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Notebook[{

Cell[CellGroupData[{
Cell["\<\

The Lagarias multidimensional geodesic continued fraction algorithm
\
\>", "Title"],

Cell["\<\

This algorithm finds all Hermite best approximations in simpler rationals 

(p_1/q,p_2/q,...p_n/q) to a given rational vector in n dimensions. 


\
\>", "Subtitle"],

Cell["\<\



An Hermite approximation to a vector v in n dimensions is a rational vector 
r=(p_1/q,p_2/q,...p_n/q) so that, for all t in some interval 0<a<t<b, 
(r-v).(r-v)+t^2q^2 is minimal among all choices of integers p_1, p_2, ... p_n \

and q. 
\
\>", "Subsubtitle"],

Cell[TextData[{
  "\nThe procedures here culminate with a procedure getcfx which takes two \
inputs, \na rational target vector v in some dimension greater than 1, and a \
positive \ninteger n which stipulates the depth to which the procedure shall \
be carried \nif it doesn't terminate due to having hit upon the input vector \
itself. \n\nAccompanying this we offer two variants that produce simpler \
output. \nThe first, \"gethermites\", takes as input a listtheta of rational \
numbers in (-1/2,1/2) \nand a positive integer, and returns all the Hermite \
n-tuples (p_1,...p_{n-1},q) \nsuch that q is a Hermite denominator for the \
target, theta, and such that the p_j's \nare the integers nearest q theta_j. \
The second, hermites, takes the same kind of inputs\nbut returns only the \
Hermite denominators themselves. \n\nWe also provide a simpler algorithm, \
lllcfx, which takes the same kind of input but returns \na sequence of \
integers that arise out of executing ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  "'s variant of the Lenstra-Lenstra-Lovasz \nalgorithm. If the target list \
`u' has length n-1, one takes an n-1 by n-1 identity matrix, appends -theta \
as\na final row, then pads each row but the last with a final zero, appending \
instead 2^(-k) to that last row, u.\nThe resulting n by n matrix is \
LLL-reduced and the successive integers that appear as absolute values of the \
last entry in \nthe top row are noted. These are more easily computed than \
the Hermite approximations, and they afford denominators q that lead to \
serviceable simultaneous diophantine approximations of u. \n\n\n\nSticking to \
rational inputs has the advantage that the algorithm executes \ncleanly. \
There is never any doubt as to whether one number is greater than \nanother \
when dealing with rationals, and so branching decisions are not \ncontingent \
upon the vagaries of floating point arithmetic. \n\n\n\nWe illustrate the \
algorithm by examining what happens when v is a {very good} \nrational \
approximation to a certain kind of algebraic input. Suppose P(x) is \nan \
irreducible polynomial with integer coefficients, of degree n. Let u be \nthe \
vector  (x,x^2,...x^(n-1)). As Drmota and Tichy have shown, u is badly \n\
approximable, meaning that there is some positive epsilon so that for all \n\
rational approximations r=(p1/q,...p_n/q), q^(1/n)|u-r|>epsilon. \n\n\n\nIf \
we take for v a rational approximation to u of the form \n\
v=10^(-N)Map[Floor,10^N u], then we may expect that the execution history of \
\nthe geodesic continued fraction algorithm on input v, will track to \n\
considerable depth that of what would abstractly happen if we worked with u \n\
itself. The resulting sequence of integers q_j, and approximations r_j,  can \
\nfor the case n=3 be inspected by evaluating the corresponding points \n\
q_j^(1/2)(q_j v mod 1), where the interval (-1/2,1/2] is used to select a \n\
representative element mod 1. These will, on theoretical grounds, all fall \n\
within an annulus about the origin. The surprise is that they all \nfall onto \
a finite set of ellipses, concentric and similar, about the origin. \n\n\n\n\n\
\n\n\n"
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(?getcfx\)\)], "Input"],

