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

Before modern computers

18th century

1733 - The French naturalist Comte de Buffon poses his needle problem.^[1]^[2]

19th century

Babbage in 1822, began work on a machine made to compute/calculate values of polynomial functions automatically by using the method of finite differences. This was eventually called the Difference engine.
Lovelace's note G on the Analytical Engine (1842) describes an algorithm for generating Bernoulli numbers. It is considered the first algorithm ever specifically tailored for implementation on a computer, and thus the first-ever computer programme.^[3]^[4] The engine was never completed, however, so her code was never tested.^[5]

1900s

1900 - Carl Runge and Martin Kutta invent the Runge-Kutta method for approximating integration for differential equations.^[6]^[7]

1920s

1922 - Lewis Fry Richardson introduces numerical weather forecasting by manual calculation, using methods originally developed by Vilhelm Bjerknes as early as 1895.^[8]^[9]
1926 - Grete Hermann publishes foundational paper for computer algebra, which established the existence of algorithms (including complexity bounds) for many of the basic problems of abstract algebra, such as ideal membership for polynomial rings.^[10]
1927 - Douglas Hartree creates what is later known as the Hartree–Fock method, the first ab initio quantum chemistry methods. However, manual solutions of the Hartree–Fock equations for a medium-sized atom were laborious and small molecules required computational resources far beyond what was available before 1950.

1930s

This decade marks the first major strides to a modern computer, and hence the start of the modern era.

Fermi's Rome physics research group (informal name I ragazzi di Via Panisperna) develop statistical algorithms based on Comte de Buffon's work, that would later become the foundation of the Monte Carlo method. See also FERMIAC.
John Vincent Atanasoff and Clifford Berry create the first electronic non-programmable, digital computing device, the Atanasoff–Berry Computer, from 1937-42.
Complex number calculator created by Stibitz.

1940s

Monte Carlo simulation (voted one of the top 10 algorithms of the 20th century) invented at Los Alamos by von Neumann, Ulam and Metropolis.^[11]^[12]^[13]
George Dantzig introduces the simplex method (voted one of the top 10 algorithms of the 20th century) in 1947.^[14]
Ulam and von Neumann introduce the notion of cellular automata.^[15]
Turing formulated the LU decomposition method.^[16]
Philips creates (invents?) the MONIAC hydraulic computer at LSE, better known as "Philip's Economic Computer".^[17]^[18]
First hydro simulations occurred at Los Alamos.^[19]^[20]

1950s

First successful weather predictions on a computer occurred.^[21]^[22]
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.^[23]^[24]^[25]^[26] Voted one of the top 10 algorithms of the 20th century.
Equations of State Calculations by Fast Computing Machines introduces the Metropolis–Hastings algorithm.^[27]
Molecular dynamics invented by Bernie Alder and Wainwright ^[28]^[29]
Householder invents his eponymous matrices and transformation method (voted one of the top 10 algorithms of the 20th century).^[30]
John G.F. Francis^[31]^[32] and Vera Kublanovskaya^[33] invent QR factorization (voted one of the top 10 algorithms of the 20th century).
A team led by Backus develops the FORTRAN compiler and programming language at IBM's research centre in San Jose, California. This sped the adoption of scientific programming,^[34]^[35]^[36] and is one of the oldest extant programming languages, as well as one of the most popular in science and engineering.

1960s

First recorded use of the term "finite element method" by Ray Clough to describe the earlier methods of Richard Courant, Alexander Hrennikoff and Olgierd Zienkiewicz in structural analysis.^[37]
Edward Lorenz discovers the butterfly effect on a computer, attracting interest in chaos theory.^[38]
Enrico Fermi, John Pasta, Stanislaw Ulam, and Mary Tsingou discover the Fermi–Pasta–Ulam–Tsingou problem through computer simulations of a vibrating string.^[39]
Using computational investigations of the 3 body problem, Michael Minovitch formulates the gravity assist method.^[40]^[41]
Molecular dynamics invented independently by Aneesur Rahman.^[42]
Fast Fourier Transform developed by James W. Cooley and John W. Tukey.^[43]
Walter Kohn, with Lu Jeu Sham and Pierre Hohenberg, instigates the development of density functional theory,^[44]^[45] for which he shares the 1998 Nobel Chemistry Prize with John Pople.^[46] This contribution is arguably the first Nobel given for a computer program or computational technique.

