20世纪十大算法

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1.1946: John von Neumann, Stan Ulam, and Nick Metropolis, all at the Los Alamos Scientific Laboratory, cook up the Metropolisalgorithm, also known as the Monte Carlo method.The Metropolis algorithm aims to obtain approximate solutions to numerical problems with unmanageably many degrees of freedomand to combinatorial problems of factorial size, by mimicking a random process. Given the digital computer’s reputation fordeterministic calculation, it’s fitting that one of its earliest applications was the generation of random numbers.

2.1947: George Dantzig, at the RAND Corporation, creates the simplex method for linear programming.In terms of widespread application, Dantzig’s algorithm is one of the most successful of all time: Linearprogramming dominates the world of industry, where economic survival depends on the ability to optimizewithin budgetary and other constraints. (Of course, the “real” problems of industry are often nonlinear; the useof linear programming is sometimes dictated by the computational budget.) The simplex method is an elegantway of arriving at optimal answers. Although theoretically susceptible to exponential delays, the algorithmin practice is highly efficient—which in itself says something interesting about the nature of computation.In terms of widespread use, George Dantzig’s simplex method is among the most successful algorithms of all time.

3.1950: Magnus Hestenes, Eduard Stiefel, and Cornelius Lanczos, all from the Institute for Numerical Analysisat the National Bureau of Standards, initiate the development of Krylov subspace iteration methods.These algorithms address the seemingly simple task of solving equations of the form Ax = b. The catch,of course, is that A is a huge n x n matrix, so that the algebraic answer x = b/A is not so easy to compute.(Indeed, matrix “division” is not a particularly useful concept.) Iterative methods—such as solving equations ofthe form Kxi + 1 = Kxi + b – Axi with a simpler matrix K that’s ideally “close” to A—lead to the study of Krylov subspaces. Named for the Russian mathematician Nikolai Krylov, Krylov subspaces are spanned by powers of a matrix applied to an initial“remainder” vector r0 = b – Ax0. Lanczos found a nifty way to generate an orthogonal basis for such a subspace when the matrix is symmetric. Hestenes and Stiefel proposed an even niftier method, known as the conjugate gradient method, for systems that are both symmetric and positive definite. Over the last 50 years, numerous researchers have improved and extended these algorithms.The current suite includes techniques for non-symmetric systems, with acronyms like GMRES and Bi-CGSTAB. (GMRES andBi-CGSTAB premiered in SIAM Journal on Scientific and Statistical Computing, in 1986 and 1992,respectively.)

4.1951: Alston Householder of Oak Ridge National Laboratory formalizes the decompositional approachto matrix computations.The ability to factor matrices into triangular, diagonal, orthogonal, and other special forms has turnedout to be extremely useful. The decompositional approach has enabled software developers to produceflexible and efficient matrix packages. It also facilitates the analysis of rounding errors, one of the bigbugbears of numerical linear algebra. (In 1961, James Wilkinson of the National Physical Laboratory inLondon published a seminal paper in the Journal of the ACM, titled “Error Analysis of Direct Methodsof Matrix Inversion,” based on the LU decomposition of a matrix as a product of lower and uppertriangular factors.)

5.1957: John Backus leads a team at IBM in developing the Fortran optimizing compiler.The creation of Fortran may rank as the single most important event in the history of computer programming: Finally, scientists (and others) could tell the computer what they wanted it to do, without having to descend into the netherworld of machine code.Although modest by modern compiler standards—Fortran I consisted of a mere 23,500 assembly-language instructions—the earlycompiler was nonetheless capable of surprisingly sophisticated computations. As Backus himself recalls in a recent history ofFortran I, II, and III, published in 1998 in the IEEE Annals of the History of Computing, the compiler “produced code of such efficiency that its output would startle the programmers who studied it.”

