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Jean Pierre Demailly

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58 pages
On the computational complexity of mathematical functions Jean-Pierre Demailly Institut Fourier, Universite de Grenoble I & Academie des Sciences, Paris (France) November 26, 2011 KVPY conference at Vijyoshi Camp Jean-Pierre Demailly (Grenoble I), November 26, 2011 On the computational complexity of mathematical functions

  • decimal numeral

  • computational complexity

  • academie des sciences

  • vijyoshi camp

  • evaluations used polygon

  • chinese mathematicians


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Jn-eaylG(amliereDiPrevemb),NobleIrenoocehtnO1102,62replomlcnaioatutmpflactcnusnoi
November 26, 2011 KVPY conference at Vijyoshi Camp
InstitutFourier,Universit´edeGrenobleI &Acade´miedesSciences,Paris(France)
On the computational complexity of mathematical functions
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Jean-Pierre Demailly
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Babylonian mathematical tablet allowing the computation of 2 (1800 – 1600 BC)
Decimal numeral system invented in India (500BC ?) :
nalcatioexitompltaehoymfacflamitoCnrecnocg,inutmpldyoeravnore(GlyilmaDerereiP-naeJmputheco1Ont,201re62evbm,)oNlbIe
MadhavasofmrlufaroπeiP-naeJenGry(llaiemeDrrpmocixelitatlanoatemalicoftythmavomeeb2rboel)IN,thecompu6,2011Onontincfu
Early calculations ofπwere done by Greek mathematicians several centuries BC.
and Indian
s
ierran-PJenebo(yrGiallDeme6,r2beemov,NI)leatupmocehtnO1102itnolaocpmelixytofmathematicalfuitcnsno
Early calculations ofπwere done by Greek and Indian mathematicians several centuries BC. These early evaluations used polygon approximations and Pythagoras theorem. In this way, using 96 sides, Archimedes got3 +1701< π <3 +1070whose average is31418(c. 230 BC). Chinese mathematicians reached 7 decimal places in 480 AD.
roπvasadhaulafformM
avahdaMlamuorsfrπfo,)oNevbmeronlbIemailly(GPierreDeJ-nae
The next progress was the discovery of the firstinfinite series formula by Madhava (circa 1350 – 1450), a prominent mathematician-astronomer from Kerala (formula rediscovered in the XVIIe century by Leibniz and Gregory) :
Early calculations ofπwere done by Greek and Indian mathematicians several centuries BC. These early evaluations used polygon approximations and Pythagoras theorem. In this way, using 96 sides, Archimedes got3 +1071< π <3 +1070whose average is31418(c. 230 BC). Chinese mathematicians reached 7 decimal places in 480 AD.
unctcalf
Convergence is unfortunately very slow, but Madhava was able to improve convergence and reached in this way 11 decimal places.
4π= 13+15117+19111+  (+2n)11++  
n +  
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