Cell["Global`getcfx", "Print",
  CellTags->"Info3339663997-2203256"],

Cell[BoxData[
    InterpretationBox[GridBox[{
          {GridBox[{
                {\(getcfx[v_List, stopwhen_] := 
                    Module[{n, k, m, t, u, e, s, z, sofar, q, qq}, 
                      n = Length[v]; m = Table[esubi[k, n + 1], {k, 1, n}]; 
                      AppendTo[m, Append[\(-v\), 1]]; t = 1; z = {}; 
                      u = Table[esubi[k, n + 1], {k, 1, n + 1}]; e = u; 
                      s = {m, t, u, e}; sofar = 0; qq = {}; 
                      While[\(! timetoquit[
                              v, \(s\[LeftDoubleBracket]3\[RightDoubleBracket]\
\)\[LeftDoubleBracket]1\[RightDoubleBracket]]\) && sofar < stopwhen, 
                        AppendTo[z, s]; s = step[s]; 
                        q = \(\(s\[LeftDoubleBracket]1\[RightDoubleBracket]\)\
\[LeftDoubleBracket]1\[RightDoubleBracket]\)\[LeftDoubleBracket]\(-1\)\
\[RightDoubleBracket]; qq = qq \[Union] {Abs[q]}; qq >> "qq.m"; 
                        s >> "mtue.m"; 
                        sofar >> "sofar.m"; \((z >> "z.m"; \(sofar++\))\)]; 
                      z]\)}
                },
              GridBaseline->{Baseline, {1, 1}},
              ColumnWidths->0.999,
              ColumnAlignments->{Left}]}
          },
        GridBaseline->{Baseline, {1, 1}},
        ColumnAlignments->{Left}],
      Definition[ "getcfx"],
      Editable->False]], "Print",
  CellTags->"Info3339663997-2203256"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(hermites[{\(-31\)/101, 50/139}, 400]\)], "Input"],

Cell[BoxData[
    \({0, 1, 3, 39, 75, 114, 303}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(lllcfx[{\(-31\)/101, 50/139}, 30]\)], "Input"],

Cell[BoxData[
    \({1, 3, 39, 114, 303, 14039}\)], "Output"]
}, Open  ]],

Cell[BoxData[""], "Input"],

Cell[BoxData[
    \(\(\(\[IndentingNewLine]\)\(esubi[k_, n_] := 
      Join[Table[0, {k - 1}], {1}, Table[0, {n - k}]]\)\)\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    \(reddit[u_] := 
      Module[{v, w}, v = LatticeReduce[u]; w = v . Inverse[u]; 
        Abs[\(w[\([1]\)]\)[\([\(-1\)]\)]]]\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    \(lllcfx[u_, n_] := 
      Module[{j, k, r}, r = Length[u]; 
        Drop[Union[
            Table[reddit[
                Join[Table[
                    esubi[k, r + 1], {k, 1, r}], {Join[
                      u, {2^\((\(-j\))\)}]}]], {j, 0, n}]], 1]]\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    \(switchmatrix[{j_, k_, n_}] := 
      Module[{r}, 
        Join[Table[esubi[r, n], {r, 1, j - 1}], {esubi[k, n]}, 
          Table[esubi[r, n], {r, j + 1, k - 1}], {esubi[j, n]}, 
          Table[esubi[r, n], {r, k + 1, n}]]]\)], "Input",
  InitializationCell->True],

Cell[CellGroupData[{

Cell[BoxData[
    \(norm[b_, t_] := 
      Module[{b1, b2}, b1 = Drop[b, \(-1\)]; b2 = b[\([\(-1\)]\)]; 
        b1 . b1 + t\ b2^2]\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    RowBox[{\(General::"spell1"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \
error: new symbol name \\\"\\!\\(norm\\)\\\" is similar to existing symbol \\\
\"\\!\\(Norm\\)\\\". \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \
ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \
ButtonData:>\\\"General::spell1\\\"]\\)\"\>"}]], "Message"]
}, Open  ]],

