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Before modern computers

18th century

• Simpson rediscovers Simpson's rule, a century after Johannes Kepler (who derived it in 1615 after seeing it used for wine barrels).
• 1733 – The French naturalist Comte de Buffon poses his needle problem.¹²
• Euler comes up with a simple numerical method for integrands.³⁴⁵

19th century

• First formulation of Gram-Schmidt orthogonalisation by Laplace,⁶ to be further improved decades later.⁷⁸⁹¹⁰
• 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.¹¹¹² The engine was never completed, however, so her code was never tested.¹³
• Adams-Bashforth method published.¹⁴
• In applied mathematics, Jacobi develops technique for solving numerical equations.¹⁵¹⁶¹⁷
• Gauss Seidel first published.
• To help with computing tides, Harmonic Analyser is built in 1886.

1900s (decade)

• 1900 – Runge's work followed by Martin Kutta to invent the Runge-Kutta method for approximating integration for differential equations.¹⁸¹⁹

1910s (decade)

• 1910 – A-M Cholesky creates a matrix decomposition scheme.²⁰²¹
• Richardson extrapolation introduced.

1920s

• 1922 – Lewis Fry Richardson introduces numerical weather forecasting by manual calculation, using methods originally developed by Vilhelm Bjerknes as early as 1895.²²²³
• 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.²⁴
• 1926 Adams-Moulton method.
• 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.
• Shannon explains how to use electric circuits to do Boolean algebra in "A Symbolic Analysis of Relay and Switching Circuits"
• 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

• 1947 – Metropolis algorithm for Monte Carlo simulation (named one of the top-10 algorithms of the 20th century)²⁵ invented at Los Alamos by von Neumann, Ulam and Metropolis.²⁶²⁷²⁸
• George Dantzig introduces the simplex method (named one of the top 10 algorithms of the 20th century)²⁹ in 1947.³⁰
• Ulam and von Neumann introduce the notion of cellular automata.³¹
• Turing formulated the LU decomposition method.³²
• A. W. H. Phillips invents the MONIAC hydraulic computer at LSE, better known as "Phillips Hydraulic Computer".³³³⁴
• First hydro simulations occurred at Los Alamos.³⁵³⁶

1950s

• First successful weather predictions on a computer occurred.³⁷³⁸
• 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.^39404142 Named one of the top 10 algorithms of the 20th century.⁴³
• Equations of State Calculations by Fast Computing Machines introduces the Metropolis–Hastings algorithm.⁴⁴
• Molecular dynamics invented by Bernie Alder and Wainwright ⁴⁵⁴⁶
• A S Householder invents his eponymous matrices and transformation method (voted one of the top 10 algorithms of the 20th century).⁴⁷
• 1953 – Enrico Fermi, John Pasta, Stanislaw Ulam, and Mary Tsingou discover the Fermi–Pasta–Ulam–Tsingou problem through computer simulations of a vibrating string.⁴⁸
• A team led by John Backus develops the FORTRAN compiler and programming language at IBM's research centre in San Jose, California. This sped the adoption of scientific programming,⁴⁹⁵⁰⁵¹ and is one of the oldest extant programming languages, as well as one of the most popular in science and engineering.

1960s

• 1960 – 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.⁵²
• 1961 – John G.F. Francis⁵³⁵⁴ and Vera Kublanovskaya⁵⁵ invent QR factorization (voted one of the top 10 algorithms of the 20th century).
• 1963 – Edward Lorenz discovers the butterfly effect on a computer, attracting interest in chaos theory.⁵⁶
• 1961 – Using computational investigations of the 3-body problem, Michael Minovitch formulates the gravity assist method.⁵⁷⁵⁸
• 1964 – Molecular dynamics invented independently by Aneesur Rahman.⁵⁹
• 1965 – fast Fourier transform developed by James W. Cooley and John W. Tukey.⁶⁰
• 1964 – Walter Kohn, with Lu Jeu Sham and Pierre Hohenberg, instigates the development of density functional theory,⁶¹⁶² for which he shares the 1998 Nobel Chemistry Prize with John Pople.⁶³ This contribution is arguably the earliest work to which Nobels were given for a computer program or computational technique.
• First regression calculations in economics.

