double computeBinWidth(const MatrixXd& positions) {
// assumes first col of positions corresponds to dominant eigenvect
ArrayXd firstCol = positions.col(0).array();
firstCol -= firstCol.mean();
double SSE = firstCol.matrix().squaredNorm();
double variance = SSE / firstCol.size();
double std = sqrt(variance);
double targetBinsPerStd = (MAX_HASH_VALUE - HASH_VALUE_OFFSET) / TARGET_HASH_SPREAD_STDS;
return std / targetBinsPerStd;
}
MatrixXd NumInt::GuassQaudrature(const int& N, double& a, double& b) {
int N0=N-1;
const int N1 = N0+1;
const int N2 = N0+2;
VectorXd xu;
xu.setLinSpaced(N1,-1.0,1.0);
// Legendre-Gauss-Vandermonde Matrix
//Matrix<double,N1,N2> L = Matrix<double,N1,N2>::Zero();
MatrixXd L(N1,N2);
L = MatrixXd::Zero(N1,N2);
// Derivative of Legendre-Gauss-Vandermonde Matrix
//Matrix<double,N1,1> Lp = Matrix<double,N1,1>::Zero();
VectorXd Lp(N1);
Lp = VectorXd::Zero(N1);
VectorXd dum;
dum.setLinSpaced(N1,0.0,N0);
ArrayXd y;
y = cos((2*dum.array()+1)*M_PI/(2*N0+2))+(0.27/N1)*sin(M_PI*xu.array()*N0/N2);
double deps = std::numeric_limits<double>::epsilon();
//Initial Guess
//Array<double,N1,1> y0 = Array<double,N1,1>::Constant(2);
ArrayXd y0 = ArrayXd::Constant(N1,2);
while ((y-y0).abs().matrix().maxCoeff() > deps) {
// L.col(0) = Matrix<double,N1,1>::Constant(1);
L.col(0) = VectorXd::Constant(N1,1);
//Lp = Matrix<double,N1,1>::Zero();
Lp = VectorXd::Zero(N1);
L.col(1) = y;
for (int k=1; k!=N1; k++)
{
L.col(k+1) = ((2*k+1)*L.col(k).cwiseProduct(y.matrix())-k*L.col(k-1))/(k+1);
}
Lp = (N2)*(L.col(N0)-L.col(N1).cwiseProduct(y.matrix())).cwiseQuotient((1-y.square()).matrix());
y0 = y;
y = y0-(L.col(N1).cwiseQuotient(Lp)).array();
}
// Gauss Points
//Matrix<double,N1,1> z = ((a*(1-y)+b*(1+y))/2).matrix();
VectorXd z(N1);
z = ((a*(1-y)+b*(1+y))/2).matrix();
// Gauss Weights
//Matrix<double,N1,1> w;
VectorXd w(N1);
w = (b-a)/(((1-y.square()).matrix()).cwiseProduct(Lp.cwiseProduct(Lp))).array()*pow((double)N2/N1,2);
// Store
//Matrix<double,N1,2> zw;
Matrix<double,Dynamic,Dynamic> zw(N1,2);
zw.col(0)=z;
zw.col(1)=w;
return zw;
}
int polyDeg(std::vector<double>& xVec,
std::vector<double>& yVec,
std::vector<double>& ysVec)
{
Map<MatrixXd> x(&xVec[0], xVec.size(), 1);
Map<MatrixXd> y(&yVec[0], yVec.size(), 1);
int N = (int)xVec.size();
double meanY = y.mean();
MatrixXd ys = MatrixXd::Constant(N,1,meanY);
// cout << "ys = " << ys << endl;
// cout << "y = " << y << endl;
double sum_ys_y_2 = (ys - y).array().square().matrix().sum();
double tobelog = 2*M_PI*sum_ys_y_2/N;
double logtobelog = log(tobelog);
double AIC = 2. + N*(logtobelog +1.) + 4./(N-2.);
// cout << "AIC= " << AIC << endl;
MatrixXd p = MatrixXd::Constant(2,2, 0);
p(0,1) = x.mean();
// cout << "AIC = " << AIC << endl;
// cout << "p = " << p << endl;
MatrixXd PL = MatrixXd::Constant(N,2, 1);
// cout << "PL = " << PL << endl;
PL.col(1) = x.array() - p(0,1);
// cout << "PL = " << PL << endl;
int n = 1;
int nit = 0;
MatrixXd ys1 = ys;
MatrixXd ys2 = ys;
MatrixXd ys3 = ys;
MatrixXd ys4 = ys;
while (nit<3) {
int ni = n - 1;
if (ni>0) {
ArrayXd x_PL = x.array() * (PL.col(ni).array().square());
double ppa = x_PL.sum();
