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Hronologija Izračunavanja Π Prije 1400

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Hronologija izračunavanja π Prije 1400 Decimalna mjesta Datum Ko Formulacija Vrijednost π (svijetski rekord podebljan) 2000? 2 Stari Egipćani 3.16045... 1 BCЕ 4 × (8 / 9) 2000? Babilonci 3 + 1 / 8 3.125 1 BCЕ 1200? Kina 3 0 BCЕ 550? Biblija "...rastopljeno more, deset lakata od jednog kraja do 3 0 BCЕ (1 Kraljevi 7:23) drugog: bilo je sveobuhvatno, ... linija od trideset lakata ga je obuhvatila u cijelosti..." Prvi Helen koji se 434 Anaksagora nije dao nikakvo bavio kvadraturom Kompas i lenjir 0 BCE rešenje kruga je Anaksagora Sulbasutras, 350? (6/(2 + 2 )) 3.088311 … 0 BCЕ indijski sveti tekst 2 √ c. 250 223 / 71 < π < 22 / 7 3.140845...<π< 3.142857... 2 BCE Arhimed 15 25 / 8 3.125 1 BCE Marko Vitruvije 5 Liu Xin Tačna metoda je nepoznata 3.1457 2 130 Zhang Heng 10 = 3.162277…. 3.146551... 2 730/232 √ 150 Ptolomej 377 / 120 3.141666... 3 250 Wang Fan 142 / 45 3.155555... 1 3.141024 < π < 3.142074 263 Liu Hui 3927 / 1250 3.14159 5 400 He Chengtia 111035 / 35329 3.142885... 2 3.1415926 < π < 3.1415927 Zuov 480 Zu Chongzhi odnos 355 / 113 3.1415926 7 499 Aryabhata 62832 / 20000 3.1416 3 640 Brahmagupta 10 3.162277... 1 800 Al Khwarizmi 3.1416 3 √ 1150 Bhāskara II 3927/1250 i 754/240 3.1416 3 1220 Fibonači 3.141818 3 1320 Zhao Youqin 3.1415926 7 Od 1400 pa nadalje Decimalna mjesta (svijetski Datum Ko Bilješka rekord podebljan) Svizapisiod1400.godinenavodetačanbrojdecimalnihmesta. Madhava Vjerovatno je otkrivena pomoću beskonačnog reda π. Danas 1400 10 Sangamagrama poznata kao Lajbnicova formula za π. Jamshīd al- 1424 16 Kāshī 1573 Valentinus Otho 355/113 6 1579 Fransoa Vijet 9 Adriaan van 1593 15 Roomen 1596 Ludolph van 20 1615 Ceulen 32 Willebrord Snell 1621 Učenik Van Ceulena 35 (Snellius) Christoph 1630 38 Grienberger 1665 Isaac Newton 16 Takakazu 11 1681 Seki 16 Abraham 1699 Izračunaoπ do 72 cifre ali nisusvebile tačne. 71 Sharp 1706 JohnMachin 100 1706 William Jones Uveo Grčko slovo 'π' Thomas Fantet 1719 Izračunao do 127 decimalnih mjesta ali nisu sve bila tačna. 112 de Lagny Toshikiyo 1722 24 Kamata 1722 Katahiro Takebe 41 Yoshisuke 1739 51 Matsunaga Koristio Grčko slovo 'π' u svojoj knjizi Uvod u Beskonačnu analizu i time 1748 Leonard Ojler značajno doprinjeo njenoj popularnosti. JohanHajnrih 1761 Dokazaodajeπiracionalan broj. Lambert 1775 Ojler Istakaomogućnost da bi πmogao biti transcendantan broj. 1789 Jurij Vega Izračunao do 143 d ecimalnih mjestaa li nisusve bila tačna. 126 1794 Jurij Vega Izračunao do 140 d ecimalnih mjestaa li nisusve bila tačna. 136 Adrien-Marie Pokazao da je π² (time iπ) iracionalan, i napomenuo da bi π mogao biti 1794 Legendre transcendantan broj. Kasni 18 Anonimni Dokument se pojavio u Radcliffe biblioteci u Oksfodu, Engleskoj. Otkrio 152 vijek spisi ga je F. X. von Zach, napisane su bile 154 cifre broja π, 152 tačno. 1824 Vilijam Izračunao do 240 decimalnih mjesta ali nisu sve bila tačna. 152 Rutherford Zacharias Dase 1844 i Izračunao do 205 decimalnih mjesta ali nisu sve bila tačna. 200 Strassnitzky 1847 Thomas Izračunao do 250 decimalnih mjesta ali nisu sve bila tačna. 248 Clausen 1853 Lehmann 261 1853 Rutherford 440 Vilijam Trebalo muje 15 godina da izračuna 707 decimalnih mjesta, ali nisusva 1874 527 Šanks bila tačna(grečkuje našao D. F. Ferguson 1946 godine) 1882 Ferdinand von Dokazao da je π transcendantan broj(Lindeman-Vajerštrasova teorema) Lindemann Bila je blizu da zakonom reguliše vrijednost 3.2 za π. Parlamentarni Prijedlog Američka država 1897 broj 246 je prošao jednoglasno. Zakon je ostao u sentu zbog lobiranja 1 Indijana pojedinih izdavača knjiga. Pronađeno je nekoliko brzih konvergentnih beskonačnih serija π, koji mogu Srinivasa izračunati 8 decimalnihmesta πsasvakimizrazomuseriji. Od 1980-ih, 1910 Ramanudžan njegovaserijapostalajeosnovazanajbržealgoritmekojetrenutnokoriste