RJ_Carlson Symmetric Elliptic Integrals= Faulty principle
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From: Alain Epain on 7 Jun 2010 00:31 ---- RJ_Carlson Symmetric Elliptic Integrals= Faulty principle ---- ---- Abstract: -...my English is very bad -The Oct 10, 2007, I have already posted on Mathforum, on the third Carlson Elliptic Integral RJ, it was for a 1rst problem: on the principal value of the Log -This present post is for a 2nd problem on the same RJ: the principle used by Carlson is Faulty when the Argument Z and the Parameter S verify: Z= +/- S *2**K with K=0,1,2,3,4... ---- ---- A/ Generalities on Carlson Symmetric Elliptic Integrals: RF,RD,RJ...etc : 1/ Carlson who have created, and work the subject near alone from 50 years (like Legendre). He use Multivariable Hypergeometric Functions on their form "R Functions" that Carlson has created See the Chapter 22 of the very new Web DLMF (May 2010= 1 month ago) 2/ I had studied the subjet by using uniquely: 2.1/The Weierstrass-Jacobi Elliptic Functions Theory (with my personnal-private improvements) 2.2/Operations on the Power Series: ( Euler->J.C Miller->Henrici->Knuth->...etc) ---- ---- B/ My Theory Principle on RF,RD,RJ (and the RJ_Carlson default): 1/ RF: -my principle is by successive Multiplication of the Generators by 2 (it is equivalent to this of Carlson: successive Division of the Argument Z by 2) -I refind the RF formula of Carlson 2/ RD: -my principle is by successive Multiplication of the Generators by 2 (it is equivalent to this of Carlson: successive Division of the Argument Z by 2) -my primary formulae is for the Zeta of Weierstrass from which I refind the RD formula of Carlson (which therefore is not the more direct formula) 3/ RJ: my primary formula is for EG3= -1/Pe' *Log(sigma(S+Z)/sigma(S-Z)*exp( -2*Zeta(S) *Z)) and RJ= -3 *EG3 and for this EG3, I have studied 2 principles: 3.1/ the principle 1: by successive Multiplication of the Generators by 2 (equivalent of successive Division at once of the Argument Z by 2 and the Parameter S by 2) -it is the principle I have discovered (=see nowhere) -this principle has none default -my formula RJ is different of this of Carlson 3.2/ the principle 2: "successive Division of the Argument Z by 2" (without changing the parameter S) -it is the principle used by Carlson: --and at the Jrd step of dividing Z by 2 we have something like: EG3= -1/Pe'*( +Sum_N +Rest_J) with Sum_N = Sum_N+Term_J Term_J= in my study is coeff*Log and in Carlson is coeff*RC(x,y) (the relation Log //RC being this given by Carlson) Rest_J= is "implicitly": Log( sigma(S+Z_J)/sigma(S-Z_J) *exp(-2*Zeta(S) *Z_J) ) with Z_J= Z/2**J a/primary I find a Term_Carlson_1885_J (in fact, not from Carlson, but from Weierstrass 1885) b/from which I refind the Term_Carlson_1979_J c/from which I refind the Term_Carlson_1995_J -we see that this principle is Faulty when Z= +/- S *2**K with K=0,1,2,3,4... because at J=K => Z_J= +/- S --and the "Implicit" Log( sigma(S+Z_J)/sigma(S-Z_J) *exp....): in Theory =+/- Log( sigma(2*S)/sigma(0) *exp...) in Pratice=+/- Log( sigma(2*S)/sigma(->0) *exp...) (...by the limited precision...) which shall produce (in general): --not an overflow (= a good alarm) --but an erroneous result (= without overflow = without good alarm) --the case K=0 => Z= +/- S is easily avoidable by p=0 => "non valid datas", but the cases K=1,2,3,4...are not avoidable except if at the beginning of the programme RJ we add: Z=RF(x,y,z) ;S= RF(x-p,y-p,z-p) if Z= +/- S*2**K avec K=1,2,3,4... "for these values our computation principle does not work" !!! ...which whould be not very professional --I have verified by numerical examples that the results are effectively erroneous with the James FitzSimons programs transformed from Maxima to .f90 ---- ---- Conclusion: on the RJ of Carlson: 2 problems: 1/principal value (my Oct 10, 2007 post) 2/principle Faulty (this June 7,2010 post) ---- ---- Bibliography ---- 1/http://dlmf.nist.gov/: 19.Elliptic Integrals, with Carlson 1964 Carlson 1970 Zill, Carlson 1970 Carlson 1979 Carlson 1995= the arXiv:math/9409227 2/James FitzSimons http://getnet.net/~cherry/ a/Maxima software for symmetric elliptic integrals b/test software (on same line of a/) c/test Carlson software (on the same line of a/ and b/) Nota: they are in Maxima (I do not know) but we can read in Windows with f90 by: "right click" and "save target as" and add to the name .f90 3/Mathforum: Oct 10, 2007, Alain EPAIN, RJ_Carlson Symmetric Elliptic Integral: errors ? -many attempts for to improve misprints... seen by the great Phil Carmody...!!! -errata (new): Pe'(TC,TB,Z) -> Pe'(TC,TB,S) if (RG,RJ) -> if (RD,RJ) G3 -> EG3 RJ_04_Carlson_GF -> RJ_04_Carlson -Nota: in May 2010, I have detected that some input test values of CARLSON 1995 for RJ --produce in Theory Pe'(S)=-2*sqrt((x-p)*(y-p)*(z-p))=0 = prohibited values --but in Practice Pe'(S)-> 0 (by the limited precision) explaining why its works ---- |