Bibliography for Computation of Pi

unabridged

 

  1. Blending two major techniques in order to compute Pi  
    M. Fernández Guasti
    International Journal of Mathematical Education in Science and Technology, 15 Jan - 15 Feb 2005, vol. 36, no. 1, pp. 85-92(8), Ingenta.  
  2. Pi at the limits of computation.
    Hwang, Chien-Lih
    Tamkang J. Math. 35 (2004), no. 4, 305--312, MathSciNet.  
  3. Calculation of Pi by mean of trigonometric functions. (Spanish)
    Bárcenas, Diómedes; Porras, Olga
    Divulg. Mat. 10 (2002), no. 2, 149--159, MathSciNet.  
  4. Ramanujan's elliptic functions to alternative bases and approximations to Pi.
    Chan, Heng Huat
    Number theory for the millennium, I (Urbana, IL, 2000), 197--213, A K Peters, Natick, MA, 2002, MathSciNet.  
  5. Extrapolation: from calculation of Pi to finite element method of partial differential equations.
    Shen, Xiaoping
    Applied mathematics reviews, Vol. 1, 537--558, World Sci. Publishing, River Edge, NJ, 2000, MathSciNet.  
  6. New formulas for approximation of Pi and other transcendental numbers.
    Kalantari, Bahman
    Computational methods from rational approximation theory (Wilrijk, 1999). Numer. Algorithms 24 (2000), no. 1-2, 59--81(23), Ingenta.  
  7. The computations of Pi. (Chinese)
    Xu, De Yi
    J. Central China Normal Univ. Natur. Sci. 34 (2000), no. 3, 376--378, MathSciNet.  
  8. Recursive reduction of series for multiple-precision evaluation and its application to pi calculation. (Japanese)
    Migita, Tsuyoshi; Amano, Akira; Asada, Naoki; Fujino, Seiji
    Trans. Inform. Process. Soc. Japan 40 (1999), no. 12, 4193--4200, MathSciNet.  
  9. Zur Irrationalität von Pi. (German)
    Stevens, J.
    Mitt. Math. Ges. Hamburg 18, 151-158, 1999.
  10. On the Rabbinical Approximation of Pi  
    Tsaban B.; Garber D.
    Historia Mathematica, February 1998, vol. 25, no. 1, pp. 75-84(10), Ingenta.  
  11. Calculation of Pi to 51.5 billion decimal digits on distributed memory parallel processors. (Japanese)
    Takahashi, Daisuke; Kanada, Yasumasa
    Trans. Inform. Process. Soc. Japan 39 (1998), no. 7, 2074--2083, MathSciNet.  
  12. A Simple Formula for Pi   
    Victor Adamchik; Stan Wagon
    The American Mathematical Monthly, Vol. 104, No. 9 (Nov., 1997), pp. 852-855, Jstor.   
  13. Many Correct Digits of Pi, Revisited   
    Gert Almkvist
    The American Mathematical Monthly, Vol. 104, No. 4 (Apr., 1997), pp. 351-353, Jstor.    
  14. Approximations of the constant Pi. (Spanish)
    Garrido, Luis
    Epsilon 13 (1997), no. 3(39), 243--251. 11-03, MathSciNet.  
  15. On Sungka approximation to Pi.
    Gupta, R. C.
    Gadnita-Bharati 19 (1997), no. 1-4, 101--106, MathSciNet.  
  16. The quest for Pi  
    D. H. Bailey, J. M. Borwein, P. B. Borwein, and S. Plouffle  
    The Mathematical Intelligencer 19 (1997), 50-57.
  17. Youqin and his calculation of Pi.
    Volkov, Alexeï Zhao
    Historia Math. 24 (1997), no. 3, 301--331, MathSciNet.  
  18. On Lambert's Proof of the Irrationality of Pi  
    M. Laczkovich
    The American Mathematical Monthly, Vol. 104, No. 5 (May, 1997), pp. 439-443, Jstor.  
  19. The beginnings of enri---the calculation of Pi by Katahiro Takebe. (Japanese)
    Ogawa, Tsukane
    Study of the history of mathematics (Japanese) (Kyoto, 1997). Surikaisekikenkyusho Kokyuroku No. 1019 (1997), 77--97, MathSciNet.  
  20. Improvement of the algorithms for Pi calculation: the Gauss-Legendre algorithm and the Borwein's quartically convergent algorithm. (Japanese)
    Takahashi, Daisuke; Kanada, Yasumasa
    Trans. Inform. Process. Soc. Japan 38 (1997), no. 11, 2406--2409, MathSciNet.  