Cell[BoxData[
    \(switchedspots[e_] := 
      Module[{j, k, s}, s = {}; j = 1; n = Length[e]; 
        While[Length[s] < 2 && j \[LessEqual] n, 
          If[e[\([j]\)] \[NotEqual] esubi[j, n], 
            AppendTo[s, j]; \(j++\), \(j++\)]]; s]\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    \(switchvalue[{e_, m_, t_}] := 
      Module[{j, k, m1, m2, n1at0, n2at0, n1att, n2att, tau, s}, {j, k} = 
          switchedspots[e]; m1 = m[\([j]\)]; m2 = m[\([k]\)]; 
        n1at0 = norm[m1, 0]; n2at0 = norm[m2, 0]; 
        If[n1at0 \[LessEqual] n2at0, Return[False]]; n1att = norm[m1, t]; 
        n2att = norm[m2, t]; 
        If[n1att > n2att, Return[False]]; \[IndentingNewLine]tau = 
          Solve[norm[m1, s] \[Equal] 
              norm[m2, s]]; \[IndentingNewLine]tau = \((s /. tau)\)[\([1]\)]; 
        tau]\)], "Input",
  InitializationCell->True],

Cell[CellGroupData[{

Cell[BoxData[
    \(nchoosek[0, n_] = {{}}\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    \({{}}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(nchoosek[j_, 0] = {}\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    \({}\)], "Output"]
}, Open  ]],

Cell[BoxData[""], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(plusminusvectors[0] = {{}}\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    \({{}}\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(plusminusvectors[n_] := \(plusminusvectors[n] = 
        Join[Map[Join[#, {1}] &, plusminusvectors[n - 1]], 
          Map[Join[#, {\(-1\)}] &, plusminusvectors[n - 1]]]\)\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    \(nchoosek[k_, n_] := \(nchoosek[k, n] = 
        Join[nchoosek[k, n - 1], 
          Map[Join[#, {n}] &, nchoosek[k - 1, n - 1]]]\)\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    \(settovectors[s_, n_] := 
      Module[{pm, j, k, l}, l = Length[s]; 
        pm = Select[plusminusvectors[l], #[\([\(-1\)]\)] > 0 &]; 
        Table[Sum[
            esubi[s[\([k]\)], n] \(pm[\([j]\)]\)[\([k]\)], {k, 1, l}], {j, 1, 
            Length[pm]}]]\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    \(lastentry[
        v_] := \(\(Position[v, 1]\)[\([\(-1\)]\)]\)[\([1]\)]\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    \(vectortomatrix[v_] := 
      Module[{j, k, n}, k = lastentry[v]; n = Length[v]; 
        Join[Table[esubi[j, n], {j, 1, k - 1}], {v}, 
          Table[esubi[j, n], {j, k + 1, n}]]]\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    \(\(\(movevalue[{v_, m_, t_}] := 
      Module[{e, k, m1, m2, n1at0, n2at0, n1att, n2att, s, tau}, 
        e = vectortomatrix[v]; k = lastentry[v]; m1 = m[\([k]\)]; 
        m2 = \((e . m)\)[\([k]\)]; \[IndentingNewLine]n1at0 = norm[m1, 0]; 
        n2at0 = norm[m2, 0]; \[IndentingNewLine]If[n1at0 \[LessEqual] n2at0, 
          Return[False]]; \[IndentingNewLine]n1att = norm[m1, t]; 
        n2att = norm[m2, t]; \[IndentingNewLine]If[n1att > n2att, 
          Return[False]]; \[IndentingNewLine]tau = 
          Solve[norm[m1, s] \[Equal] norm[m2, s], 
            s]; \[IndentingNewLine]tau = \((s /. tau)\)[\([1]\)]; 
        tau]\)\(\[IndentingNewLine]\)
    \)\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    \(vectors[n_] := 
      Flatten[\((settovectors[#1, n] &)\) /@ usablesubsets[n], 1]\)], "Input",\