1970s

1975 - Benoit Mandelbrot coins the term "fractal" to describe the self-similarity found in the Fatou, Julia and Mandelbrot sets. Fractals become the first mathematical visualization tool extensively explored with computing.^[47]
1977 - Kenneth Appel and Wolfgang Haken prove the four colour theorem, the first theorem to be proved by computer.^[48]^[49]^[50]

1980s

Fast multipole method (voted one of the top 10 algorithms of the 20th century) invented by Vladimir Rokhlin and Leslie Greengard.^[51]^[52]^[53]

1990s

In computational genomics and sequence analysis, the Human Genome Project, an endeavour to sequence the entire human genome, begins in 1990.
Kepler conjecture is almost all but certainly proved algorithmically by Thomas Hales in 1998.
The appearance of the first research grids using volunteer computing - GIMPS (1996), distributed.net (1997) and Seti@Home (1999).

2000s

The Human Genome Project completes a rough draft of human genome in 2000.
The Human Genome Project completed in 2003.
The BOINC architecture is launched in 2002.

2010s

Foldit players solve virus structure, one of the first cases of a game solving a scientific question.

References

^ Buffon, G. Editor's note concerning a lecture given 1733 by Mr. Le Clerc de Buffon to the Royal Academy of Sciences in Paris. Histoire de l'Acad. Roy. des Sci., pp. 43-45, 1733; according to Weisstein, Eric W. "Buffon's Needle Problem." From MathWorld--A Wolfram Web Resource. 20 Dec 2012 20 Dec 2012.
^ Buffon, G. "Essai d'arithmétique morale." Histoire naturelle, générale er particulière, Supplément 4, 46-123, 1777; according to Weisstein, Eric W. "Buffon's Needle Problem." From MathWorld--A Wolfram Web Resource. 20 Dec 2012
^ Simonite, Tom (24 March 2009). "Short Sharp Science: Celebrating Ada Lovelace: the 'world's first programmer'". New Scientist. Retrieved 14 April 2012.
^ http://www.newyorker.com/online/blogs/books/2013/08/tom-stoppards-arcadia-at-twenty.html
^ Kim, Eugene Eric; Toole, Betty Alexandra (May 1999). "Ada and the first computer". Scientific American. 280 (5): 70–71. doi:10.1038/scientificamerican0599-76.
^
MW Kutta. "Beiträge zur näherungsweisen Integration totaler Differentialgleichungen" [Contributions to the approximate integration of total differential equations] (in German). Thesis, University of Munich.
- "Reprinted", Z. Math. Phys., 46: 435–453, 1901 and in B.G Teubner, 1901.
^ Runge, C., "Über die numerische Auflösung von Differentialgleichungen" [About the numerical solution of differential equations](in German), Math. Ann. 46 (1895) 167-178.
^ L F Richardson, Weather Prediction by Numerical Process. Cambridge University Press (1922).
^ Lynch, Peter (March 2008). "The origins of computer weather prediction and climate modeling" (PDF). Journal of Computational Physics. 227 (7). University of Miami: 3431–44. Bibcode:2008JCoPh.227.3431L. doi:10.1016/j.jcp.2007.02.034. Retrieved 2010-12-23.
^ Grete Hermann (1926). "Die Frage der endlich vielen Schritte in der Theorie der Polynomideale". Mathematische Annalen. 95: 736–788. doi:10.1007/bf01206635.
^ Metropolis, N. (1987). "The Beginning of the Monte Carlo method" (PDF). Los Alamos Science. No. 15, Page 125. {{cite journal}}: |volume= has extra text (help). 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.
^ Metropolis, N.; Ulam, S. (1949). "The Monte Carlo method". Journal of the American Statistical Association. 44: 335–341. doi:10.1080/01621459.1949.10483310.
^ "SIAM News, November 1994". Retrieved 6 June 2012. Systems Optimization Laboratory, Stanford University Huang Engineering Center (site host/mirror).
^ Von Neumann, J., Theory of Self-Reproduiing Automata, Univ. of Illinois Press, Urbana, 1966.
^ A. M. Turing, Rounding-off errors in matrix processes. Quart. J Mech. Appl. Math. 1 (1948), 287–308 (according to Poole, David (2006), Linear Algebra: A Modern Introduction (2nd ed.), Canada: Thomson Brooks/Cole, ISBN 0-534-99845-3.) .
^ The computer model that once explained the British economy. Larry Elliott, The Guardian, Thursday 8 May 2008.
