phys., 23, 3 (1955).
- ↑ Stanley P. Frankel, Unrecognized Genius, HP9825.COM (accessed 29 Aug 2015).
- ↑ Fermi, E. (posthumously); Pasta, J.; Ulam, S. (1955) : Studies of Nonlinear Problems (accessed 25 Sep 2012). Los Alamos Laboratory Document LA-1940. Also appeared in 'Collected Works of Enrico Fermi', E. Segre ed. , University of Chicago Press, Vol.II,978–988,1965. Recovered 21 Dec 2012
- ↑ Ford, L. R.; Fulkerson, D. R. (1956). "Maximal flow through a network" . Canadian Journal of Mathematics. 8: 399–404.
- ↑ Householder, A. S. (1958). "Unitary Triangularization of a Nonsymmetric Matrix" (PDF).
*Journal of the ACM*.**5**(4): 339–342. doi:10.1145/320941.320947. MR 0111128. S2CID 9858625. - ↑ 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
- ↑ J. G. F. Francis, "The QR Transformation, I",
*The Computer Journal*, vol. 4, no. 3, pages 265–271 (1961, received Oct 1959) online at oxfordjournals.org;

J. G. F. Francis, "The QR Transformation, II"*The Computer Journal*, vol. 4, no. 4, pages 332–345 (1962) online at oxfordjournals.org. - ↑ 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).
- ↑ RW Clough, “The Finite Element Method in Plane Stress Analysis,” Proceedings of 2nd ASCE Conference on Electronic Computation, Pittsburgh, PA, Sept. 8, 9, 1960.
- ↑ 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).
- ↑ Christopher Riley and Dallas Campbell, Oct 22, 2012. "The maths that made Voyager possible". BBC News Science and Environment. Recovered 16 Jun 2013.
- ↑ Rahman, A (1964). "Correlations in the Motion of Atoms in Liquid Argon".
*Phys Rev*.**136**(2A): A405–A41. Bibcode:1964PhRv..136..405R. doi:10.1103/PhysRev.136.A405. - ↑ Lorenz, Edward N. (1963). "Deterministic Nonperiodic Flow" (PDF).
*Journal of the Atmospheric Sciences*.**20**(2): 130–141. Bibcode:1963JAtS...20..130L. doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2. - ↑ 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.
- ↑ http://www.merriam-webster.com/dictionary/soliton ; retrieved 3 nov 2012.
- ↑ 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.
- ↑ Bruno Buchberger: Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal (PDF; 1,8 MB). 1965
- 1 2 Verlet, Loup (1967). "Computer "Experiments" on Classical Fluids. I. Thermodynamical Properties of Lennard−Jones Molecules".
*Physical Review*.**159**(1): 98–103. Bibcode:1967PhRv..159...98V. doi: 10.1103/PhysRev.159.98 . - ↑ Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). "Section 17.4. Second-Order Conservative Equations".
*Numerical Recipes: The Art of Scientific Computing*(3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8. - ↑ 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.
- ↑ http://www.umiacs.umd.edu/~helalfy/pub/mscthesis01.pdf
^{[ bare URL PDF ]} - ↑ 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. - ↑ Mandelbrot, Benoît B.; (1983). The Fractal Geometry of Nature. San Francisco: W.H. Freeman. ISBN 0-7167-1186-9.
- ↑ Kenneth Appel and Wolfgang Haken, "Every planar map is four colorable, Part I: Discharging," Illinois Journal of Mathematics 21: 429–490, 1977.
- ↑ Appel, K. and Haken, W. "Every Planar Map is Four-Colorable, II: Reducibility." Illinois J. Math. 21, 491–567, 1977.
- ↑ Appel, K. and Haken, W. "The Solution of the Four-Color Map Problem." Sci. Amer. 237, 108–121, 1977.
- ↑ L. Greengard, The Rapid Evaluation of Potential Fields in Particle Systems, MIT, Cambridge, (1987).
- ↑ Rokhlin, Vladimir (1985). "Rapid Solution of Integral Equations of Classic Potential Theory." J. Computational Physics Vol. 60, pp. 187–207.
- ↑ L. Greengard and V. Rokhlin, "A fast algorithm for particle simulations," J. Comput. Phys., 73 (1987), no. 2, pp. 325–348.
- ↑ The Rubik's Cube Conjecture PROVEN! (Do we care?) Wednesday, September 08, 2010
- ↑ God's Number is 20.
- ↑ Math research team maps E8: Calculation on paper would cover Manhattan. MIT News. Elizabeth A. Thomson, News Office; March 18, 2007.
- ↑ E8 Media Blitz, Peter Woit.
- ↑ Mathematicians Map E8. Archived 2015-09-24 at the Wayback Machine By Armine Hareyan 2007-03-20 02:21.
- ↑ 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.
- ↑ Announcement of Completion. Flyspeck Project, Google Code.
- ↑ Proof confirmed of 400-year-old fruit-stacking problem. New Scientist, 12 August 2014.

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