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ANCIENT AND HONORABLE SOCIETY OF PI WATCHERS: 1984
REPORT THE INTEREST in long computations of 1T is holding up nicely. As was expected, the point of the investigations has altered somewhat. In the sixteen years since the last report, the subject of computational complexity has matured wonderfully. This subject is devoted to theoretical predictions of the minimum number of basic computer operations required to achieve a particular mathematical goal. Interest in long computations of 1T is now lodged firmly in that field.
But first, a couple more milestones.
Jean Gilloud and Martine Bouyer, 1976, CDC-7600
Kazunori Miyoshi and Kazuhika Nakayama, 1981, FACOM M-200
Y. Tamura and Y. Kanada, 1982, HITAC M-280H, under a VOS3 LCMP operating system,
CPU Time required: 2 hours, 53 minutes.
1,000,000 decimals.
2,000,038 decimals.
4,194,293 decimals.
My conjecture that the Russians would enter the 1T-Olympics seems not to have been borne out. Oh well, Marx predicted that the Revolution would start in Germany, so history always surprises us.
The computation by Miyoshi and Nakayama was performed at the University of Tsukuba using the formula of Klingenstiema
177
178 3,146 and All That
which is one of the family of arctangent fonnulas that have been traditionally used for such computations.
The computation by Tamura and Kanada breaks this tradition, making use of algorithms which revitalize the work of A. M. Legendre and K. F. Gauss in the early nineteenth century. These iterative methods, based upon the arithmetic-geometric mean, and related to the theory of elliptic integrals, are extraordinarily rapidly convergent. The iteration employed was proposed by E. Salamin and is as follows.
Let ao = 1, bo = 11\1'2.
Let, iteratively,
1 112 an = 2"(an-t + bn- t), bn = (an-tbn- t)
c~=a~-b~. n
Define 1Tn = 4a~+t/(1 - L zi+tcf). j=t
Then 1T = lim 1T nand
where agm = lim an . n-->OO
The number of correct decimals is essentially doubled at each iteration. The Schoenhage-Strassen algorithm is employed to do multiplication using a real version of the Fast Fourier Transfonn; it can perfonn n digit multiplication in a time that is roughly proportional to n log(n) loglog(n). This can be used to get an n log(n)loglog(n) algorithm for division and square roots. The running time for the algorithm is therefore at most proportional to n log\n)loglog(n).
Ancient and Honorable Pi Watchers 179
Here are some references: (1) E. Salamin, "Computation of 'iT using arithmetic-geometric
mean", Mathematics of Computation, vol. 135 (1976), pp. 565-570.
(2) D. J. Newman, "Rational Approximation versus Fast Computer Methods", in "Lectures on Approximation and Value Distribution", Presses de l'Universite de Montreal, 1982, pp. 149-174.
(3) J. M. Borwein and P. B. Borwein, "The Arithmetic-Geometric Mean and Fast Computation of Elementary Functions", Department of Mathematics, Dalhousie University, Halifax, Nova Scotia, 1983.
(4) Y. Tamura and Y. Kanada, "Calculation of 'iT to 4, 194, 293 Decimals Based on the Gauss-Legendre Algorithm."
Using CRA Y type vector operations, the FFT routines can be accelerated.
By the time this addendum is in print, the record will undoubtedly have been broken again.
So long until the next time.
Courtesy Peter B. Borwein
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2. The Code of the Primes
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3. Pompeiu's Magie Seven
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4. What Is an Abstraction?
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7. Number
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Bell, Eric Temple, The Development of Mathematics. New York: McGraw-Hill Book, 1945.
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Isaacs, G. L., Real Numbers. New York: McGraw-Hill Book, 1968.
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8. The Philadelphia Story
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9. Poinsot's Points and Lines
Coxeter, H. S. M., Regular Polytopes. London: Methuen, 1948. Pp. 93-94.
Davis, Philip J., and Rabinowitz, Philip, Numerical Integration. Waltham, Mass.: Blaisdell Publishing, 1967. P. 151. (For Weyt's Theorem and its application.)
Poinsot, Louis, and others, Abhandlungen uber die regelmiissigen Sternkorper, translated and edited by Robert Haussner. Leipzig: W. Engelmann, 1906. (For German version Poinsot's original article of 1810; no English translation is available.)
184 3.1416 and All That
10. Chaos and Polygons
Aaboe, Asger, Episodes from the Early History of Mathematics. New York: Random House, 1964.
Cundy, H. Martyn, and Rollett, A. P., Mathematical Models. London: Oxford University Press, 1951.
Klein, Felix, Famous Problems of Elementary Geometry. New York: Chelsea Publishing, 1955.
Kline, Morris, Mathematics-A Cultural Approach. Reading, Mass.: Addison-Wesley Publishing, 1962.
Olds, Charles D., Continued Fractions. New York: Random House, 1962.
11. Numbers, Point and Counterpoint
Andree, Richard V., Selections from Modern Abstract Mathematics. New York: Henry Holt, 1958.
Courant, Richard, and Robbins, Herbert, What Is Mathematics? New York: Oxford University Press, 1941.
Kline, Morris, Mathematics-A Cultural Approach. Reading, Mass.: Addison-Wesley Publishing, 1962.
Van der Waerden, B. L., Science Awakening. New York: John Wiley and Sons, 1963.
12. The Mathematical Beauty Contest
Hilbert, David, and Cohn-Vossen, Stefan, Geometry and the Imagination. New York: Chelsea Publishing, 1952. Chapter 4.
Kazarinoff, Nicholas, Geometric Inequalities. New York: Random House, New MatP.ematical Library, Vol. 4, 1961. Chapter 2.
Steinhaus, Hugo, Mathematical Snapshots. New York: Oxford University Press, 1960. Pp. 159-64.