  InitializationCell->True],

Cell[BoxData[
    \(usablesubsets[n_] := 
      Flatten[Join[Table[nchoosek[j, n], {j, 2, n}]], 1]\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    \(switchy[n_] := 
      Flatten[Table[{j, k, n}, {k, 2, n}, {j, 1, k - 1}], 1]\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    \(switchbunch[n_] := Map[{#, switchmatrix[#]} &, switchy[n]]\)], "Input",
  InitializationCell->True],

Cell[CellGroupData[{

Cell[BoxData[
    \(subsets[0] = {{}}\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    RowBox[{\(General::"spell1"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \
error: new symbol name \\\"\\!\\(subsets\\)\\\" is similar to existing symbol \
\\\"\\!\\(Subsets\\)\\\". \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \
ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \
ButtonData:>\\\"General::spell1\\\"]\\)\"\>"}]], "Message"],

Cell[BoxData[
    \({{}}\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(subsets[n_] := \(subsets[n] = 
        Join[subsets[n - 1], 
          Map[Join[#, {n}] &, subsets[n - 1]]]\)\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    \(usablesubsets[n_] := 
      Select[subsets[n], \((Length[#] \[GreaterEqual] 2)\) &]\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    \(vectorbunch[n_] := 
      Partition[Flatten[Map[{#, vectortomatrix[#]} &, vectors[n]], 1], 
        2]\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    \(moves[n_] := Join[switchbunch[n], vectorbunch[n]]\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    \(\(\(validswitches[{m_, t_}] := 
      Module[{n, vs}, n = Length[m]; vs = switchbunch[n]; 
        Select[vs, 
          NumberQ[switchvalue[{#[\([2]\)], m, 
                  t}]] &]]\)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)
    \)\)], "Input",
  InitializationCell->True],

Cell[BoxData[""], "Input"],

Cell[BoxData[
    \(validmoves[{m_, t_}] := 
      Module[{n, vs}, n = Length[m]; vs = vectorbunch[n]; 
        Select[vs, NumberQ[movevalue[{#[\([1]\)], m, t}]] &]]\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    \(valids[{m_, t_}] := 
      Join[Map[{switchvalue[{#[\([2]\)], m, t}], #} &, 
          validswitches[{m, 
              t}]], \[IndentingNewLine]Map[{movevalue[{#[\([1]\)], m, 
                  t}], #} &, validmoves[{m, t}]]]\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    \(bestchoices[{m_, t_}] := 
      Sort[valids[{m, t}], OrderedQ[{#1[\([1]\)], #2[\([1]\)]}] &]\)], "Input",\

  InitializationCell->True],

Cell[BoxData[
    \(step[{m_, t_, u_}] := 
      Module[{e, s, v, w}, w = bestchoices[{m, t}]; v = w[\([\(-1\)]\)]; 
        e = \(v[\([\(-1\)]\)]\)[\([2]\)]; 
        s = v[\([1]\)]; {e . m, s, e . u}]\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    \(settovectors[s_, n_] := 
      Module[{pm, j, k, l}, l = Length[s]; 
        pm = Select[plusminusvectors[l], #[\([\(-1\)]\)] > 0 &]; 
        Table[Sum[
            esubi[s[\([k]\)], n] \(pm[\([j]\)]\)[\([k]\)], {k, 1, l}], {j, 1, 
            Length[pm]}]]\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    \(\(\(setandtemplatetovectors[s_, n_, template_] := 
      Module[{pm, j, k, l}, l = Length[s]; 
        If[l \[NotEqual] Length[template], 
          Print["\<template should have same length as the set\>"]; 
          Return[False]]; \[IndentingNewLine]pm = 
          Select[plusminusvectors[
              l], #[\([\(-1\)]\)] > 0 &]; \[IndentingNewLine]Table[
          Sum[esubi[s[\([k]\)], n] 
              template[\([k]\)]\ \(pm[\([j]\)]\)[\([k]\)], {k, 1, l}], {j, 1, 
            Length[pm]}]]\)\(\[IndentingNewLine]\)
    \)\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    \(usablesubsets[n_] := 
      Flatten[Join[Table[nchoosek[j, n], {j, 2, n}]], 1]\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    \(vectorfromsubsetandtemplate[sbset_, template_, n_] := 
      Module[{j}, 
        Sum[esubi[sbset[\([j]\)], n] template[\([j]\)], {j, 1, 
            Length[sbset]}]]\)], "Input",
  InitializationCell->True],