^ Phillip's Economic Computer, 1949. Exhibit at London Science Museum.
^ Richtmyer, R. D. (1948). Proposed Numerical Method for Calculation of Shocks. Los Alamos, NM: Los Alamos Scientific Laboratory LA-671.
^ Von Neumann, J.; Richtmyer, R. D. (1950). "A Method for the Numerical Calculation of Hydrodynamic Shocks". Journal of Applied Physics. 21: 232–237. doi:10.1063/1.1699639.
^ Charney, J.; Fjørtoft, R.; von Neumann, J. (1950). "Numerical Integration of the Barotropic Vorticity Equation". Tellus. 2 (4).
^ See the review article:- Smagorinsky, J (1983). "The Beginnings of Numerical Weather Prediction and General Circulation Modelling: Early Recollections" (PDF). Advances in Geophysics. 25. Retrieved 6 June 2012.
^ 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" (PDF). Journal of Chemical Physics. 21 (6): 1087–1092. Bibcode:1953JChPh..21.1087M. doi:10.1063/1.1699114.
^ Alder, B. J.; Wainwright, T. E. (1957). "Phase Transition for a Hard Sphere System". J. Chem. Phys. 27 (5): 1208. doi:10.1063/1.1743957.
^ Alder, B. J.; Wainwright, T. E. (1962). "Phase Transition in Elastic Disks". Phys. Rev. 127 (2): 359–361. doi:10.1103/PhysRev.127.359.
^ Householder, A. S. (1958). "Unitary Triangularization of a Nonsymmetric Matrix". Journal of the ACM. 5 (4): 339–342. doi:10.1145/320941.320947. MR 0111128.
^ Francis, J.G.F. (1961). "The QR Transformation, I". The Computer Journal. 4 (3): 265–271. doi:10.1093/comjnl/4.3.265.
^ Francis, J.G.F. (1962). "The QR Transformation, II". The Computer Journal. 4 (4): 332–345. doi:10.1093/comjnl/4.4.332.
^ Kublanovskaya, Vera N. (1961). "On some algorithms for the solution of the complete eigenvalue problem". USSR Computational Mathematics and Mathematical Physics. 1 (3): 637–657. doi:10.1016/0041-5553(63)90168-X. Also published in: Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki [Journal of Computational Mathematics and Mathematical Physics], 1(4), pages 555–570 (1961).
^ W.W. McDowell Award citation: "W. Wallace McDowell Award". Retrieved April 15, 2008.
^ National Medal of Science citation: "The President's National Medal of Science: John Backus". National Science Foundation. Retrieved March 21, 2007.
^ "ACM Turing Award Citation: John Backus". Association for Computing Machinery. Archived from the original on February 4, 2007. Retrieved March 22, 2007. {{cite web}}: Unknown parameter |deadurl= ignored (|url-status= suggested) (help)
^ RW Clough, "The Finite Element Method in Plane Stress Analysis," Proceedings of 2nd ASCE Conference on Electronic Computation, Pittsburgh, PA, Sept. 8, 9, 1960.
^ Lorenz, Edward N. (1963). "Deterministic Nonperiodic Flow" (PDF). Journal of the Atmospheric Sciences. 20 (2): 130–141. doi:10.1175/1520-0469(1963)020<0130:dnf>2.0.co;2.
^ 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
^ 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.
^ Cooley, James W.; Tukey, John W. (1965). "An algorithm for the machine calculation of complex Fourier series". Math. Comput. 19: 297–301.
^ Kohn, Walter; Hohenberg, Pierre (1964). "Inhomogeneous Electron Gas". Physical Review. 136 (3B): B864–B871. Bibcode:1964PhRv..136..864H. doi:10.1103/PhysRev.136.B864.
^ Kohn, Walter; Sham, Lu Jeu (1965). "Self-Consistent Equations Including Exchange and Correlation Effects". Physical Review. 140 (4A): A1133–A1138. Bibcode:1965PhRv..140.1133K. doi:10.1103/PhysRev.136.B864.
^ "The Nobel Prize in Chemistry 1998". Nobelprize.org. Retrieved 2008-10-06.
^ 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.
^ Appel, Kenneth; Haken, Wolfgang (1977). "Every planar map is four colorable, Part I: Discharging". Illinois Journal of Mathematics. 21: 429–490.
^ Appel, K.; Haken, W. (1977). "Every Planar Map is Four-Colorable, II: Reducibility". Illinois J. Math. 21: 491–567.
^ Appel, K.; Haken, W. (1977). "The Solution of the Four-Color Map Problem". Sci. Amer. 237: 108–121. doi:10.1038/scientificamerican1077-108.
^ 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.

