"PSLQ for dilogarithm identities"의 두 판 사이의 차이

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imported>Pythagoras0
imported>Pythagoras0
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==introduction==
 
==introduction==
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* Implement the [[PSLQ algorithm]] first.
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* I found
  
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<math>-2L(1)+2L(\alpha)+2L(\alpha^{2})-L(\alpha^{4})=0</math> or
  
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<math>2L(\alpha)+2L(\alpha^{2})-L(\alpha^{4})=\frac{\pi^2}{3}</math>
  
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Implement the [[PSLQ algorithm]] first.
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#   <br> PSLQ[inx_List, prec_] :=<br>  Block[<br>   {<br>    x,<br>    n = Length[inx],<br>    \[Gamma] = 2/Sqrt[3],<br>    A, B, H, D, Dinv, t, i, j, k, l, iter,<br>    \[Alpha], \[Beta], \[Lambda], \[Delta], r, R<br>    },<br>   (*Initialize*)<br>   x = N[inx /Sqrt[inx . inx], prec];<br>   s = Sqrt[MapIndexed[Plus @@ Drop[x^2, First[#2] - 1] &, x]];<br>   A = B = IdentityMatrix[n];<br>   H = Table[Which[<br>      i > j, (-x[[i]]*x[[j]])/(s[[j]]*s[[j + 1]]),<br>      i == j, s[[i + 1]]/s[[i]],<br>      i < j, 0<br>      ], {i, 1, n}, {j, 1, n - 1}];<br>   (* Reduce H *)<br>   t = HermiteReduce[H];<br>   D = First[t];<br>   Dinv = Inverse[D];<br>   (*Update*)<br>   H = Last[t]; x = x.Dinv; A = D.A; B = B.Dinv;<br>   For[iter = 0, iter < $IterationLimit, ++iter,<br>    (* Step One *)<br>    r = MaxIndex[MapIndexed[\[Gamma]^First[#2] Abs[#1] &, Tr[H, List]]];<br>    If[r < n - 1, \[Alpha] = H[[r, r]]; \[Beta] = <br>      H[[r + 1, r]]; \[Lambda] = H[[r + 1, r + 1]]; \[Delta] = <br>      Sqrt[\[Beta]^2 + \[Lambda]^2]];<br>    R = IdentityMatrix[n]; t = R[[r]]; R[[r]] = R[[r + 1]]; <br>    R[[r + 1]] = t;<br>    x = x.R; H = R.H; A = R.A; B = B.R;<br>    (* Step Two *)<br>    If[r < n - 1,<br>     H = H.Table[<br>        Which[<br>         i == r && j == r, \[Beta]/\[Delta],<br>         i == r && j == r + 1, -\[Lambda]/\[Delta],<br>         i == r + 1 && j == r, \[Lambda]/\[Delta],<br>         i == r + 1 && j == r + 1, \[Beta]/\[Delta],<br>         i == j && j != r || i == j && j != r + 1, 1,<br>         True, 0],<br>        {i, 1, n - 1}, {j, 1, n - 1}]<br>     ];<br>    (* Step Three *)<br>    t = HermiteReduce[H];<br>    D = First[t];<br>    Dinv = Inverse[D];<br>    (*Update*)<br>    H = Last[t]; x = x.Dinv; A = D.A; B = B.Dinv;<br>    (* Step Four *)<br>    If[Min[Abs[Union[x, Tr[H, List]]]] <= 10^(-prec + 5), Break[]]<br>    ];(*Main Iteraton*)<br>   Return[Transpose[B][[MaxIndex[-Abs[x]]]]]<br>   ]<br>
 
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Then find a relation.
 
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#  Clear[a]<br> f[x_] := x^3 + x - 1<br> Solve[f[x] == 0, x]<br> N[%]<br> a := -(2/(3 (9 + Sqrt[93])))^(1/3) + (1/2 (9 + Sqrt[93]))^(1/3)/3^(2/3)<br> N[a, 20]<br> L[x_] := PolyLog[2, x] + 1/2 Log[x] Log[1 - x]<br>
 
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#  L[1] := Pi^2/6<br>
 
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#  (* choose a expected height of the dilogarithm identity *)<br>
 
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#  deg:=6<br> S := Table[L[a^i],{i,0,deg}]<br> N[S, 100]<br> PSLQ[N[S, 100], 1000]<br>
 
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#  N[N[S, 50].%, 50]<br>
 
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I found 
 
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<math>-2L(1)+2L(\alpha)+2L(\alpha^{2})-L(\alpha^{4})=0</math> or
 
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<math>2L(\alpha)+2L(\alpha^{2})-L(\alpha^{4})=\frac{\pi^2}{3}</math><br>
 
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==history==
 
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* http://www.google.com/search?hl=en&tbs=tl:1&q=
 
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==related items==
 
==related items==
  
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* [[PSLQ algorithm]]<br>
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* [[PSLQ algorithm]]
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* [[Slater 34]]<br>
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* [[Slater 34]]
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==encyclopedia==
 
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* http://en.wikipedia.org/wiki/
 
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* http://www.scholarpedia.org/
 
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* http://www.proofwiki.org/wiki/
 
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==books==
 
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* [[2010년 books and articles]]<br>
 
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* http://gigapedia.info/1/
 
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* http://gigapedia.info/1/
 
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* http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
 
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==articles==
 
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* http://www.ams.org/mathscinet
 
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* [http://www.zentralblatt-math.org/zmath/en/ ]http://www.zentralblatt-math.org/zmath/en/
 
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* http://arxiv.org/
 
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* http://www.pdf-search.org/
 
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* http://pythagoras0.springnote.com/
 
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* http://math.berkeley.edu/~reb/papers/index.html
 
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* http://dx.doi.org/
 
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==question and answers(Math Overflow)==
 
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* http://mathoverflow.net/search?q=
 
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* http://mathoverflow.net/search?q=
 
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==blogs==
 
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*  구글 블로그 검색<br>
 
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**  http://blogsearch.google.com/blogsearch?q=<br>
 
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** http://blogsearch.google.com/blogsearch?q=
 
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* http://ncatlab.org/nlab/show/HomePage
 
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==experts on the field==
 
  
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* http://arxiv.org/
 
  
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==computational resource==
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* https://docs.google.com/file/d/0B8XXo8Tve1cxNW0ta2hCUVJPdmc/edit
  
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==links==
 
  
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* [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier]
 
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* [http://pythagoras0.springnote.com/pages/1947378 수식표현 안내]
 
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* [http://www.research.att.com/~njas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]
 
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* http://functions.wolfram.com/
 
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[[분류:개인노트]]
 
 
[[분류:개인노트]]
 
[[분류:개인노트]]
 
[[Category:계산]]
 
[[Category:계산]]
 
[[Category:quantum dimensions]]
 
[[Category:quantum dimensions]]
 
[[분류:dilogarithm]]
 
[[분류:dilogarithm]]

2013년 8월 9일 (금) 02:14 판

introduction

\(-2L(1)+2L(\alpha)+2L(\alpha^{2})-L(\alpha^{4})=0\) or

\(2L(\alpha)+2L(\alpha^{2})-L(\alpha^{4})=\frac{\pi^2}{3}\)



related items


computational resource

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