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In 1993, Prof. L.N. Trefethen published a NA-net posting with a list of thirteen paper he used for teaching the seminar Classic Papers in Numerical Analysis. In Trefethen's words, ... this course provided a satisfying vison of the broad scope of numerical analysis and the sense of excitement at what a diversity of beautiful and powerful ideas have been invented in this field.

Prof. Trefethen's list (links):

  1. Cooley & Tukey (1965) the Fast Fourier Transform
  2. Courant, Friedrichs & Lewy (1928) finite difference methods for PDE
  3. Householder (1958) QR factorization of matrices
  4. Curtiss & Hirschfelder (1952) stiffness of ODEs; BD formulas
  5. de Boor (1972) calculations with B-splines
  6. Courant (1943) finite element methods for PDE
  7. Golub & Kahan (1965) the singular value decomposition
  8. Brandt (1977) multigrid algorithms
  9. Hestenes & Stiefel (1952) the conjugate gradient iteration
  10. Fletcher & Powell (1963) optimization via quasi-Newton updates
  11. Wanner, Hairer & Norsett (1978) order stars and applications to ODE
  12. Karmarkar (1984) interior pt. methods for linear prog.
  13. Greengard & Rokhlin (1987) multipole methods for particles

Most readers of this note, according Prof. Trefethen, will have thought of other classic authors and papers that should have been on the list.

The question is: In your opinion, what are other classic authors and papers that should be in a must read list of papers in numerical analysis?

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See a Who is Who of numerical analysis in Trefethen's Who invented the great numerical algorithms?. – lhf Jul 17 '12 at 1:55
Dear PaPiro: Please read the guidelines at and rewrite your question. – S. Carnahan Jul 17 '12 at 2:26
@Carnahan, done! Thanks. – Papiro Jul 17 '12 at 2:39

11 Answers 11

J. von Neumann, H.H. Goldstine, Numerical Inverting of Matrices of High Order, Bull. Amer. Math. Soc., Vol. 53, No. 11, pp. 1021-1099, 1947.

According to J. Grcar, Birthday of Modern Numerical Analysis, available here or here, this paper is considered the first paper in modern analysis because it is the first to study rounding error and because much of the paper is an essay on scientific computing (albeit with an emphasis on numerical linear algebra).

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Metropolis N, Ulam S (1949) The Monte Carlo method J. Am. Stat. Assoc. 44:335-341

Marsaglia G (1968) Random numbers fall mainly in the planes. Proc. Natl. Acad. Sci. USA 61:25-28

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Fermi, Pasta, Ulam, Studies of Nonlinear Problems, unpublished report from Los Alamos National Laboratory, 1955.

This paper arguably started the numerical investigation of nonlinear dynamics and chaos, which has become a huge field. See for example Ford, The Fermi-Pasta-Ulam problem: Paradox turns discovery, Physics Reports 213 (2002) pp. 271-310 for a review.

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The Fermi, Pasta and Ulam report is available at – Papiro Jul 17 '12 at 14:58

P. D. Lax & B. Wendroff. On the stability of difference schemes, Communications on Pure and Applied Mathematics 15, pp 363–371 (1962).

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2 a comment why it is "must read " - I think would be worth for many. From abstract I see that "they covered many previously known results", but for outsider this is not informative. – Alexander Chervov Jul 17 '12 at 6:38

S.K. Godunov, A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics, Matematicheskii Sbornik (1959). (in Russian)

Virtually all modern methods for nonlinear hyperbolic PDEs are based on this.

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Gordunov's paper (in Russian) is available at… – Papiro Jul 17 '12 at 9:44

Jinchao Xu, Iterative methods by space decomposition and subspace correction, SIAM Review 34(4):581-613, 1992.

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Edward N. Lorenz (1963). "Deterministic Nonperiodic Flow". Journal of the Atmospheric Sciences 20 (2): 130–141.

This demonstrated the butterfly effect.

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This is not a numerical analysis paper. – timur Jul 31 '12 at 3:01

L.F. Richardson, The Approximate Arithmetical Solution by Finite Differences of Physical Problems Involving Differential Equations, with an Application to the Stresses in a Masonry Dam, Proceedings of the Royal Society of London, 1910, available here.

From the Abstract,

The object of this paper is to develop methods whereby the differential equations of physics may be applied more freely than hitherto in the approximate form of difference equations to problems concerning irregular bodies. (... ...).

Both for engineering and for many of the less exact sciences, such as biology, there is a demand for rapid methods, easy to be understood and applicable to unusual equations and irregular bodies. If they can be accurate, so much the better ; but 1 per cent, would suffice for many purposes. It is hoped that the methods put forward in this paper will help to supply this demand.

According CFD-Online, this 50 page paper is a key paper in Computational Fluid Dynamics.

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Any mention of de Boor should also include Jos Stam's work:

(Especially his siggraph 1999 course notes, which cover a number of useful approaches for dealing with b-splines.)

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It is not a classic paper, but I would add Xiu, D. and Karniadakis, G.E, "The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations," which can be found here.

I mention this paper because Generalized Polynomial Chaos is still underrepresented in many fields, though it has found much use in engineering applications. It is not precisely a magic bullet, but in some cases it can drastically reduce computational demand in uncertainty analysis. The aforementioned paper is a good summary of the method.

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