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.⁶⁴
• 1977 – Kenneth Appel and Wolfgang Haken prove the four colour theorem, the first theorem to be proved by computer.⁶⁵⁶⁶⁶⁷

1980s

• Fast multipole method (voted one of the top 10 algorithms of the 20th century) invented by Vladimir Rokhlin and Leslie Greengard.⁶⁸⁶⁹⁷⁰
• Car–Parrinello molecular dynamics developed by Roberto Car and Michele Parrinello

1990s

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

2000s

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

2010s

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

Further information: Timeline of computing 2020–2029 and List of years in science § 2020s

See also

• Computational science
• History of computing
• History of mathematics
• Timeline of mathematics
• Timeline of algorithms
• Timeline of computational physics
• Timeline of computational mathematics
• Timeline of numerical analysis after 1945
• History of computing hardware

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 (7083): 399–405. Bibcode:2006Natur.440..399R. doi:10.1038/440399a. PMID 16554772. S2CID 21967804.
• Anderson, H. L. (1986). "Scientific Uses of the MANIAC". Journal of Statistical Physics. 43 (5–6): 731–748. Bibcode:1986JSP....43..731A. doi:10.1007/BF02628301. S2CID 122676398.
• IEEE Milestones

References

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. http://mathworld.wolfram.com/BuffonsNeedleProblem.html ↩

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 http://mathworld.wolfram.com/BuffonsNeedleProblem.html ↩

3. Euler, L. Institutionum calculi integralis. Impensis Academiae Imperialis Scientiarum, 1768. /wiki/Leonhard_Euler ↩

4. Butcher, John C. (2003), Numerical Methods for Ordinary Differential Equations, New York: John Wiley & Sons, ISBN 978-0-471-96758-3. /wiki/ISBN_(identifier) ↩

5. Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993), Solving ordinary differential equations I: Nonstiff problems, Berlin, New York: Springer-Verlag, ISBN 978-3-540-56670-0. /wiki/ISBN_(identifier) ↩

6. Laplace, PS. (1816). Théorie Analytique des Probabilités :First Supplement, p. 497ff. ↩

7. Gram, J. P. (1883). "Ueber die Entwickelung reeler Funtionen in Reihen mittelst der Methode der kleinsten Quadrate". JRNL. Für die reine und angewandte Math. 94: 71–73. ↩

8. Schmidt, E. "Zur Theorie der linearen und nichtlinearen Integralgleichungen. I. Teil: Entwicklung willkürlicher Funktionen nach Systemen vorgeschriebener". Math. Ann. 63: 1907. ↩

9. Earliest Known Uses of Some of the Words of Mathematics (G). As of Aug 2017. http://jeff560.tripod.com/g.html#GRAM-SCHMIDT%20ORTHOGONALIZATION ↩

10. Farebrother, RW (1988). Linear Least Squares Computations. CRC Press. ISBN 9780824776619. Retrieved 19 August 2017. 9780824776619 ↩

11. Simonite, Tom (24 March 2009). "Short Sharp Science: Celebrating Ada Lovelace: the 'world's first programmer'". New Scientist. Retrieved 14 April 2012. https://www.newscientist.com/blogs/shortsharpscience/2009/03/ada-lovelace-day.html ↩

12. Tom Stoppard's “Arcadia,” at Twenty. By Brad Leithauser. The New Yorker, August 8, 2013. http://www.newyorker.com/online/blogs/books/2013/08/tom-stoppards-arcadia-at-twenty.html ↩