double ppb = PL.col(ni).array().square().sum();
ArrayXd x_PL2 = x.array() * PL.col(ni-1).array() * PL.col(ni).array();
p.conservativeResize(p.rows(), p.cols()+1);
p(0,ni+1) = ppa / ppb;
p(1,ni+1) = x_PL2.sum() / PL.col(ni-1).array().square().sum();
PL.conservativeResize(PL.rows(), PL.cols()+1);
PL.col(ni+1) = (x.array() - p(0,ni+1)) * PL.col(ni).array()
- p(1,ni+1)* PL.col(ni-1).array();
//cout << "PL = " << PL << endl;
}
MatrixXd bsxfun_time_y_pl = y.asDiagonal() * PL;
MatrixXd pl_square = PL.array().square();
ArrayXd tmp = bsxfun_time_y_pl.colwise().sum().array() / pl_square.colwise().sum().array();
ys = (PL * tmp.matrix().asDiagonal()).rowwise().sum();
MatrixXd ys_minus_y = ys - y;
ArrayXd ys_minus_y_square = ys_minus_y.array().square();
sum_ys_y_2 = ys_minus_y_square.matrix().sum();
double log_v = log(2.*M_PI*sum_ys_y_2/N);
double aic = 2.*(n+1) + N*(log_v + 1.) +
2.*(n+1)*(n+2.)/(N-n-2.); // correction for small sample sizes
// cout << "aic = " << aic << endl;
if (aic>=AIC) {
++nit;
} else {
nit = 0;
AIC = aic;
}
++n;
ys1 = ys2;
ys2 = ys3;
ys3 = ys4;
ys4 = ys;
if (n>=N) {
ys1 = ys;
break;
}
}
ysVec.resize(ys1.rows());
Map<MatrixXd> dest(&ysVec[0],ys1.rows(),1);
dest = ys1;
return n - nit - 1;
}
int main() {
// The eigen approach
ArrayXd n = ArrayXd::LinSpaced(N+1,0,N);
double multiplier = M_PI/N;
Array<double, 1, N+1> nT = n.transpose();
ArrayXd x = - cos(multiplier*n);
ArrayXd xsub = x.middleRows(1, N-1);
ArrayXd ysub = (x1-x0)/2*xsub + (x1+x0)/2;
ArrayXXd T = cos((acos(x).matrix()*nT.matrix()).array());
ArrayXXd Tsub = cos((acos(xsub).matrix()*nT.matrix()).array());
ArrayXd sqinx = 1/sqrt(1-xsub*xsub);
MatrixXd inv1x2 = (sqinx).matrix().asDiagonal();
// Can't use the following to test elements of inv1x2
// std::cout << inv1x2(0,0) << "\n";
MatrixXd Usub = inv1x2 * sin(((acos(xsub).matrix())*nT.matrix()).array()).matrix();
MatrixXd dTsub = Usub*(n.matrix().asDiagonal());
MatrixXd d2Tsub = ((sqinx*sqinx).matrix().asDiagonal())*((xsub.matrix().asDiagonal()) * (dTsub.matrix()) - (Tsub.matrix()) * ((n*n).matrix().asDiagonal()));
MatrixXd d2T(N+1, N+1);
RowVectorXd a = (pow((-1),nT))*(nT*nT+1)*(nT*nT)/3;
RowVectorXd b = (nT*nT+1)*(nT*nT)/3;
d2T.middleRows(1,N-1) = d2Tsub;
d2T.row(0) = a;
d2T.row(N) = b;
MatrixXd D2 = d2T.matrix() * ((T.matrix()).inverse());
MatrixXd E2 = D2.middleRows(1,N-1).middleCols(1,N-1);
MatrixXd Y = ysub.matrix().asDiagonal();
MatrixXd H = - (4 / ((x1-x0)*(x1-x0))) * E2 + k*Y;
Eigen::EigenSolver<Eigen::MatrixXd> HE(H);
VectorXcd D = HE.eigenvalues();
MatrixXcd V = HE.eigenvectors();
std::cout << HE.info() << std::endl;
// Open ofstream
ofstream Dfile;
Dfile.open("D-output.txt");
ofstream Vfile;
Vfile.open("V-output.txt");
ofstream V544file;
V544file.open("V544-output.txt");
Dfile.precision(15);
Dfile << D.real() << "\n";
Vfile.precision(15);
Vfile << V.real() << "\n";
V544file.precision(15);
for(int i = 1; i<N-1; i++)
{
V544file << ysub[i-1];
V544file << " " << V.col(544).row(i-1).real() << "\n";
}
Dfile.close();
Vfile.close();
V544file.close();
system("gnuplot -p plot.gp");
system("rsvg-convert -w 2000 -o V544-plot.png V544-plot.svg");
}