External links

SIAM (Society for Industrial and Applied Mathematics) News. Top 10 Algorithms of the 20th Century.
The History of Numerical Analysis and Scientific Computing @ SIAM (Society for Industrial and Applied Mathematics)
Ruttimann, Jacqueline (2006). "2020 computing: Milestones in scientific computing". Nature. 440: 399–405. doi:10.1038/440399a. PMID 16554772.
Anderson, H. L. "Scientific Uses of the MANIAC". Journal of Statistical Physics. 43 (5–6): 731–748. Bibcode:1986JSP....43..731A. doi:10.1007/BF02628301.

[1] Buffon, G. Editor's note concerning a lecture given 1733 by Mr. Le Clerc de Buffon to the Royal Academy of Sciences in Paris. Histoire de l'Acad. Roy. des Sci., pp. 43-45, 1733; according to Weisstein, Eric W. "Buffon's Needle Problem." From MathWorld--A Wolfram Web Resource. 20 Dec 2012 20 Dec 2012.

[2] Buffon, G. "Essai d'arithmétique morale." Histoire naturelle, générale er particulière, Supplément 4, 46-123, 1777; according to Weisstein, Eric W. "Buffon's Needle Problem." From MathWorld--A Wolfram Web Resource. 20 Dec 2012

[3] Simonite, Tom (24 March 2009). "Short Sharp Science: Celebrating Ada Lovelace: the 'world's first programmer'". New Scientist. Retrieved 14 April 2012.

[newyorker'13-4] ttp://www.newyorker.com/online/blogs/books/2013/08/tom-stoppards-arcadia-at-twenty.html

[KimToole1999-5] Kim, Eugene Eric; Toole, Betty Alexandra (May 1999). "Ada and the first computer". Scientific American. 280 (5): 70–71. doi:10.1038/scientificamerican0599-76.

[6] MW Kutta. "Beiträge zur näherungsweisen Integration totaler Differentialgleichungen" [Contributions to the approximate integration of total differential equations] (in German). Thesis, University of Munich.
"Reprinted", Z. Math. Phys., 46: 435–453, 1901 and in B.G Teubner, 1901.

[7] "Reprinted", Z. Math. Phys., 46: 435–453, 1901 and in B.G Teubner, 1901.

[7] Runge, C., "Über die numerische Auflösung von Differentialgleichungen" [About the numerical solution of differential equations](in German), Math. Ann. 46 (1895) 167-178.

[8] L F Richardson, Weather Prediction by Numerical Process. Cambridge University Press (1922).

[Lynch_JCP-9] Lynch, Peter (March 2008). "The origins of computer weather prediction and climate modeling" (PDF). Journal of Computational Physics. 227 (7). University of Miami: 3431–44. Bibcode:2008JCoPh.227.3431L. doi:10.1016/j.jcp.2007.02.034. Retrieved 2010-12-23.

[10] Grete Hermann (1926). "Die Frage der endlich vielen Schritte in der Theorie der Polynomideale". Mathematische Annalen. 95: 736–788. doi:10.1007/bf01206635.

[11] Metropolis, N. (1987). "The Beginning of the Monte Carlo method" (PDF). Los Alamos Science. No. 15, Page 125. {{cite journal}}: |volume= has extra text (help). Accessed 5 may 2012.