13. Kim, Eugene Eric; Toole, Betty Alexandra (May 1999). "Ada and the first computer". Scientific American. 280 (5): 70–71. Bibcode:1999SciAm.280e..76E. doi:10.1038/scientificamerican0599-76. /wiki/Bibcode_(identifier) ↩

14. Bashforth, Francis (1883), An Attempt to test the Theories of Capillary Action by comparing the theoretical and measured forms of drops of fluid. With an explanation of the method of integration employed in constructing the tables which give the theoretical forms of such drops, by J. C. Adams, Cambridge. ↩

15. Jacobi's Ideas on Eigenvalue Computation in a modern context, Henk van der Vorst. http://canum2006.univ-rennes1.fr/exposes_et_posters/vandervorst.pdf ↩

16. Jacobi method, Encyclopedia of Mathematics. http://www.encyclopediaofmath.org/index.php/Jacobi_method ↩

17. The Early History of Matrix Iterations: With a Focus on the Italian Contribution, Michele Benzi, 26 October 2009. SIAM Conference on Applied Linear Algebra, Monterey Bay – Seaside, California. http://www.siam.org/meetings/la09/talks/benzi.pdf ↩

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

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

20. Commandant Benoit (1924). "Note sur une méthode de résolution des équations normales provenant de l'application de la méthode des moindres carrés à un système d'équations linéaires en nombre inférieur à celui des inconnues (Procédé du Commandant Cholesky)". Bulletin Géodésique. 2: 67–77. doi:10.1007/BF03031308. /wiki/Doi_(identifier) ↩

21. Cholesky (1910). Sur la résolution numérique des systèmes d'équations linéaires. /wiki/Andr%C3%A9-Louis_Cholesky ↩

22. L F Richardson, Weather Prediction by Numerical Process. Cambridge University Press (1922). /wiki/Cambridge_University_Press ↩

23. 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. Archived from the original (PDF) on 2010-07-08. Retrieved 2010-12-23. /wiki/Peter_Lynch_(meteorologist) ↩

24. Grete Hermann (1926). "Die Frage der endlich vielen Schritte in der Theorie der Polynomideale". Mathematische Annalen. 95: 736–788. doi:10.1007/bf01206635. S2CID 115897210. Archived from the original on 2016-10-09. Retrieved 2017-05-05. https://web.archive.org/web/20161009190144/http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0095&DMDID=DMDLOG_0044&L=1 ↩

25. Dongarra, J.; Sullivan, F. (January 2000). "Guest Editors Introduction: the Top 10 Algorithms". Computing in Science & Engineering. 2 (1): 22–23. Bibcode:2000CSE.....2a..22D. doi:10.1109/MCISE.2000.814652. ISSN 1521-9615. https://ieeexplore.ieee.org/document/814652 ↩

26. Metropolis, N. (1987). "The Beginning of the Monte Carlo method" (PDF). Los Alamos Science. 15: 125.. Accessed 5 May 2012. http://library.lanl.gov/cgi-bin/getfile?15-12.pdf ↩

27. S. Ulam, R. D. Richtmyer, and J. von Neumann(1947). Statistical methods in neutron diffusion. Los Alamos Scientific Laboratory report LAMS–551. http://library.lanl.gov/cgi-bin/getfile?00329286.pdf ↩

28. Metropolis, N.; Ulam, S. (1949). "The Monte Carlo method". Journal of the American Statistical Association. 44 (247): 335–341. doi:10.1080/01621459.1949.10483310. PMID 18139350. /wiki/Doi_(identifier) ↩

29. Dongarra, J.; Sullivan, F. (January 2000). "Guest Editors Introduction: the Top 10 Algorithms". Computing in Science & Engineering. 2 (1): 22–23. Bibcode:2000CSE.....2a..22D. doi:10.1109/MCISE.2000.814652. ISSN 1521-9615. https://ieeexplore.ieee.org/document/814652 ↩