  21. The Quest for Pi  
    David H. Bailey; Jonathan M. Borwein;     Peter B. Borwein; Simon Plouffe  
    NAS Technical Report NAS-96-015 June 1996, Moffett Field, CA
  22. Algorithms for the approximation of Pi.
    van Leijenhorst, D. C.
    Nieuw Arch. Wisk. (4) 14 (1996), no. 2, 255--274, MathSciNet.  
  23. Exploring Pi Using the Computer in Middle School Mathematics.
    Pyzdrowski, Laura; Holtan, Boyd
    School Science and Mathematics v96 n7 p378-81 Nov 1996, First Search.
  24. The computation of Pi to 10,000,000 decimal digits. (Chinese)
    Wei, Gong Yi; Yang, Zi Giang; Sun, Jia Chang; Li, Jia Kai
    J. Numer. Methods Comput. Appl. 17 (1996), no. 1, 78--81; translation in Chinese J. Numer. Math. Appl. 18 (1996), no. 3, 96--100, MathSciNet.  
  25. A Spigot Algorithm for the Digits of Pi  
    Stanley Rabinowitz; Stan Wagon
    The American Mathematical Monthly, Vol. 102, No. 3 (Mar., 1995), pp. 195-203, Jstor.   
  26. A New Formula for Picking off Pieces of Pi
    D. H. Bailey, P. B. Borwein, and S. Plouffe  
    Science News, v 148 (Oct 28, 1995), p 279.
  27. Spying Pi in the sky
    Peterson, I.
    Science News, 147 (May 20, 1995): 319.
  28. Calculation of Pi as a tool to think about the meaning of FGHC programs. (Japanese)
    Hirata, Keiji
    The theory of parallel computation and its applications (Japanese) (Kyoto, 1994). Surikaisekikenkyusho Kokyuroku No. 902 (1995), 117--132, MathSciNet.  
  29. Lazzarini's Lucky Approximation of Pi   
    Lee Badger
    Mathematics Magazine, Vol. 67, No. 2 (Apr., 1994), pp. 83-91, Jstor.   
  30. Calculation of Pi in ancient China: from Liu Hui to Zu Chongzhi.
    Volkov, Alexeï
    Historia Sci. (2) 4 (1994), no. 2, 139--157, MathSciNet.  
  31. As Easy as Pi.
    Chan, J.
    Math Horizons, pp. 18-19, Winter 1993.
  32. Class number three Ramanujan type series for 1/Pi
    Borwein, J.M.; Borwein, P.B.
    Journal of Computational and Applied Mathematics, v 46, n 1-2, Jun 14, 1993, p 281, Compendex.
  33. Rational approximations to Pi and some other numbers.
    Hata, Masayoshi
    Acta Arith. 63 (1993), no. 4, 335--349, MathSciNet.  
  34. Numerische Berechnung von Schranken für Pi. (German)
    Krämer, Walter
    [Numerical computation of bounds for Pi]
    Jahrbuch Überblicke Mathematik, 1993, 57--72, Vieweg, Braunschweig, 1993, MathSciNet.  
  35. Better approximation of Pi/ root3 by rational fractions
    Dubitskaya, A.K.
    Vestnik Moskovskogo Universiteta, Seriya 1 (Matematika Mekhanika), n 6, Nov-Dec, 1993, p 76-77 Language: Russian, Compendex.
  36. Joint approximations of Pi and logarithms of algebraic numbers
    Fel'dman, N.I.
    Vestnik Moskovskogo Universiteta, Seriya 1 (Matematika Mekhanika), n 6, Nov-Dec, 1993, p 47-50 Language: Russian, Compendex.
  37. Mountains of Pi.
    Preston, R.
    New Yorker 68, 36-67, Mar. 2, 1992.
  38. Improving an approximation for Pi  
    Shanks, Daniel
    Amer. Math. Monthly 99 (1992), no. 3, 263, MathSciNet.  
  39. Wallis formula for Pi, including two new expressions involving irrationals
    Mavromatis, Harry A.  
    International Journal of Computer Mathematics, v 43, n 3-4, 1992, p 197-203, Compendex.
  40. A complement to: "On the approximation of Pi by some particular algebraic numbers" (French)  
    Diaz, Guy
    Compositio Math. 79 (1991), no. 3, 255--270, MathSciNet.  
  41. The Discovery of the Series Formula for Pi by Leibniz, Gregory and Nilakantha  
    Ranjan Roy
    Mathematics Magazine, Vol. 63, No. 5 (Dec., 1990), pp. 291-306, Jstor.   
  42. N-dimensional harmonic oscillator yields monotonic series for the mathematical constant p
    Lynch, R.; Mavromatis, H.A.