6.1959–61: J.G.F. Francis of Ferranti Ltd., London, finds a stable method for computing eigenvalues, known as the QR algorithm.Eigenvalues are arguably the most important numbers associated with matrices—and they can be the trickiest to compute. It’srelatively easy to transform a square matrix into a matrix that’s “almost” upper triangular, meaning one with a single extra set of nonzero entries just below the main diagonal. But chipping away those final nonzeros, without launching an avalanche of error, is nontrivial. The QR algorithm is just the ticket. Based on the QR decomposition, which writes A as the product of an orthogonal matrix Q and an upper triangular matrix R, this approach iteratively changes Ai = QR into Ai + 1 = RQ, with a few bells and whistles for accelerating convergence to upper triangular form. By the mid-1960s, the QR algorithm had turned once-formidable eigenvalue problems into routine calculations.

7.1962: Tony Hoare of Elliott Brothers, Ltd., London, presents Quicksort.Putting N things in numerical or alphabetical order is mind-numbingly mundane. The intellectual challenge lies in devising ways of doing so quickly. Hoare’s algorithm uses the age-old recursive strategy of divide and conquer to solve the problem: Pick one element as a “pivot,” separate the rest into piles of “big” and “small” elements (as compared with the pivot), and then repeat this procedure on each pile. Although it’s possible to get stuck doing all N(N – 1)/2 comparisons (especially if you use as your pivot the first item on a list that’s already sorted!), Quicksort runs on average with O(N log N) efficiency. Its elegant simplicity has made Quicksort the pos-terchild of computational complexity.

8.1965: James Cooley of the IBM T.J. Watson Research Center and John Tukey of Princeton University and AT&T Bell Laboratories unveil the fast Fourier transform. Easily the most far-reaching algo-rithm in applied mathematics, the FFT revolutionized signal processing. The underlying idea goes back to Gauss (who needed to calculate orbits of asteroids), but it was the Cooley–Tukey paper that made it clear how easily Fourier transforms can be computed. Like Quicksort, the FFT relies on a divide-and-conquer strategy to reduce an ostensibly O(N2) chore to an O(N log N) frolic. But unlike Quick- sort, the implementation is (at first sight) nonintuitive and less than straightforward. This in itself gave computer science an impetus to investigate the inherent complexity of computational problems and algorithms.

9.1977: Helaman Ferguson and Rodney Forcade of Brigham Young University advance an integer relation detection algorithm.The problem is an old one: Given a bunch of real numbers, say x1, x2, . . . , xn, are there integers a1, a2, . . . , an (not all 0) for which a1x1 + a2x2 + . . . + anxn = 0? For n = 2, the venerable Euclidean algorithm does the job, computing terms in the continued-fraction expansion of x1/x2. If x1/x2 is rational, the expansion terminates and, with proper unraveling, gives the “smallest” integers a1 and a2.If the Euclidean algorithm doesn’t terminate—or if you simply get tired of computing it—then the unraveling procedure at least provides lower bounds on the size of the smallest integer relation. Ferguson and Forcade’s generalization, although much more difficult to implement (and to understand), is also more powerful. Their detection algorithm, for example, has been used to find the precise coefficients of the polynomials satisfied by the third and fourth bifurcation points, B3 = 3.544090 and B4 = 3.564407, of the logistic map. (The latter polynomial is of degree 120; its largest coefficient is 25730.) It has also proved useful in simplifying calculations with Feynman diagrams in quantum field theory.

10.1987: Leslie Greengard and Vladimir Rokhlin of Yale University invent the fast multipole algorithm.This algorithm overcomes one of the biggest headaches of N-body simulations: the fact that accurate calculations of the motions of N particles interacting via gravitational or electrostatic forces (think stars in a galaxy, or atoms in a protein) would seem to require O(N2) computations—one for each pair of particles. The fast multipole algorithm gets by with O(N) computations. It does so by using multipole expansions (net charge or mass, dipole moment, quadrupole, and so forth) to approximate the effects of a distant group of particles on a local group. A hierarchical decomposition of space is used to define ever-larger groups as distances increase.One of the distinct advantages of the fast multipole algorithm is that it comes equipped with rigorous error estimates, a feature that many methods lack.

What new insights and algorithms will the 21st century bring? The complete answer obviously won’t be known for anotherhundred years. One thing seems certain, however. As Sullivan writes in the introduction to the top-10 list, “The new century is not going to be very restful for us, but it is not going to be dull either!”

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