Thanks to Computing in Science and Engineering for this one. Here are the ten best algorithms:
John von Neumann, Stan Ulam, and Nick Metropolis’ Metropolis algorithm (the Monte Carlo method).
George Dantzig’s Simplex method for linear programming.
Magnus Hestenes, Eduard Stiefel, and Cornelius Lanczos’ Krylov subspace iteration methods.
Alston Householder’s Decompositional approach to matrix computations.
John Backus’ Fortran optimizing compiler.
J.G.F. Francis’ QR algorithm.
Tony Hoare’s Quicksort.
James Cooley and John Tukey’s Fast Fourier transform.
Helaman Ferguson and Rodney Forcade’s Integer relation detection algorithm.
Leslie Greengard and Vladimir Rokhlin\s Fast multipole algorithm.

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