PSLQ for dilogarithm identities

introduction

 

 

Implement the PSLQ algorithm first.

 

1.  
   PSLQ[inx_List, prec_] :=
    Block[
     {
      x,
      n = Length[inx],
      \[Gamma] = 2/Sqrt[3],
      A, B, H, D, Dinv, t, i, j, k, l, iter,
      \[Alpha], \[Beta], \[Lambda], \[Delta], r, R
      },
     (*Initialize*)
     x = N[inx /Sqrt[inx . inx], prec];
     s = Sqrt[MapIndexed[Plus @@ Drop[x^2, First[#2] - 1] &, x]];
     A = B = IdentityMatrix[n];
     H = Table[Which[
        i > j, (-xi*xj)/(sj*sj + 1),
        i == j, si + 1/si,
        i < j, 0
        ], {i, 1, n}, {j, 1, n - 1}];
     (* Reduce H *)
     t = HermiteReduce[H];
     D = First[t];
     Dinv = Inverse[D];
     (*Update*)
     H = Last[t]; x = x.Dinv; A = D.A; B = B.Dinv;
     For[iter = 0, iter < $IterationLimit, ++iter,
      (* Step One *)
      r = MaxIndex[MapIndexed[\[Gamma]^First[#2] Abs[#1] &, Tr[H, List]]];
      If[r < n - 1, \[Alpha] = Hr, r; \[Beta] = 
        Hr + 1, r; \[Lambda] = Hr + 1, r + 1; \[Delta] = 
        Sqrt[\[Beta]^2 + \[Lambda]^2]];
      R = IdentityMatrix[n]; t = Rr; Rr = Rr + 1; 
      Rr + 1 = t;
      x = x.R; H = R.H; A = R.A; B = B.R;
      (* Step Two *)
      If[r < n - 1,
       H = H.Table[
          Which[
           i == r && j == r, \[Beta]/\[Delta],
           i == r && j == r + 1, -\[Lambda]/\[Delta],
           i == r + 1 && j == r, \[Lambda]/\[Delta],
           i == r + 1 && j == r + 1, \[Beta]/\[Delta],
           i == j && j != r || i == j && j != r + 1, 1,
           True, 0],
          {i, 1, n - 1}, {j, 1, n - 1}]
       ];
      (* Step Three *)
      t = HermiteReduce[H];
      D = First[t];
      Dinv = Inverse[D];
      (*Update*)
      H = Last[t]; x = x.Dinv; A = D.A; B = B.Dinv;
      (* Step Four *)
      If[Min[Abs[Union[x, Tr[H, List]]]] <= 10^(-prec + 5), Break[]]
      ];(*Main Iteraton*)
     Return[Transpose[B][[MaxIndex[-Abs[x]]]]]
     ]
   

 

Then find a relation.

 

1. Clear[a]
   f[x_] := x^3 + x - 1
   Solve[f[x] == 0, x]
   N[%]
   a := -(2/(3 (9 + Sqrt[93])))^(1/3) + (1/2 (9 + Sqrt[93]))^(1/3)/3^(2/3)
   N[a, 20]
   L[x_] := PolyLog[2, x] + 1/2 Log[x] Log[1 - x]
   
2. L[1] := Pi^2/6
   
3. (* choose a expected height of the dilogarithm identity *)
   
4. deg:=6
   S := Table[L[a^i],{i,0,deg}]
   N[S, 100]
   PSLQ[N[S, 100], 1000]
   
5. N[N[S, 50].%, 50]
   

 

 

I found 

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

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

 

 

 

 

history

• http://www.google.com/search?hl=en&tbs=tl:1&q=

 

 

related items

• PSLQ algorithm
  
• Slater 34
  

 

 

encyclopedia

• http://en.wikipedia.org/wiki/
• http://www.scholarpedia.org/
• http://www.proofwiki.org/wiki/
• Princeton companion to mathematics(Companion_to_Mathematics.pdf)

 

 

books

 

• 2010년 books and articles
  
• http://gigapedia.info/1/
• http://gigapedia.info/1/
• http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=

4909919

 

 

articles

 

• http://www.ams.org/mathscinet
• [1]http://www.zentralblatt-math.org/zmath/en/
• http://arxiv.org/
• http://www.pdf-search.org/
• http://pythagoras0.springnote.com/
• http://math.berkeley.edu/~reb/papers/index.html
• http://dx.doi.org/

 

 

question and answers(Math Overflow)

• http://mathoverflow.net/search?q=
• http://mathoverflow.net/search?q=

 

 

blogs

• 구글 블로그 검색
  
  • http://blogsearch.google.com/blogsearch?q=
    
  • http://blogsearch.google.com/blogsearch?q=
• http://ncatlab.org/nlab/show/HomePage

 

 

experts on the field

• http://arxiv.org/

 

 

links

• Detexify2 - LaTeX symbol classifier
• 수식표현 안내
• The On-Line Encyclopedia of Integer Sequences
• http://functions.wolfram.com/