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.

- 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] } - Crank–Nicolson method was developed by Crank and Nicolson.
^{ [4] } - Dantzig introduces the simplex method (voted one of the top 10 algorithms of the 20th century) in 1947.
^{ [5] } - Turing formulated the LU decomposition method.
^{ [6] }

- Successive over-relaxation was devised simultaneously by D.M. Young, Jr.
^{ [7] }and by H. Frankel in 1950. - 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.
^{ [8] }^{ [9] }^{ [10] }^{ [11] }Voted one of the top 10 algorithms of the 20th century. *Equations of State Calculations by Fast Computing Machines*introduces the Metropolis–Hastings algorithm.^{ [12] }- In numerical differential equations, Lax and Friedrichs invent the Lax-Friedrichs method.
^{ [13] }^{ [14] } - Householder invents his eponymous matrices and transformation method (voted one of the top 10 algorithms of the 20th century).
^{ [15] } - Romberg integration
^{ [16] } - John G.F. Francis
^{ [17] }and Vera Kublanovskaya^{ [18] }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,
^{ [19] }to describe the methods of Courant, Hrenikoff, Galerkin and Zienkiewicz, among others. See also here. - Exponential integration by Certaine and Pope.
- In computational fluid dynamics and numerical differential equations, Lax and Wendroff invent the Lax-Wendroff method.
^{ [20] } - Fast Fourier Transform (voted one of the top 10 algorithms of the 20th century) invented by Cooley and Tukey.
^{ [21] } - First edition of
*Handbook of Mathematical Functions*by Abramowitz and Stegun, both of the U.S.National Bureau of Standards.^{ [22] } - Broyden does new quasi-Newton method for finding roots in 1965.
- The MacCormack method, for the numerical solution of hyperbolic partial differential equations in computational fluid dynamics, is introduced by MacCormack in 1969.
^{ [23] } - Verlet (re)discovers a numerical integration algorithm, (first used in 1791 by Delambre, by Cowell and Crommelin in 1909, and by Carl Fredrik Störmer in 1907, hence the alternative names Störmer's method or the Verlet-Störmer method) for dynamics.

Creation of LINPACK and associated benchmark by Dongarra et al.,^{ [24] }^{ [25] } as well as BLAS.

- Progress in digital wavelet theory throughout the decade, led by Daubechies et al.
- Creation of MINPACK
- Fast multipole method (voted one of the top 10 algorithms of the 20th century) invented by Rokhlin and Greengard.
^{ [26] }^{ [27] }^{ [28] } - First edition of
*Numerical Recipes*by Press, Teukolsky, et al.^{ [29] } - In numerical linear algebra, the GMRES algorithm invented in 1986.
^{ [30] }

**Numerical analysis** is the study of algorithms that use numerical approximation for the problems of mathematical analysis. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics, numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living cells in medicine and biology.

In the mathematical subfield of numerical analysis, **numerical stability** is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context. One is numerical linear algebra and the other is algorithms for solving ordinary and partial differential equations by discrete approximation.

**Jack Joseph Dongarra** is an American computer scientist. He is the American University Distinguished Professor of Computer Science in the Electrical Engineering and Computer Science Department at the University of Tennessee. He holds the position of a Distinguished Research Staff member in the Computer Science and Mathematics Division at Oak Ridge National Laboratory, Turing Fellowship in the School of Mathematics at the University of Manchester, and is an adjunct professor in the Computer Science Department at Rice University. He served as a faculty fellow at the Texas A&M University Institute for Advanced Study (2014–2018). Dongarra is the founding director of the Innovative Computing Laboratory at the University of Tennessee.

**Basic Linear Algebra Subprograms** (**BLAS**) is a specification that prescribes a set of low-level routines for performing common linear algebra operations such as vector addition, scalar multiplication, dot products, linear combinations, and matrix multiplication. They are the *de facto* standard low-level routines for linear algebra libraries; the routines have bindings for both C and Fortran. Although the BLAS specification is general, BLAS implementations are often optimized for speed on a particular machine, so using them can bring substantial performance benefits. BLAS implementations will take advantage of special floating point hardware such as vector registers or SIMD instructions.

**Computational science**, also known as **scientific computing** or **scientific computation** (**SC**), is a field in mathematics that uses advanced computing capabilities to understand and solve complex problems. It is an area of science that spans many disciplines, but at its core, it involves the development of models and simulations to understand natural systems.

**Cornelius (Cornel) Lanczos** was a Hungarian-American and later Hungarian-Irish mathematician and physicist, who was born in Székesfehérvár, Fejér County, Kingdom of Hungary on February 2, 1893, and died on June 25, 1974. According to György Marx he was one of The Martians.

In numerical analysis, the **Lax equivalence theorem** is a fundamental theorem in the analysis of finite difference methods for the numerical solution of partial differential equations. It states that for a consistent finite difference method for a well-posed linear initial value problem, the method is convergent if and only if it is stable.

**Numerical linear algebra**, sometimes called **applied linear algebra**, is the study of how matrix operations can be used to create computer algorithms which efficiently and accurately provide approximate answers to questions in continuous mathematics. It is a subfield of numerical analysis, and a type of linear algebra. Computers use floating-point arithmetic and cannot exactly represent irrational data, so when a computer algorithm is applied to a matrix of data, it can sometimes increase the difference between a number stored in the computer and the true number that it is an approximation of. Numerical linear algebra uses properties of vectors and matrices to develop computer algorithms that minimize the error introduced by the computer, and is also concerned with ensuring that the algorithm is as efficient as possible.