[12] S. Ulam, R. D. Richtmyer, and J. von Neumann(1947). Statistical methods in neutron diffusion. Los Alamos Scientific Laboratory report LAMS–551.

[13] Metropolis, N.; Ulam, S. (1949). "The Monte Carlo method". Journal of the American Statistical Association. 44: 335–341. doi:10.1080/01621459.1949.10483310.

[14] "SIAM News, November 1994". Retrieved 6 June 2012. Systems Optimization Laboratory, Stanford University Huang Engineering Center (site host/mirror).

[15] Von Neumann, J., Theory of Self-Reproduiing Automata, Univ. of Illinois Press, Urbana, 1966.

[16] A. M. Turing, Rounding-off errors in matrix processes. Quart. J Mech. Appl. Math. 1 (1948), 287–308 (according to Poole, David (2006), Linear Algebra: A Modern Introduction (2nd ed.), Canada: Thomson Brooks/Cole, ISBN 0-534-99845-3.) .

[17] The computer model that once explained the British economy. Larry Elliott, The Guardian, Thursday 8 May 2008.

[18] Phillip's Economic Computer, 1949. Exhibit at London Science Museum.

[19] Richtmyer, R. D. (1948). Proposed Numerical Method for Calculation of Shocks. Los Alamos, NM: Los Alamos Scientific Laboratory LA-671.

[20] Von Neumann, J.; Richtmyer, R. D. (1950). "A Method for the Numerical Calculation of Hydrodynamic Shocks". Journal of Applied Physics. 21: 232–237. doi:10.1063/1.1699639.

[21] Charney, J.; Fjørtoft, R.; von Neumann, J. (1950). "Numerical Integration of the Barotropic Vorticity Equation". Tellus. 2 (4).

[22] See the review article:- Smagorinsky, J (1983). "The Beginnings of Numerical Weather Prediction and General Circulation Modelling: Early Recollections" (PDF). Advances in Geophysics. 25. Retrieved 6 June 2012.

[23] Magnus R. Hestenes and Eduard Stiefel, Methods of Conjugate Gradients for Solving Linear Systems, J. Res. Natl. Bur. Stand. 49, 409-436 (1952).

[24] Eduard Stiefel,U¨ ber einige Methoden der Relaxationsrechnung (in German), Z. Angew. Math. Phys. 3, 1-33 (1952).

[25] Cornelius Lanczos, Solution of Systems of Linear Equations by Minimized Iterations, J. Res. Natl. Bur. Stand. 49, 33-53 (1952).

[26] 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).

[27] Metropolis, N.; Rosenbluth, A.W.; Rosenbluth, M.N.; Teller, A.H.; Teller, E. (1953). "Equations of State Calculations by Fast Computing Machines" (PDF). Journal of Chemical Physics. 21 (6): 1087–1092. Bibcode:1953JChPh..21.1087M. doi:10.1063/1.1699114.

[28] Alder, B. J.; Wainwright, T. E. (1957). "Phase Transition for a Hard Sphere System". J. Chem. Phys. 27 (5): 1208. doi:10.1063/1.1743957.

[29] Alder, B. J.; Wainwright, T. E. (1962). "Phase Transition in Elastic Disks". Phys. Rev. 127 (2): 359–361. doi:10.1103/PhysRev.127.359.

[30] Householder, A. S. (1958). "Unitary Triangularization of a Nonsymmetric Matrix". Journal of the ACM. 5 (4): 339–342. doi:10.1145/320941.320947. MR 0111128.

[31] Francis, J.G.F. (1961). "The QR Transformation, I". The Computer Journal. 4 (3): 265–271. doi:10.1093/comjnl/4.3.265.

[32] Francis, J.G.F. (1962). "The QR Transformation, II". The Computer Journal. 4 (4): 332–345. doi:10.1093/comjnl/4.4.332.

[33] Kublanovskaya, Vera N. (1961). "On some algorithms for the solution of the complete eigenvalue problem". USSR Computational Mathematics and Mathematical Physics. 1 (3): 637–657. doi:10.1016/0041-5553(63)90168-X. Also published in: Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki [Journal of Computational Mathematics and Mathematical Physics], 1(4), pages 555–570 (1961).

