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(* ::Package:: *)
(*Setup*)
prec = 200;
(* A matrix with constant anti-diagonals given by the list bs *)
antiBandMatrix[bs_] := Module[
{n = Ceiling[Length[bs]/2]},
Reverse[Normal[
SparseArray[
Join[
Table[Band[{i, 1}] -> bs[[n - i + 1]], {i, n}],
Table[Band[{1, i}] -> bs[[n + i - 1]], {i, 2, n}]],
{n, n}]]]];
(* DampedRational[c, {p1, p2, ...}, b, x] stands for c b^x / ((x-p1)(x-p2)...) *)
(* It satisfies the following identities *)
DampedRational[const_, poles_, base_, x + a_] :=
DampedRational[base^a const, # - a & /@ poles, base, x];
DampedRational[const_, poles_, base_, a_ /; FreeQ[a, x]] :=
const base^a/Product[a - p, {p, poles}];
DampedRational/:x DampedRational[const_, poles_ /; MemberQ[poles, 0], base_, x] :=
DampedRational[const, DeleteCases[poles, 0], base, x];
DampedRational/:DampedRational[c1_,p1_,b1_,x] DampedRational[c2_,p2_,b2_,x] :=
DampedRational[c1 c2, Join[p1, p2], b1 b2, x];
(* bilinearForm[f, m] = Integral[x^m f[x], {x, 0, Infinity}] *)
(* The special case when f[x] has no poles *)
bilinearForm[DampedRational[const_, {}, base_, x], m_] :=
const Gamma[1+m] (-Log[base])^(-1-m);
(*memoizeGamma[a_,b_]:=memoizeGamma[a,b]=Gamma[a,b];*)
(* The case where f[x] has only single poles *)
(*bilinearForm[DampedRational[const_, poles_, base_, x], m_] :=
const Sum[
((-poles[[i]])^m) ( base^poles[[i]]) Gamma[1 + m] memoizeGamma[-m, poles[[i]] Log[base]]/
Product[poles[[i]] - p, {p, Delete[poles, i]}],
{i, Length[poles]}];*)
(* The case where f[x] can have single or double poles *)
bilinearForm[DampedRational[c_, poles_, b_, x_], m_] := Module[
{
gatheredPoles = Gather[poles],
quotientCoeffs = CoefficientList[PolynomialQuotient[x^m, Product[x-p, {p, poles}], x], x],
integral, p, rest
},
integral[a_,1] := b^a Gamma[0, a Log[b]];
integral[a_,2] := -1/a + b^a Gamma[0, a Log[b]] Log[b];
c (Sum[
p = gatheredPoles[[n,1]];
rest = x^m / Product[x-q, {q, Join@@Delete[gatheredPoles, n]}];
Switch[Length[gatheredPoles[[n]]],
1, integral[p,1] rest /. x->p,
2, integral[p,2] rest + integral[p,1] D[rest, x] /. x->p],
{n, Length[gatheredPoles]}] +
Sum[
quotientCoeffs[[n+1]] Gamma[1+n] (-Log[b])^(-1-n),
{n, 0, Length[quotientCoeffs]-1}])];
(* orthogonalPolynomials[f, n] is a set of polynomials with degree 0
through n which are orthogonal with respect to the measure f[x] dx *)
orthogonalPolynomials[const_ /; FreeQ[const, x], 0] := {1/Sqrt[const]};
orthogonalPolynomials[const_ /; FreeQ[const, x], degree_] :=
error["can't get orthogonal polynomials of nonzero degree for constant measure"];
orthogonalPolynomials[DampedRational[const_, poles_, base_, x], degree_] :=
Table[x^m, {m, 0, degree}] . Inverse[
CholeskyDecomposition[
antiBandMatrix[
Table[bilinearForm[DampedRational[const, Select[poles, # < 0&], base, x], m],
{m, 0, 2 degree}]]]];
(* Preparing SDP for Export *)
rhoCrossing = SetPrecision[3-2 Sqrt[2], prec];
rescaledLaguerreSamplePoints[n_] := Table[
SetPrecision[\[Pi]^2 (-1+4k)^2/(-64n Log[rhoCrossing]), prec],
{k,0,n-1}];
maxIndexBy[l_,f_] := SortBy[
Transpose[{l,Range[Length[l]]}],
-f[First[#]]&][[1,2]];
(* finds v' such that a . v = First[v'] + a' . Rest[v'] when normalization . a == 1, where a' is a vector of length one less than a *)
reshuffleWithNormalization[normalization_, v_] := Module[
{j = maxIndexBy[normalization, Abs], const},
const = v[[j]]/normalization[[j]];
Prepend[Delete[v - normalization*const, j], const]];
(* XML Exporting *)
nf[x_Integer] := x;
nf[x_] := NumberForm[SetPrecision[x,prec],prec,ExponentFunction->(Null&)];
safeCoefficientList[p_, x_] := Module[
{coeffs = CoefficientList[p, x]},
If[Length[coeffs] > 0, coeffs, {0}]];
WriteBootstrapSDP[file_, SDP[objective_, normalization_, positiveMatricesWithPrefactors_]] := Module[
{
stream = OpenWrite[file],
node, real, int, vector, polynomial,
polynomialVector, polynomialVectorMatrix,
affineObjective, polynomialVectorMatrices
},
(* write a single XML node to file. children is a routine that writes child nodes when run. *)
node[name_, children_] := (
WriteString[stream, "<", name, ">"];
children[];
WriteString[stream, "</", name, ">\n"];
);
real[r_][] := WriteString[stream, nf[r]];
int[i_][] := WriteString[stream, i];
vector[v_][] := Do[node["elt", real[c]], {c, v}];
polynomial[p_][] := Do[node["coeff", real[c]], {c, safeCoefficientList[p,x]}];
polynomialVector[v_][] := Do[node["polynomial", polynomial[p]], {p, v}];
polynomialVectorMatrix[PositiveMatrixWithPrefactor[prefactor_, m_]][] := Module[
{degree = Max[Exponent[m, x]], samplePoints, sampleScalings, bilinearBasis},
samplePoints = rescaledLaguerreSamplePoints[degree + 1];
sampleScalings = Table[prefactor /. x -> a, {a, samplePoints}];
bilinearBasis = orthogonalPolynomials[prefactor, Floor[degree/2]];
node["rows", int[Length[m]]];
node["cols", int[Length[First[m]]]];
node["elements", Function[
{},
Do[node[
"polynomialVector",
polynomialVector[reshuffleWithNormalization[normalization,pv]]],
{row, m}, {pv, row}]]];
node["samplePoints", vector[samplePoints]];
node["sampleScalings", vector[sampleScalings]];
node["bilinearBasis", polynomialVector[bilinearBasis]];
];
node["sdp", Function[
{},
node["objective", vector[reshuffleWithNormalization[normalization, objective]]];
node["polynomialVectorMatrices", Function[
{},
Do[node["polynomialVectorMatrix", polynomialVectorMatrix[pvm]], {pvm, positiveMatricesWithPrefactors}];
]];
]];
Close[stream];
];
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