The Top 10 Computational Methods of the 20th Century
Givoli

In response to the Dongarra and Sullivan list, Dan Givoli published a list of the 20th Century numerical algorithms and methods that have most influenced the field of computational mechanics:

1. the Finite Element Method (including the Boundary Element Method);
2. Iterative Linear Algebraic Solvers, include Krylov Spaces, Conjugate Gradient Methods, and GMRES;
3. Algebraic Eigenvalue Solvers, including the Lanczos and QR methods;
4. Matrix Decomposition Methods, including spectral and polar decomposition;
5. Finite Difference Methods for Wave Problems, including the methods of Newmark, Lax-Wendroff, Hilbert-Hughes-Taylor, the shock wave techniques of Godunov, upwinding and flux-splitting;
6. Nonlinear Algebraic Solvers, including Quasi Newton methods such as BFGS, and arclength or continuation methods;
7. the Fast Fourier Transform;
8. Nonlinear Programming, in particular, Quadratic Programming
9. Soft Computing Methods, such as neural networks, genetic algorithms, and fuzzy logic.
10. Multiscale methods, including the multigrid method and wavelets.

Reference:

1. Barry Cipra,
   The Best of the 20th Century: Editors Name Top 10 Algorithms
   SIAM News,
   Volume 33, Number 4, May 2000, page 1.
2. Jack Dongarra, Francis Sullivan,
   Top Ten Algorithms of the Century,
   Computing in Science and Engineering,
   Volume 2, Number 1, January/February 2000, pages 22-23.
3. Dan Givoli,
   The Top 10 Computational Methods of the 20th Century,
   IACM Expressions,
   Number 11, September 2001, pages 5-9.