30. "SIAM News, November 1994". Archived from the original on 16 April 2009. Retrieved 6 June 2012. Systems Optimization Laboratory, Stanford University Huang Engineering Center (site host/mirror). https://web.archive.org/web/20090416184457/http://www.stanford.edu/group/SOL/dantzig.html ↩

31. Von Neumann, J., Theory of Self-Reproducing Automata, Univ. of Illinois Press, Urbana, 1966. ↩

32. 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.) . /wiki/ISBN_(identifier) ↩

33. The computer model that once explained the British economy. Larry Elliott, The Guardian, Thursday 8 May 2008. https://www.theguardian.com/business/2008/may/08/bankofenglandgovernor.economics ↩

34. Phillip's Economic Computer, 1949. Archived 2014-10-03 at the Wayback Machine Exhibit at London Science Museum. http://www.sciencemuseum.org.uk/objects/computing_and_data_processing/1995-210.aspx ↩

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

36. Von Neumann, J.; Richtmyer, R. D. (1950). "A Method for the Numerical Calculation of Hydrodynamic Shocks". Journal of Applied Physics. 21 (3): 232–237. Bibcode:1950JAP....21..232V. doi:10.1063/1.1699639. /wiki/Bibcode_(identifier) ↩

37. Charney, J.; Fjørtoft, R.; von Neumann, J. (1950). "Numerical Integration of the Barotropic Vorticity Equation". Tellus. 2 (4): 237–254. Bibcode:1950Tell....2..237C. doi:10.1111/j.2153-3490.1950.tb00336.x. /wiki/Bibcode_(identifier) ↩

38. See the review article:- Smagorinsky, J (1983). "The Beginnings of Numerical Weather Prediction and General Circulation Modelling: Early Recollections" (PDF). Advances in Geophysics. 25: 3–37. Bibcode:1983AdGeo..25....3S. doi:10.1016/S0065-2687(08)60170-3. ISBN 9780120188253. Retrieved 6 June 2012. 9780120188253 ↩

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

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

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

42. 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). ↩

43. Dongarra, J.; Sullivan, F. (January 2000). "Guest Editors Introduction: the Top 10 Algorithms". Computing in Science & Engineering. 2 (1): 22–23. Bibcode:2000CSE.....2a..22D. doi:10.1109/MCISE.2000.814652. ISSN 1521-9615. https://ieeexplore.ieee.org/document/814652 ↩

44. 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. OSTI 4390578. S2CID 1046577. http://www.aliquote.org/pub/metropolis-et-al-1953.pdf ↩

45. Alder, B. J.; Wainwright, T. E. (1957). "Phase Transition for a Hard Sphere System". J. Chem. Phys. 27 (5): 1208. Bibcode:1957JChPh..27.1208A. doi:10.1063/1.1743957. S2CID 10791650. /wiki/Bibcode_(identifier) ↩

46. Alder, B. J.; Wainwright, T. E. (1962). "Phase Transition in Elastic Disks". Phys. Rev. 127 (2): 359–361. Bibcode:1962PhRv..127..359A. doi:10.1103/PhysRev.127.359. OSTI 4798469. /wiki/Bibcode_(identifier) ↩

47. 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. https://hal.archives-ouvertes.fr/hal-01316095/file/p339householderb.pdf ↩

48. 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 https://www.osti.gov/accomplishments/documents/fullText/ACC0041.pdf ↩

49. W.W. McDowell Award citation: "W. Wallace McDowell Award". Archived from the original on September 29, 2007. Retrieved April 15, 2008. https://web.archive.org/web/20070929133553/http://www.computer.org/portal/site/ieeecs/menuitem.c5efb9b8ade9096b8a9ca0108bcd45f3/index.jsp?&pName=ieeecs_level1&path=ieeecs%2Fabout%2Fawards&file=WallaceMcD_recipients.xml&xsl=generic.xsl& ↩

50. National Medal of Science citation: "The President's National Medal of Science: John Backus". National Science Foundation. Retrieved March 21, 2007. https://www.nsf.gov/od/nms/recip_details.cfm?recip_id=25 ↩