    Journal of Computational and Applied Mathematics, v 30, n 2, May 28, 1990, p 127-137, Compendex.
  43. Sur l'approximation de Pi par des nombres algébriques particuliers. (French)
    [On the approximation of Pi by some particular algebraic numbers]
    Diaz, Guy
    Compositio Math. 74 (1990), no. 3, 285--298, MathSciNet.  
  44. Ramanujan, Modular Equations, and Approximations to Pi or How to Compute One Billion Digits of Pi  
    J. M. Borwein; P. B. Borwein; D. H. Bailey
    The American Mathematical Monthly, Vol. 96, No. 3 (Mar., 1989), pp. 201-219, Jstor.  
  45. An asymptotic formula for Pi.
    Kiss, Péter; Mátyás, Ferenc
    J. Number Theory 31 (1989), no. 3, 255--259, MathSciNet.  
  46. The Computation of p to 29,360,000 Decimal Digits Using Borweins' Quartically Convergent Algorithm   
    David H. Bailey
    Mathematics of Computation, Vol. 50, No. 181 (Jan., 1988), pp. 283-296, Jstor.  
  47. The Ubiquitous Pi  
    Dario Castellanos
    Mathematics Magazine, Vol. 61, No. 3 (Jun., 1988), pp. 148-163, Jstor.  
  48. The Ubiquitous Pi  
    Dario Castellanos
    Mathematics Magazine, Vol. 61, No. 2 (Apr., 1988), pp. 67-98, Jstor.  
  49. Estimating Pi by Microcomputer.
    Donahue, Richard J.
    Mathematics Teacher v81 n3 p203-06,226 Mar 1988, First Search.
  50. Recent computations of Pi.
    Hurley, Donal
    Irish Math. Soc. Bull. No. 21 (1988), 38--44, MathSciNet.  
  51. Algorithm of the Bi-Month: Computing Pi  
    Harley Flanders
    The College Mathematics Journal, Vol. 18, No. 3 (May, 1987), pp. 230-235, Jstor.   
  52. An Elementary Proof of Pi
    Boo Rim Choe
    The American Mathematical Monthly, Vol. 94, No. 7 (Aug., 1987), pp. 662-663, Jstor.   
  53. Ramanujan's Rational and Algebraic Series for 1/Pi.
    Borwein, J. M. and Borwein, P. B.
    Indian J. Math. 51, 147-160, 1987.
  54. Three Familiar Formulas for Pi via Geometry  
    Norman Schaumberger
    The College Mathematics Journal, Vol. 17, No. 4 (Sep., 1986), p. 339, Jstor.   
  55. A New Formula for Pi  
    Yuri V. Matiyasevich; Richard K. Guy
    The American Mathematical Monthly, Vol. 93, No. 8 (Oct., 1986), pp. 631-635, Jstor.   
  56. More Quadratically Converging Algorithms for Pi   
    J. M. Borwein; P. B. Borwein
    Mathematics of Computation, Vol. 46, No. 173 (Jan., 1986), pp. 247-253, Jstor.   
  57. Some Modular Identities of Ramanujan useful in Approximating Pi  
    Jon Borwein
    Proceedings of the American Mathematical Society, Vol. 95, No. 3 (Nov., 1985), pp. 365-371, Jstor.   
  58. The legal values of Pi  
    David Singmaster  
    The Mathematical Intelligencer, Vol. 7, No. 2, 1985.
  59. A remarkable approximation to Pi.
    Stern, M. D.
    Math. Gaz. 69 (1985), no. 449, 218--219, MathSciNet.  
  60. Is Pi normal?
    Stan Wagon
    The Mathematical Intelligencer, Vol. 7, No. 3, 1985.
  61. Mathematical-educational aspects of the computation of Pi  
    Shlomo Breuer and Gideon Zwas
    Int. J. Math. Educ. Sci. Technol., Vol. 15, No. 2, 1984, pp. 231-244.
  62. An Infinite Series for Pi with Determinants
    Nobuo Adachi
    Mathematics Magazine, Vol. 57, No. 4 (Sep., 1984), pp. 215-216, Jstor.  
  63. On a Sequence Arising in Series for Pi   
    Morris Newman; Daniel Shanks
    Mathematics of Computation, Vol. 42, No. 165 (Jan., 1984), pp. 199-217, Jstor.   
  64. The arithmetic-geometric mean and fast computation of elementary functions  
    J. M. Borwein and P.B. Borwein  
    SIAM Review, Vol. 26, 1984, pp. 351-366.