Iasumasa Kanada i braća Čudnovski za izračunavanje π. 1946 D. F. Ferguson Ručni kalkulator 620 1947 Ivan Niven Dao veoma jednostavan dokaz da je π iracionalan. Januar D. F. Ferguson Ručni kalkulator 710 1947 Septembar D. F. Ferguson Ručni kalkulator 808 1947 D. F. Ferguson 1949 i John Ručni kalkulator 1,120 Wrench Doba kompjutera (od 1949 pa nadalje) Decimalna mjesta (svijetski Vrijeme Datum Ko Implementacija rekord podebljan) Svi rezultati od 1949 nadalje su izra;unati pomoću elektronskog kompjutera John Wrench i Bili suprvikojisukoristilielektronski kompjuter ENIAC da 70 1949 2,037 L. R. Smith izračunaju π sati 1953 Kurt Mahler Pokazao da π nije Liouville broj S. C. Nicholson i J. 13 1954 Koristili su NORC 3,093 Jeenel minuta Ferranti Pegasus kompjuter (London), izračunao 10.021 1957 7,480 George E. Felton cifara, ali nisusve bile tačne. Januar 1.7 Francois Genuys IBM 704 10,000 1958 sati 33 Maj 1958 George E. Felton Pegasus kompjuter (London) 10,021 hours 4.3 1959 Francois Genuys IBM 704 (Pariz) 16,167 sati Daniel Shanks i 8.7 1961 IBM 7090 (Njujork) 100,265 John Wrench sati 39 1961 J.M. Gerard IBM 7090 (London) 20,000 minuta Jean Guilloud i 28 1966 IBM 7030 (Pariz) 250,000 J. Filliatre sati Jean Guilloud i 28 1967 CDC 6600 (Pariz) 500,000 M. Dichampt sati Jean Guilloud i 23.3 1973 CDC 7600 1,001,250 Martin Bouyer sati Kazunori Miyoshi 1981 i Yasumasa FACOM M-200 2,000,036 Kanada 1981 Jean Guilloud Nepoznata 2,000,050 1982 Yoshiaki Tamura MELCOM 900II 2,097,144 Yoshiaki Tamura 2.9 1982 i Yasumasa HITAC M-280H 4,194,288 sati Kanada Yoshiaki Tamura 1982 i Yasumasa HITAC M-280H 8,388,576 Kanada Yasumasa Kanada,Sayaka HITAC M-280H 16,777,206 1983 Yoshinoi Yoshiaki Tamura Yasunori Ushiro Oktobar i Yasumasa 1983 HITAC S-810/20 10,013,395 Kanada Oktobar Bill Gosper Symbolics 3670 17,526,200 1985 Januar David H. Bailey CRAY-2 29,360,111 1986 Septembar Yasumasa Kanada HITAC S-810/20 33,554,414 1986 i Yoshiaki Tamura Oktobar Yasumasa HITAC S-810/20 67,108,839 1986 Kanada i Yoshiaki Tamura Yasumasa Kanada, Januar Yoshiaki Tamura, NEC SX-2 134,214,700 1987 Yoshinobu Kubo i drugi Januar Yasumasa HITAC S-820/80 201,326,551 1988 Kanada i Yoshiaki Tamura Gregory V. Chudnovsky i Maj 1989 CRAY-2 i IBM 3090/VF 480,000,000 David V. Chudnovsky Gregory V. Jun 1989 Chudnovsky i IBM 3090 535,339,270 David V. Chudnovsky Yasumasa Jul 1989 Kanada i HITAC S-820/80 536,870,898 Yoshiaki Tamura Gregory V. Avgust Chudnovsky i IBM 3090 1,011,196,691 1989 David V. Chudnovsky 19 Yasumasa Novembar Kanada i HITAC S-820/80 1,073,740,799 1989 Yoshiaki Tamura Gregory V. Avgust Paralelni kompjuter kućne radinosti (detalji nepoznati, nisu Chudnovsky i 2,260,000,000 1991 David V. verifikovani) Chudnovsky Gregory V. 18Maj Novi paralelni kompjuter kućne radinosti (detalji nepoznati, Chudnovsky i 4,044,000,000 1994 David V. nisu verifikovani) Chudnovsky Yasumasa Kanada 26 Jun i Daisuke HITAC S-3800/480(dualni CPU) 3,221,220,000 1995 Takahashi Pronašao formulu koja omogućava izračunavanje n-te 1995 Simon Plouffe heksadecimalne cifre π bez izračunavanja prethodnih cifara. Yasumasa Kanada 28 Avgust i Daisuke HITAC S-3800/480(dualni CPU) 4,294,960,000 1995 Takahashi 11 Yasumasa Kanada Oktobar i Daisuke HITAC S-3800/480(dualni CPU) 6,442,450,000 1995 Takahashi 6Jul Yasumasa Kanada i Daisuke 51,539,600,000 1997 HITACHI SR2201(1024 CPU) Takahashi Yasumasa Kanada 5April i Daisuke HITACHI SR8000 (64 of 128 čvorova) 68,719,470,000 1999 Takahashi 20 Yasumasa Kanada Septembar i Daisuke HITACHI SR8000/MPP (sa 128 čvorova) 206,158,430,000 1999 Takahashi 24 HITACHI SR8000 / MPP (64 čvorova), Odsek za Yasumasa Kanada 600 Novembar i tim od 9 ljudi informacione nauke na Tokio Universiti u Tokiju, Japan 1,241,100,000,000 2002 sati T2KOpen Supercomputer(640 čvorova), brzina jednogčvoraje 29 April Daisuke Takahashi 147.2 gigaflopsa, računarska memorija je 13.5 terabajta, 29.09 2,576,980,377,524 2009 iostali algoritam Gauss-Legendre, Centar za računarske nauke na sati Univerzitetu Tsukuba u Tsukuba, Japan .