Cell[CellGroupData[{

Cell[BoxData[
    \(signedvectors[{}] = {{}}\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    \({{}}\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(signedvectors[v_] := \(signedvectors[v] = 
        Union[Map[Join[#, {v[\([\(-1\)]\)]}] &, 
            signedvectors[Drop[v, \(-1\)]]], \(-Map[
              Join[#, {v[\([\(-1\)]\)]}] &, 
              signedvectors[Drop[v, \(-1\)]]]\)]\)\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    \(goodsignedvectors[v_] := 
      Union[Select[signedvectors[v], goodvector]]\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    \(goodvector[v_] := 
      Module[{k}, v . Table[3^k, {k, 1, Length[v]}] > 0]\)], "Input",
  InitializationCell->True],

Cell[BoxData[""], "Input"],

Cell[BoxData[""], "Input"],

Cell[BoxData[
    \(step[{m_, t_, u_, f_}] := 
      Module[{e, s, v, w}, w = bestchoices[{m, t}]; v = w[\([\(-1\)]\)]; 
        e = \(v[\([\(-1\)]\)]\)[\([2]\)]; 
        s = v[\([1]\)]; {e . m, s, e . u, e}]\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    \(rationalchop[x_, precision_] := 
      Module[{y, z}, y = Floor[10^precision\ \ x]/10^precision; 
        z = remnant[y]]\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    \(\(\(parallel[u_List, v_List] := 
      Module[{n, k, y, z}, n = Length[u]; 
        If[Length[v] \[NotEqual] n, Return[False]]; k = 1; 
        While[u[\([k]\)] \[Equal] 0, 
          If[k \[Equal] n, Return[True]]; \(k++\)]; 
        z = u[\([k]\)]; \[IndentingNewLine]y = 
          v[\([k]\)]/u[\([k]\)]; \[IndentingNewLine]If[v \[NotEqual] y\ u, 
          Return[False], Return[True]]]\)\(\[IndentingNewLine]\)
    \)\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    \(timetoquit[v_List, eu_List] := 
      Module[{n, w, x}, n = Length[v]; w = Append[v, 1]; 
        parallel[w, eu]]\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    \(getcfx[v_List, stopwhen_] := 
      Module[{n, k, m, t, u, e, s, z, sofar, q, qq}, 
        n = Length[v]; \[IndentingNewLine]\t
        m = Table[esubi[k, n + 1], {k, 1, n}]; \[IndentingNewLine]\t
        AppendTo[m, Append[\(-v\), 1]]; \[IndentingNewLine]t = 1; 
        z = {}; \[IndentingNewLine]u = Table[esubi[k, n + 1], {k, 1, n + 1}]; 
        e = u; \[IndentingNewLine]s = {m, t, u, e}; sofar = 0; 
        qq = {}; \[IndentingNewLine]While[
          Not[timetoquit[v, \(s[\([3]\)]\)[\([1]\)]]] && 
            sofar < stopwhen, \[IndentingNewLine]AppendTo[z, s]; s = step[s]; 
          q = \(\(s[\([1]\)]\)[\([1]\)]\)[\([\(-1\)]\)]; 
          qq = Union[qq, {Abs[q]}]; qq >> qq.m; s >> mtue.m; 
          sofar >> sofar.m; \(z >> 
            z.m;  (*\(Print["\<q=\>", q, "\<sofar=\>", 
                sofar];\)*) \(sofar++\)\)]; z]\)], "Input",
  InitializationCell->True],