The **fast multipole method** (**FMM**) is a numerical technique that was developed to speed up the calculation of long-ranged forces in the *n*-body problem. It does this by expanding the system Green's function using a multipole expansion, which allows one to group sources that lie close together and treat them as if they are a single source.

**Leslie Fox** was a British mathematician noted for his contribution to numerical analysis.

**Burton Wendroff** is an American applied mathematician known for his contributions to the development of numerical methods for the solution of hyperbolic partial differential equations. The Lax–Wendroff method for the solution of hyperbolic PDE is named for Wendroff.

**Andrew Knyazev** is an American mathematician. He graduated from the Faculty of Computational Mathematics and Cybernetics of Moscow State University under the supervision of Evgenii Georgievich D'yakonov in 1981 and obtained his PhD in Numerical Mathematics at the Russian Academy of Sciences under the supervision of Vyacheslav Ivanovich Lebedev in 1985. He worked at the Kurchatov Institute between 1981–1983, and then to 1992 at the Marchuk Institute of Numerical Mathematics of the Russian Academy of Sciences, headed by Gury Marchuk.

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**.

This is a timeline of key developments in computational mathematics.

**Parareal** is a parallel algorithm from numerical analysis and used for the solution of initial value problems. It was introduced in 2001 by Lions, Maday and Turinici. Since then, it has become one of the most widely studied parallel-in-time integration methods.

**Beresford Neill Parlett** is an English applied mathematician, specializing in numerical analysis and scientific computation.

**Probabilistic numerics** is a scientific field at the intersection of statistics, machine learning and applied mathematics, where tasks in numerical analysis including finding numerical solutions for integration, linear algebra, optimisation and differential equations are seen as problems of statistical, probabilistic, or Bayesian inference.

- ↑ 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.
- ↑ 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. - ↑ Crank, J. (John); Nicolson, P. (Phyllis) (1947). "A practical method for numerical evaluation of solutions of partial differential equations of the heat conduction type".
*Proc. Camb. Phil. Soc*.**43**(1): 50–67. doi:10.1007/BF02127704. S2CID 16676040. - ↑ "SIAM News, November 1994" . Retrieved 6 June 2012. Hosted at Systems Optimization Laboratory, Stanford University, Huang Engineering Center Archived 12 November 2012 at the Wayback Machine .
- ↑ 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.) .
- ↑ Young, David M. (1 May 1950),
*Iterative methods for solving partial difference equations of elliptical type*(PDF), PhD thesis, Harvard University, retrieved 15 June 2009 - ↑ 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). "Equation 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. - ↑ Lax, PD (1954). "Weak solutions of nonlinear hyperbolic equations and their numerical approximation".
*Comm. Pure Appl. Math*.**7**: 159–193. doi:10.1002/cpa.3160070112. - ↑ Friedrichs, KO (1954). "Symmetric hyperbolic linear differential equations".
*Comm. Pure Appl. Math*.**7**(2): 345–392. doi:10.1002/cpa.3160070206. - ↑ 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. - ↑ 1955
- ↑ J.G.F. Francis, "The QR Transformation, I",
*The Computer Journal*, 4(3), pages 265–271 (1961, received October 1959) online at oxfordjournals.org;J.G.F. Francis, "The QR Transformation, II"*The Computer Journal*, 4(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.
- ↑ P.D Lax; B. Wendroff (1960). "Systems of conservation laws".
*Commun. Pure Appl. Math*.**13**(2): 217–237. doi:10.1002/cpa.3160130205. Archived from the original on 25 September 2017. - ↑ 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 . - ↑ M Abramowitz and I Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Publisher: Dover Publications. Publication date: 1964; ISBN 0-486-61272-4;OCLC Number:18003605 .
- ↑ MacCormack, R. W., The Effect of viscosity in hypervelocity impact cratering, AIAA Paper, 69-354 (1969).
- ↑ J. Bunch; G. W. Stewart.; Cleve Moler; Jack J. Dongarra (1979). "LINPACK User's Guide". Philadelphia, PA: SIAM.
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(help) - ↑ The LINPACK Benchmark:Past,Present,and Future. Jack J. Dongarra, Piotr Luszczeky, and Antoine Petitetz. December 2001.
- ↑ 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.
- ↑ 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. - ↑ Press, William H.; Teukolsky, Saul A.; Vetterling, William T.; Flannery, Brian P. (1986). Numerical Recipes: The Art of Scientific Computing. New York: Cambridge University Press. ISBN 0-521-30811-9.
- ↑ Saad, Y.; Schultz, M.H. (1986). "GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems".
*SIAM J. Sci. Stat. Comput*.**7**(3): 856–869. CiteSeerX 10.1.1.476.951 . doi:10.1137/0907058.

- Cipra, Barry Arthur (2000). "Top 10 Algorithms of the 20th Century".
*SIAM News*. Society for Industrial and Applied Mathematics (SIAM). Retrieved 1 December 2012.

- 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. - The Monte Carlo Method: Classic Papers
- Monte Carlo Landmark Papers
- “Must read” papers in numerical analysis. Discussion at MathOverflow based upon a selected reading list on Lloyd N. Trefethen's personal site.

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