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    Agarwal et al. Advances in Difference Equations 2013, 2013:100 http://www.advancesindifferenceequations.com/content/2013/1/100 R E S E A R C H Open Access Birth, growth and computation of pi to ten trillion digits Ravi P Agarwal1*, Hans Agarwal2 and Syamal K Sen3 *Correspondence: [email protected] Abstract 1Department of Mathematics, Texas A&M University-Kingsville, Kingsville, The universal real constant pi, the ratio of the circumference of any circle and its TX, 78363, USA diameter, has no exact numerical representation in a finite number of digits in any Full list of author information is number/radix system. It has conjured up tremendous interest in mathematicians and available at the end of the article non-mathematicians alike, who spent countless hours over millennia to explore its beauty and varied applications in science and engineering. The article attempts to record the pi exploration over centuries including its successive computation to ever increasing number of digits and its remarkable usages, the list of which is not yet closed. Keywords: circle; error-free; history of pi; Matlab; random sequence; stability of a computer; trillion digits All circles have the same shape, and traditionally represent the infinite, immeasurable and even spiritual world. Some circles may be large and some small, but their ‘circleness’,their perfect roundness, is immediately evident. Mathematicians say that all circles are similar. Before dismissing this as an utterly trivial observation, we note by way of contrast that not all triangles have the same shape, nor all rectangles, nor all people. We can easily imag- ine tall narrow rectangles or tall narrow people, but a tall narrow circle is not a circle at all.
  • Node Architecture and Performance Evaluation of the Hitachi Super Technical Server SR8000

    Node Architecture and Performance Evaluation of the Hitachi Super Technical Server SR8000

    Node Architecture and Performance Evaluation of the Hitachi Super Technical Server SR8000 Yoshiko Tamaki, Naonobu Sukegawa, Masanao Ito, Yoshikazu Tanaka, Masakazu Fukagawa*, Tsutomu Sumimoto*, and Nobuhiro Ioki** Central Research Laboratory, Hitachi, Ltd. 1-280 Higashi-Koigakubo, Kokubunji, Tokyo, 185-8601, Japan. Tel: +81-423-23-1111, Fax: +81-423-27-7743 e-mail: [email protected] * General Purpose Computer Division, Hitachi, Ltd. 1 Horiyamashita, Hadano, Kanagawa, Japan. ** Software Division, Hitachi, Ltd. 549-6 Shinanochou, Totsuka-ku, Yokohama, Kanagawa, Japan. Abstract systems are high performance when processing smaller size data than the cache size, necessity of source tuning, A new architecture for the Hitachi super technical such as blocking to achieve high performance when server SR8000 has been developed. The performance of processing large-size data, and necessity of large the SR8000 is 8 GFLOPS in each node, which is a RISC- granularity of DO-loops to achieve efficient parallel based SMP, and 1 TFLOPS when up to 128 nodes are execution. connected with an interconnect network. A node of the SR8000 provides a new COMPAS (CO-operative Micro- In developing the super technical server SR8000, Processors in single Address Space) architecture and we aimed to adopt RISC-based SMP nodes and make improved PVP (Pseudo Vector Processing) architecture. them achieve equivalent or higher performance than that COMPAS provides rapid simultaneous start up of all of a vector processor when processing not only smaller but microprocessors in a node. And PVP provides stable and also larger size data than the cache size. We therefore high data-reference throughput even when the node have to construct a node to satisfy the following processes a larger data set than the cache size.