13. The House That Geometry Built
Bing, R. H., Elementary Point Set Topology, Herbert Ellsworth Siaught Memorial Papers No.8. Menasha, Wisc.: Mathematical Association of America, 1960.
Birkhoff, Garrett, and MacLane, Saunders, A Survey of Modern Algebra, revised edition. New York: Macmillan, 1953.
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Courant, Richard, and Robbins, Herbert, What Is Mathematics? New York: Oxford University Press, 1941.
Eves, Howard, and Newsom, Carroll V., An Introduction to the Foundations and Fundamental Concepts of Mathematics. New York: Rinehart, 1958.
Meserve, Bruce E., Fundamental Concepts of Geometry. Cambridge, Mass.: Addison-Wesley Publishing, 1955.
14. Explorers of the Nth Dimension
Coxeter, H. S. M., Introduction to Geometry. New York: John Wiley and Sons, 1961. Chapter 22.
Synge, J. L., "The Geometry of Many Dimensions," Mathematical Gazette, Vol. 33 (1949), pp. 249-63.
Ulam, Stanislaw M., A Collection of Mathematical Problems. New York: Interscience Publishers, 1960. Chapter 8, "Computing Machines as a Heuristic Aid."
15. The Band-Aid Principle
Kline, Morris, Mathematics and the Physical World. New York: Thomas Y. Crowell, 1959. Chapter 25.
Kreyszig, Erwin, Differential Geometry. Toronto: University of Toronto Press, 1959.
Lyusternik, Lazar A., Shortest Paths. New York: Macmillan, 1963.
Stackel, Paul G., "Bemerkungen zur Geschichte der geodatische Linien," Sachsische Gesellschaft der Wissenschaften, Berichte, Math-Physische Klasse, Vol. 45 (1893), pp. 444-67.
16. The Spider and the Fly
Courant, Richard, and Robbins, Herbert, What Is Mathematics? New York: Oxford University Press, 1941.
Eves, Howard, and Newsom, Carroll V., An Introduction to the.Foundations and Fundamental Concepts of Mathematics. New York: Rinehart, 1958.
Freund, John E., A Modern Introduction to Mathematics. Englewood Cliffs, N.J.: Prentice-Hall, 1956.
Kline, Morris, Mathematics-A Cultural Approach. Reading, Mass.: Addison-Wesley Publishing, 1962.
186 3.1416 and All That
17. A Walk in the Neighborhood
Chinn, William G., and Steenrod, Norman E., First Concepts of Topology. New York: Random House, 1966.
Court, Nathan A., Mathematics in Fun and Earnest. New York: New American Library of World Literature, 1961.
Dantzig, Tobias, Number-The Language of Science. Garden City, N.Y.: Doubleday, 1956.
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Meserve, Bruce E., Fundamental Concepts of Geometry. Cambridge, Mass.: Addison-Wesley Publishing, 1955.
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Zippin, Leo, Uses of Infinity. New York: Random House, New Mathe-matical Library, Vol. 7, 1962.
18. Division in the Cellar
Isaacson, Eugene, and Keller, Herbert B., Analysis of Numerical Methods. New York: John Wiley and Sons, 1966.
Kuo, Shan Sun, Numerical Methods and Computers. Reading, Mass.: Addison-Wesley Publishing, 1965.
Vilenkin, N. Y., Successive Approximation. New York: Macmillan, 1964.
19. The Art of Squeezing
Chinn, William G., and Steenrod, Norman E., First Concepts of Topology. New York: Random House, 1966.
Courant, Richard, and Robbins, Herbert, What Is Mathematics? New York: Oxford University Press, 1941.
Dubisch, Roy, The Nature of Number. New York: Ronald Press, 1952.
Goodstein, Reuben L., Fundamental Concepts of Mathematics. New York: Pergamon Press, 1962.
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20. The Business of Inequalities
Hadley, George, Linear Algebra. Reading, Mass.: Addison-Wesley Publishing, 1961.
Johnston, John B., Price, G. Baley, and Van Vleck" Fred S., Linear Equations and Matrices. Reading, Mass.: Addison-Wesley Publishing, 1966.
Kemeny, John G., Snell, J. Laurie, and Thompson, Gerald L., Introduction to Finite Mathematics. Englewood Cliffs, N.J.: PrenticeHall,1955.
MacDonald, John Dennis, Strategy in Poker, Business and War. New York: W. W. Norton, 1950.
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Price, G. Baley, "Progress in Mathematics and Its Implications for the Schools," The Revolution in School Mathematics. Washington: National Council of Teachers of Mathematics, 1961.
21. The Abacus and the Slipstick
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22. Of Maps and Mathematics
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188 3.1416 and All That
23. "Mr. Milton, Mr. Bradley-Meet Andrey Andreyevicb Markov"
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24. 3.1416 and AU That
Davis, Philip J., The Lore of Large Numbers. New York: Random House, New Mathematical Library, 1961. Chapter 17.
Hobson, Ernest W., Squaring the Circle. New York: Chelsea Publishing, 1958.
Shanks, D., and Wrench, J. W., Jr., "Calculation of 'IT to 100,000 DeCImals," Mathematics of Computation, Vol. 16 (1962).
ABOUT THE AUTHORS
William G. Chinn has taught in the San Francisco schools for 37 years, including 19 years at the City College of San Francisco from which he recently retired. He has authored seven texts for the School Mathematics Study Group and three other books including First Concepts of Topology. He served as second vice-president of the Mathematics Association of America in 1981-82. He resides in San Francisco with his wife, Grace.
Philip J. Davis teaches mathematics at Brown University and is the author of many books in that field. "The Mathematical Experience", written jointly with Reuben Hersh, won an American Book Award.