[34] W.W. McDowell Award citation: "W. Wallace McDowell Award". Retrieved April 15, 2008.

[National_Science_Foundation-35] National Medal of Science citation: "The President's National Medal of Science: John Backus". National Science Foundation. Retrieved March 21, 2007.

[36] "ACM Turing Award Citation: John Backus". Association for Computing Machinery. Archived from the original on February 4, 2007. Retrieved March 22, 2007. {{cite web}}: Unknown parameter |deadurl= ignored (|url-status= suggested) (help)

[37] RW Clough, "The Finite Element Method in Plane Stress Analysis," Proceedings of 2nd ASCE Conference on Electronic Computation, Pittsburgh, PA, Sept. 8, 9, 1960.

[38] Lorenz, Edward N. (1963). "Deterministic Nonperiodic Flow" (PDF). Journal of the Atmospheric Sciences. 20 (2): 130–141. doi:10.1175/1520-0469(1963)020<0130:dnf>2.0.co;2.

[39] 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

[40] 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).

[41] Christopher Riley and Dallas Campbell, Oct 22, 2012. "The maths that made Voyager possible". BBC News Science and Environment. Recovered 16 Jun 2013.

[42] 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.

[43] Cooley, James W.; Tukey, John W. (1965). "An algorithm for the machine calculation of complex Fourier series". Math. Comput. 19: 297–301.

[44] Kohn, Walter; Hohenberg, Pierre (1964). "Inhomogeneous Electron Gas". Physical Review. 136 (3B): B864–B871. Bibcode:1964PhRv..136..864H. doi:10.1103/PhysRev.136.B864.

[45] Kohn, Walter; Sham, Lu Jeu (1965). "Self-Consistent Equations Including Exchange and Correlation Effects". Physical Review. 140 (4A): A1133–A1138. Bibcode:1965PhRv..140.1133K. doi:10.1103/PhysRev.136.B864.

[46] "The Nobel Prize in Chemistry 1998". Nobelprize.org. Retrieved 2008-10-06.

[47] 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.

[48] Appel, Kenneth; Haken, Wolfgang (1977). "Every planar map is four colorable, Part I: Discharging". Illinois Journal of Mathematics. 21: 429–490.

[49] Appel, K.; Haken, W. (1977). "Every Planar Map is Four-Colorable, II: Reducibility". Illinois J. Math. 21: 491–567.

[50] Appel, K.; Haken, W. (1977). "The Solution of the Four-Color Map Problem". Sci. Amer. 237: 108–121. doi:10.1038/scientificamerican1077-108.

[51] L. Greengard, The Rapid Evaluation of Potential Fields in Particle Systems, MIT, Cambridge, (1987).

[52] Rokhlin, Vladimir (1985). "Rapid Solution of Integral Equations of Classic Potential Theory." J. Computational Physics Vol. 60, pp. 187-207.

[53] L. Greengard and V. Rokhlin, "A fast algorithm for particle simulations," J. Comput. Phys., 73 (1987), no. 2, pp. 325–348.