Cell[BoxData[""], "Input"],

Cell[BoxData[
    \(getmorecfx[v_, stopwhen_] := 
      Module[{s, z, sofar, q, qq, oldq}, s = << mtue.m; sofar = << sofar.m; 
        qq = << qq.m; z = {}; 
        oldq = 0; \[IndentingNewLine]While[
          Not[timetoquit[v, \(s[\([3]\)]\)[\([1]\)]]] && 
            sofar < stopwhen, \[IndentingNewLine]AppendTo[z, s]; s = step[s]; 
          q = \(\(s[\([1]\)]\)[\([1]\)]\)[\([\(-1\)]\)]; 
          qq = Union[qq, {Abs[q]}]; qq >> qq.m; s >> mtue.m; 
          sofar >> sofar.m; 
          If[q \[NotEqual] oldq, 
            oldq = q; \[IndentingNewLine]Print["\<q=\>", q, "\< sofar=\>", 
              sofar]]; \(sofar++\)]; z]\)], "Input",
  InitializationCell->True],

Cell["\<\

Minor point: If this getmorecfx routine is used to grind out a \
multidimensional continued fraction expansion of a several-dimensional vector \
over many days, starting and stopping so as to free the computer for other \
tasks between times, then on each restart, q will have been reset to zero \
while qq retains the correct values and is read to the hard drive and back \
from it. Thus q will update from zero to its old value unless a new value \
occurs immediately, and the program will print out the new value of sofar \
along with whatever q is. \
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(gethermites[{\(-22\)/71, \(-31\)/109}, 2000]\)], "Input"],

Cell[BoxData[
    \({{1, 0, 0}, {22\/71, 31\/109, 1}, {5\/71, 16\/109, 3}, {\(-\(1\/71\)\), 
        6\/109, 42}, {\(-\(1\/71\)\), 2\/109, 313}, {0, 1\/109, 
        1846}}\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(gethermites[v_List, stopwhen_] := 
      Map[Join[Drop[#, \(-1\)], {Abs[#[\([\(-1\)]\)]]}] &, 
        Sort[Union[
            Map[\(#[\([1]\)]\)[\([1]\)] &, 
              getcfx[v, stopwhen]]], \((Abs[#1[\([\(-1\)]\)]] < 
                Abs[#2[\([\(-1\)]\)]])\) &]]\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    \(hermites[uvals_, n_] := 
      Module[{hs}, hs = getcfx[uvals, n]; 
        Union[Abs[
            Map[\(\(#[\([1]\)]\)[\([1]\)]\)[\([\(-1\)]\)] &, 
              hs]]]]\)], "Input",
  InitializationCell->True],

Cell[CellGroupData[{

Cell[BoxData[
    \(hermites[{\(-22\)/71, \(-31\)/109}, 2000]\)], "Input"],

Cell[BoxData[
    \({0, 1, 3, 42, 313, 1846}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(71*109\)], "Input"],