  65. On a sequence of normal approximations to Pi/4 and the Brouwer conjecture.
    Stoneham, R. G.
    Acta Arith. 42 (1983), no. 3, 265--279, MathSciNet.  
  66. Dihedral quartic approximations and series for Pi.
    Shanks, Daniel
    J. Number Theory 14 (1982), no. 3, 397--423, MathSciNet.  
  67. Rational approximations to Pi.
    Neild, D. A.
    New Zealand Math. Mag. 18 (1981/82), no. 3-4, 99--100, MathSciNet.  
  68. Are p, e, and v2 Equally Difficult to Compute?  
    L. Baxter
    The American Mathematical Monthly, Vol. 88, No. 1 (Jan., 1981), pp. 50-51, Jstor.  
  69. An Irrational Calculation of Pi.
    Lowry, Pat G.
    Creative Computing v7 n12 p238-39 Dec 1981, First Search.
  70. John Ward's method for the calculation of Pi.
    Cohen, G. L.; Shannon, A. G.
    Historia Math. 8 (1981), no. 2, 133--144, MathSciNet.  
  71. On the approximation of Pi by special algebraic numbers.
    Braune, Erhard
    Glasgow Math. J. 21 (1980), no. 1, 51--56, MathSciNet.  
  72. An Algorithm for the Calculation of Pi   
    George Miel
    The American Mathematical Monthly, Vol. 86, No. 8 (Oct., 1979), pp. 694-697, Jstor.   
  73. Using A Minicalculator to Find An Approximate Value for Pi
    Bolduc, E. J.
    School Science and Mathematics 77, 8, 689-91, Dec 77, First Search.
  74. Table of Simple Continued Fraction for Pi and the Derived Decimal Approximation.
    Gosper, R. W.
    Stanford, CA: Artificial Intelligence Laboratory, Stanford University, Oct. 1975. Reviewed in Math. Comput. 31, 1044, 1977.
  75. Gauss' formula for Pi. (Dutch)
    Buissant des Amorie, E. C.
    Nieuw Tijdschr. Wisk. 64 (1976/77), no. 4, 200--204, MathSciNet.  
  76. Computation of Pi Using Arithmetic-Geometric Mean   
    Eugene Salamin
    Mathematics of Computation, Vol. 30, No. 135 (Jul., 1976), pp. 565-570, Jstor.  
  77. A Simple Proof of the Formula for Pi  
    Ioannis Papadimitriou
    The American Mathematical Monthly, Vol. 80, No. 4 (Apr., 1973), pp. 424-425, Jstor.   
  78. A Method for Approximating the Value of Pi With a Computer Application
    Einhorn, Erwin
    Mathematics Teacher 66, 5, 427-430, May 73, First Search.  
  79. Series approximations to powers of Pi.
    Henning, H. B.
    Portugal. Math. 32 (1973), 171--177, MathSciNet.  
  80. Still Another Elementary Proof That Pi
    Daniel P. Giesy
    Mathematics Magazine, Vol. 45, No. 3 (May, 1972), pp. 148-149, Jstor.    
  81. On Huygens' Approximation to Pi   
    T. S. Nanjundiah
    Mathematics Magazine, Vol. 44, No. 4 (Sep., 1971), pp. 221-223, Jstor.   
  82. Rational Approximations to Pi   
    K. Y. Choong; D. E. Daykin; C. R. Rathbone
    Mathematics of Computation, Vol. 25, No. 114 (Apr., 1971), pp. 387-392, Jstor.   
  83. A Further Note on Machine Computation for Pi
    Moakes, A. J.
    Mathematical Gazette 55, 393, 306-310, Jun 71, First Search.
  84. The Calculation of Pi
    Moakes, A. J.
    Mathematical Gazette 54, 389, 261-264, Oct '70, First Search.
  85. Leibniz' formula for Pi deduced by a "mapping" of the circular disc.
    Brun, Viggo
    Nordisk Mat. Tidskr. 18 1970 73--81, 120, MathSciNet.  
  86. Another Proof of the Formula for Pi   
    E. L. Stark
    The American Mathematical Monthly, Vol. 76, No. 5 (May, 1969), pp. 552-553, Jstor.   
  87. Calculation of p to 100,000 Decimals   
    Daniel Shanks; John W. Wrench, Jr.
    Mathematics of Computation, Vol. 16, No. 77 (Jan., 1962), pp. 76-99, Jstor.    
  88. A Statistical Study of Randomness Among the First 10,000 Digits of Pi  
    R. K. Pathria
    Mathematics of Computation, Vol. 16, No. 78 (Apr., 1962), pp. 188-197, Jstor.    