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  |volume=5 |issue=4 |year=1958 |pages=339&ndash;342|doi=10.1145/320941.320947 |mr=0111128}}</ref>
 * [[John G.F. Francis]]<ref>{{cite journal | last1 = Francis | first1 = J.G.F. | year = 1961 | title = The QR Transformation, I | url = | journal = The Computer Journal | volume = 4 | issue = 3| pages = 265–271 | doi=10.1093/comjnl/4.3.265}}</ref><ref>{{cite journal | last1 = Francis | first1 = J.G.F. | year = 1962 | title = The QR Transformation, II | url = | journal = The Computer Journal | volume = 4 | issue = 4| pages = 332–345 | doi=10.1093/comjnl/4.4.332}}</ref> and [[Vera Kublanovskaya]]<ref>{{cite journal | last1 = Kublanovskaya | first1 = Vera N. | year = 1961 | title = On some algorithms for the solution of the complete eigenvalue problem | url = | journal = USSR Computational Mathematics and Mathematical Physics | volume = 1 | issue = 3| pages = 637–657 | doi = 10.1016/0041-5553(63)90168-X }} 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).
-* A team led by Backus develops the FORTRAN compiler and programming language  at [[IBM]]'s research centre in [[San Jose, California]]. This sped the adoption of scientific programming,<ref>W.W. McDowell Award citation: {{cite web| title=W. Wallace McDowell Award| url=http://www.computer.org/portal/site/ieeecs/menuitem.c5efb9b8ade9096b8a9ca0108bcd45f3/index.jsp?&pName=ieeecs_level1&path=ieeecs/about/awards&file=WallaceMcD_recipients.xml&xsl=generic.xsl&| accessdate=April 15, 2008}}</ref><ref name="National Science Foundation">National Medal of Science citation: {{cite web | title = The President's National Medal of Science: John Backus | publisher = National Science Foundation | url = http://www.nsf.gov/od/nms/recip_details.cfm?recip_id=25 | accessdate =March 21, 2007}}</ref><ref>{{cite web | title = ACM Turing Award Citation: John Backus | publisher = [[Association for Computing Machinery]] | url = http://www.acm.org/awards/turing_citations/backus.html | accessdate =March 22, 2007 |archiveurl = http://web.archive.org/web/20070204114319/http://www.acm.org/awards/turing_citations/backus.html <!-- Bot retrieved archive --> |archivedate = February 4, 2007}}</ref> and is one of the [[Timeline of programming languages|oldest extant programming languages]], as well as one of the [[Measuring programming language popularity|most popular in science and engineering]].
+* A team led by Backus develops the FORTRAN compiler and programming language  at [[IBM]]'s research centre in [[San Jose, California]]. This sped the adoption of scientific programming,<ref>W.W. McDowell Award citation: {{cite web| title=W. Wallace McDowell Award| url=http://www.computer.org/portal/site/ieeecs/menuitem.c5efb9b8ade9096b8a9ca0108bcd45f3/index.jsp?&pName=ieeecs_level1&path=ieeecs/about/awards&file=WallaceMcD_recipients.xml&xsl=generic.xsl&| accessdate=April 15, 2008}}</ref><ref name="National Science Foundation">National Medal of Science citation: {{cite web | title = The President's National Medal of Science: John Backus | publisher = National Science Foundation | url = http://www.nsf.gov/od/nms/recip_details.cfm?recip_id=25 | accessdate =March 21, 2007}}</ref><ref>{{cite web|title=ACM Turing Award Citation: John Backus |publisher=[[Association for Computing Machinery]] |url=http://www.acm.org/awards/turing_citations/backus.html |accessdate=March 22, 2007 |archiveurl=https://web.archive.org/web/20070204114319/http://www.acm.org/awards/turing_citations/backus.html |archivedate=February 4, 2007 |deadurl=yes |df= }}</ref> and is one of the [[Timeline of programming languages|oldest extant programming languages]], as well as one of the [[Measuring programming language popularity|most popular in science and engineering]].
 ==1960s==
@@ Line 107: / Line 107: @@
 ==External links==
 * SIAM (Society for Industrial and Applied Mathematics) News. [http://www.siam.org/news/news.php?id=637 Top 10 Algorithms of the 20th Century].
-* [http://history.siam.org/ The History of Numerical Analysis and Scientific Computing @ SIAM (Society for Industrial and Applied Mathematics)]
+* [https://web.archive.org/web/20130125035840/http://history.siam.org/ The History of Numerical Analysis and Scientific Computing @ SIAM (Society for Industrial and Applied Mathematics)]
 * {{cite journal | last1 = Ruttimann | first1 = Jacqueline | year = 2006 | title = 2020 computing: Milestones in scientific computing | url = http://www.nature.com/nature/journal/v440/n7083/full/440399a.html | journal = Nature | volume = 440 | issue = | pages = 399–405 | doi = 10.1038/440399a | pmid=16554772}}
 * {{cite journal | last1 = Anderson | first1 = H. L. | year = | title = Scientific Uses of the MANIAC | url = http://adsabs.harvard.edu/abs/1986JSP....43..731A | journal = Journal of Statistical Physics | volume = 43 | issue = 5–6| pages = 731–748 | doi = 10.1007/BF02628301 | bibcode=1986JSP....43..731A}}