Cell[BoxData[
    \(7739\)], "Output"]
}, Open  ]],

Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[""], "Input"],

Cell[BoxData[
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Cell[BoxData[
    \(\(\(\[IndentingNewLine]\)\(normalizeit[v_] := v/Sqrt[v . v]; 
    crossproduct[a_List, 
        b_List] := {a[\([2]\)] b[\([3]\)] - a[\([3]\)] b[\([2]\)], 
        a[\([3]\)] b[\([1]\)] - a[\([1]\)] b[\([3]\)], 
        a[\([1]\)] b[\([2]\)] - a[\([2]\)] b[\([1]\)]};\[IndentingNewLine]
    \(normalizedcrossproduct[a_List, b_List] := 
        normalizeit[crossproduct[a, b]];\)\[IndentingNewLine]
    el[v_] := {v[\([1]\)]^2, v[\([1]\)] v[\([2]\)], 
        v[\([2]\)]^2}\)\)\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    \(getacf[cubic_, n_] := 
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        v = Select[v, realnumber]; v = v[\([1]\)]; w = N[v, n]; z = w^2; 
        u1 = remnant[w]; u2 = remnant[z]; u1 = rationalchop[u1, n]; 
        u2 = rationalchop[u2, n]; 
        Print["\<cubic==\>", cubic, "\< target\[Equal]\>", 
          N[{u1, u2}]]; {{u1, u2}, getcfx[{u1, u2}, 6\ n]}]\)], "Input",
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Cell[BoxData[
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Cell[BoxData[
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      Module[{a, b, c, d, pts, qs, u, cf, m, els, pls, u1, u2, u3, u4, v1, 
          v2, v3, v4, nc1, nc2, nc3, nc4}, cf = getacf[cubic, n]; 
        u = cf[\([1]\)]; 
        qs = Map[\(\(#[\([3]\)]\)[\([1]\)]\)[\([3]\)] &, 
            cf[\([2]\)]]; \[IndentingNewLine]qs = Union[Abs[qs]]; 
        Print["\<early qs=\>", 
          Take[qs, Min[Length[qs], 10]]]; \[IndentingNewLine]m = Length[qs]; 
        Print["\<m=\>", m]; \[IndentingNewLine]pts = 
          Map[scaledtofinal, 
            Table[Join[Map[remnant, qs[\([k]\)] u], {qs[\([k]\)]}], {k, 
                1 + Floor[m/2], m}]]; \[IndentingNewLine]ListPlot[pts, 
          AspectRatio \[Rule] Automatic]; \[IndentingNewLine]els = 
          Map[el, pts]; pls = Sort[els, \((#1 . #1 < #2 . #2)\) &]; 
        els >> els.m; \[IndentingNewLine] (*ScatterPlot3D[els]; 
          enable\ this\ only\ if\ your\ machine\ has\ ScatterPlot3D\
*) \[IndentingNewLine]u1 = pls[\([2]\)] - pls[\([1]\)]; 
        u2 = pls[\([3]\)] - pls[\([1]\)]; u3 = pls[\([4]\)] - pls[\([1]\)]; 
        v1 = pls[\([\(-2\)]\)] - pls[\([\(-1\)]\)]; 
        v2 = pls[\([\(-3\)]\)] - pls[\([\(-1\)]\)]; 
        v3 = pls[\([\(-4\)]\)] - pls[\([\(-1\)]\)]; \[IndentingNewLine]nc1 = 
          normalizedcrossproduct[u1, u2]; 
        nc2 = normalizedcrossproduct[u1, u3]; \[IndentingNewLine]nc3 = 
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        nc4 = normalizedcrossproduct[v1, 
            v3]; \[IndentingNewLine]abc = {{nc1, nc2, nc3, nc4}, 
            nc1 . pls[\([1]\)], 
            nc4 . pls[\([\(-1\)]\)]}]\)\(\[IndentingNewLine]\)\(\
\[IndentingNewLine]\)
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      SequenceForm[ "cubic==", 
        Plus[ -1, 
          Times[ -1, x], 
          Power[ x, 3]], " target\[Equal]", {
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      "early qs=", {0, 1, 3, 12, 37, 151, 465, 1897, 5842, 17991}],
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Cell["\<\
Now let's see how this thing works at the acid test: coming up with a \
simultaneous approximation to a ten-dimensional vector. We are beyond reach \
of the Tammela results, so the algorithm will have to do things its own way, \
finding nearest lattice points etc., in order to grind ahead. 

It doesn't seem to much matter to the algorithm what the current value of q \
is, so progress in terms of getting a new q proceeds at a reasonably steady \
pace. Thus the agonizingly slow progress initially, with many hours devoted \
to finding that q=13 gives a good approximation for example, is deceiving. \
The algorithm has got a matrix of good approximations and later q's will come \
faster, ultimately, than just checking one q after another would go. 
\
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