from SIAM News, Volume 33, Number 4

The Best of the 20th Century: Editors Name Top 10 Algorithms

By Barry A. Cipra

Algos is the Greek word for pain. Algor is Latin, to be cold. Neither is the root for algorithm, which stems instead from al-Khwarizmi, the name of the ninth-century Arab scholar whose book al-jabr wa’l muqabalah devolved into today’s high schoolalgebra textbooks. Al-Khwarizmi stressed the importance of methodical procedures for solving problems. Were he around today,he’d no doubt be impressed by the advances in his eponymous approach.

Some of the very best algorithms of the computer age are highlighted in the January/February 2000 issue of Computing in Science& Engineering, a joint publication of the American Institute of Physics and the IEEE Computer Society. Guest editors Jack Don-garra of theUniversity of Tennessee and Oak Ridge National Laboratory and Fran-cis Sullivan of the Center for Comput-ing Sciences at the Institute forDefense Analyses put togeth-er a list they call the “Top Ten Algorithms of the Century.”

“We tried to assemble the 10 al-gorithms with the greatest influence on the development and practice of science and engineeringin the 20th century,” Dongarra and Sullivan write. As with any top-10 list, their selections—and non-selections—are bound to becontroversial, they acknowledge. When it comes to picking the algorithmic best, there seems to be no best algorithm.

Without further ado, here’s the CiSE top-10 list, in chronological order. (Dates and names associated with the algorithms should be readas first-order approximations. Most algorithms take shape over time, with many contributors.)

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.

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 algorithm

in practice is highly efficient—which in itself says something interesting about the nature of computation.1950: Magnus Hestenes, Eduard Stiefel, and Cornelius Lanczos, all from the Institute for Numerical Analysis

In terms of wide-spread use, GeorgeDantzig’s simplexmethod is among themost successful al-gorithms of all time.

at 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 ́ 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 of+ b – Axi with a simpler matrix K that’ s ideally “close” to A—lead to the study of Krylov subspaces. Namedfor 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 matrixis symmetric. Hestenes and Stiefel proposed an even niftier method, known as the conjugate gradient method, for systems that areboth 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 and

the form Kxi + 1 = Kxi

Bi-CGSTAB premiered in SIAM Journal on Scientific and Statistical Computing, in 1986 and 1992,respectively.)

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.)

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

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Alston Householder

(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 suchefficiency that its output would startle the programmers who studied it.”

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 ofnonzero 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 orthogonalmatrix Q and an upper triangular matrix R, this approach iteratively changes Ai = QR into Ai + 1 = RQ, with a few bells and whistlesfor accelerating convergence to upper triangular form. By the mid-1960s, the QR algorithm had turned once-formidable eigenvalue

problems into routine calculations.

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 oneelement as a “pivot,” separate the rest into piles of “big” and “small” elements (as compared with the pivot), and then repeat thisprocedureoneachpile.Althoughit’spossibletogetstuckdoingallN(N – 1)/2comparisons(especiallyifyouuseasyourpivotthefirstitem on a list that’s already sorted!), Quicksort runs on average with O(N log N) efficiency. Its elegant simplicity has made Quicksortthe pos-terchild of computational complexity.

James Cooley

1965: James Cooley of the IBM T.J. Watson Research Center and John Tukey of PrincetonUniversity and AT&T Bell Laboratories unveil the fast Fourier transform.

Easily the most far-reaching algo-rithm in applied mathematics, the FFT revolutionizedsignal processing. The underlying idea goes back to Gauss (who needed to calculate orbitsof asteroids), but it was the Cooley–Tukey paper that made it clear how easily Fouriertransforms can be computed. Like Quicksort, the FFT relies on a divide-and-conquerstrategy to reduce an ostensibly O(N 2) 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 itselfgave computer science an impetus to investigate the inherent complexity of computationalproblems and algorithms.

John Tukey

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 = 3.564407,30

ofthelogisticmap.(Thelatterpolynomialisofdegree120;itslargestcoefficientis257 .)Ithasalsoprovedusefulinsimplifyingcalculations with Feynman diagrams in quantum field theory.

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 require2

O(N ) computations—one for each pair of particles. The fast multipole algorithm gets by with O(N) computations. It does so byusing multipole expansions (net charge or mass, dipole moment, quadrupole, and so forth) to approximate the effects of a distantgroup 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 thatmany 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 notgoing to be very restful for us, but it is not going to be dull either!”

Barry A. Cipra is a mathematician and writer based in Northfield, Minnesota.

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