51. "ACM Turing Award Citation: John Backus". Association for Computing Machinery. Archived from the original on February 4, 2007. Retrieved March 22, 2007. https://web.archive.org/web/20070204114319/http://www.acm.org/awards/turing_citations/backus.html ↩

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

53. Francis, J.G.F. (1961). "The QR Transformation, I". The Computer Journal. 4 (3): 265–271. doi:10.1093/comjnl/4.3.265. https://doi.org/10.1093%2Fcomjnl%2F4.3.265 ↩

54. Francis, J.G.F. (1962). "The QR Transformation, II". The Computer Journal. 4 (4): 332–345. doi:10.1093/comjnl/4.4.332. https://doi.org/10.1093%2Fcomjnl%2F4.4.332 ↩

55. 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). /wiki/Doi_(identifier) ↩

56. 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. http://www.nd.edu/~powers/ame.60611/lorenz.article.pdf ↩

57. 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). ↩

58. Christopher Riley and Dallas Campbell, Oct 22, 2012. "The maths that made Voyager possible". BBC News Science and Environment. Recovered 16 Jun 2013. http://www.bbc.co.uk%2Fnews%2Fscience-environment-20033940&ei=j-29UZ6sNIexPInBgfAG&usg=AFQjCNEj30660hWJWTpfDJohrZek5KxAFA ↩

59. 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. /wiki/Bibcode_(identifier) ↩

60. Cooley, James W.; Tukey, John W. (1965). "An algorithm for the machine calculation of complex Fourier series" (PDF). Math. Comput. 19 (90): 297–301. doi:10.1090/s0025-5718-1965-0178586-1.[permanent dead link] http://attach3.bdwm.net/attach/0Announce/groups/GROUP_3/MathTools/D6714701A/D69595345/M.1089260001.A/CooleyJ_AlgMCC.pdf ↩

61. Kohn, Walter; Hohenberg, Pierre (1964). "Inhomogeneous Electron Gas". Physical Review. 136 (3B): B864 – B871. Bibcode:1964PhRv..136..864H. doi:10.1103/PhysRev.136.B864. https://doi.org/10.1103%2FPhysRev.136.B864 ↩

62. 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.140.A1133. https://doi.org/10.1103%2FPHYSREV.140.A1133 ↩

63. "The Nobel Prize in Chemistry 1998". Nobelprize.org. Retrieved 2008-10-06. http://nobelprize.org/nobel_prizes/chemistry/laureates/1998/index.html ↩

64. 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. /wiki/ISBN_(identifier) ↩

65. Appel, Kenneth; Haken, Wolfgang (1977). "Every planar map is four colorable, Part I: Discharging". Illinois Journal of Mathematics. 21 (3): 429–490. doi:10.1215/ijm/1256049011. https://doi.org/10.1215%2Fijm%2F1256049011 ↩

66. Appel, K.; Haken, W. (1977). "Every Planar Map is Four-Colorable, II: Reducibility". Illinois J. Math. 21: 491–567. doi:10.1215/ijm/1256049012. https://doi.org/10.1215%2Fijm%2F1256049012 ↩

67. Appel, K.; Haken, W. (1977). "The Solution of the Four-Color Map Problem". Sci. Am. 237 (4): 108–121. Bibcode:1977SciAm.237d.108A. doi:10.1038/scientificamerican1077-108. /wiki/Bibcode_(identifier) ↩

68. L. Greengard, The Rapid Evaluation of Potential Fields in Particle Systems, MIT, Cambridge, (1987). ↩

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

70. Greengard, L.; Rokhlin, V. (1987). "A fast algorithm for particle simulations". J. Comput. Phys. 73 (2): 325–348. Bibcode:1987JCoPh..73..325G. doi:10.1016/0021-9991(87)90140-9. /wiki/Bibcode_(identifier) ↩