  89. Calculation of Pi to 100,000 decimals.
    Shanks, Daniel; Wrench, John W., Jr.
    Math. Comp. 16 1962 76--99, MathSciNet.  
  90. An Elementary Proof of the Formula for Pi
    Yoshio Matsuoka
    The American Mathematical Monthly, Vol. 68, No. 5 (May, 1961), pp. 485-487, Jstor.   
  91. On Wallis' Formula  
    John Gurland
    The American Mathematical Monthly, Vol. 63, No. 9 (Nov., 1956), pp. 643-645, Jstor.  
  92. Some comments on a NORC computation of Pi.
    Nicholson, S. C.; Jeenel, J.
    Math. Tables Aids Comput. 9 (1955), 162--164, MathSciNet.  
  93. A Proof of the Irrationality of Pi
    Robert Breusch
    The American Mathematical Monthly, Vol. 61, No. 9 (Nov., 1954), pp. 631-632, Jstor.   
  94. On the approximation of Pi.
    Mahler, K.
    Nederl. Akad. Wetensch. Proc. Ser. A. 56=Indagationes Math. 15, (1953). 30--42, MathSciNet.  
  95. An elementary derivation of the formulas of Wallis, Leibnitz and Euler for the number Pi. (Russian)
    Yaglom, A. M.; Yaglom, I. M.
    Uspehi Matem. Nauk (N.S.) 8, (1953). no. 5(57), 181--187, MathSciNet.  
  96. A Hindu approximation to Pi.
    Rajagopal, C. T.; Vedamurti Aiyar, T. V.
    Scripta Math. 18, (1952). 25--30, MathSciNet.  
  97. Approximations exceeding 1300 decimals for sqrt 3, 1/sqrt 3, Pi/3 and distribution of digits in them.
    Uhler, Horace S.
    Proc. Nat. Acad. Sci. U. S. A. 37, (1951), MathSciNet.  
  98. Statistical Treatment of Values of First 2,000 Decimal Digits of e and of Pi Calculated on the ENIAC
    Olga Taussky
    Mathematical Tables and Other Aids to Computation, Vol. 4, No. 30 (Apr., 1950), pp. 109-112, Jstor.    
  99. A New Approximation to Pi   
    Mathematical Tables and Other Aids to Computation, Vol. 2, No. 18 (Apr., 1947), pp. 245-248, Jstor.   
  100. A Simple Proof that  is Irrational.
    Niven, I.
    Bull. Amer. Math. Soc. 53, 509, 1947.
  101. Some approximations for Pi. (Norwegian)
    Selmer, Ernst S.
    Norsk Mat. Tidsskr. 29, (1947). 9--20, MathSciNet.  
  102. On the approximation to Pi/4 in Egyptian geometry. (Dutch)
    Bruins, E. M.
    Nederl. Akad. Wetensch., Proc. 48, (1945) 206--210 = Indagationes Math. 7, 11--15 (1945), MathSciNet.  
  103. The Three Infinite Harmonic Series and their Sums (with Topical Reference to the Newton and Leibniz Series for Pi)  
    F. Soddy
    Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 182, No. 991 (Jun., 1944), p. 416, Jstor.   
  104. The Three Infinite Harmonic Series and their Sums (with Topical Reference to the Newton and Leibniz Series for Pi)  
    F. Soddy
    Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 182, No. 989 (Dec., 1943), pp. 113-129, Jstor.   
  105. Formula for Computing Pi to a Thousand Places  
    J. P. Ballantine
    The American Mathematical Monthly, Vol. 46, No. 8 (Oct., 1939), pp. 499-501, Jstor.   
  106. A Simple Approximation for Pi   
    M. G. Gaba
    The American Mathematical Monthly, Vol. 45, No. 6 (Jun., 1938), pp. 373-375, Jstor.   
  107. On Arccotangent Relations for Pi  
    D. H. Lehmer
    The American Mathematical Monthly, Vol. 45, No. 10 (Dec., 1938), pp. 657-664, Jstor.  
  108. A Series Useful in the Computation of Pi  
    J. S. Frame
    The American Mathematical Monthly, Vol. 42, No. 8 (Oct., 1935), pp. 499-501, Jstor.  
  109. A Sixteenth Century Chinese Approximation for Pi  
    J. M. Barbour
    The American Mathematical Monthly, Vol. 40, No. 2 (Feb., 1933), pp. 69-73, Jstor.   
  110. Two New Arctangent Relations for Pi   
    A. A. Bennett
    The American Mathematical Monthly, Vol. 32, No. 5 (May, 1925), pp. 253-255, Jstor.  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2006