void KVFlowTensor::Calculate()
{
// Calculate eigenvalues & eigenvectors of tensor
TMatrixT<double> evectors = fTensor.EigenVectors(fEVal);
for (int i = 0; i < 3; i++) {
TVectorD col = TMatrixDColumn(evectors, i);
fEVec[i].SetXYZ(col[0], col[1], col[2]);
}
// check orientation of flow axis
// by symmetry/convention, we allow FlowAngle between 0 & 90° i.e. flow vector always
// points in the forward beam direction.
if (fEVec[0].Theta() > TMath::PiOver2()) {
fEVec[0] = -fEVec[0];
}
// check we have an orthonormal basis
// we choose evec[0]=>'z' evec[1]=>'y' evec[2]=>'x'
// therefore we should have evec[2].Cross(evec[1])=evec[0]
// if not we change the sign of evec[2]
if (fEVec[2].Cross(fEVec[1]) != fEVec[0]) fEVec[2] = -fEVec[2];
// set up azimuthal rotation of CM axes around beam axis in order to put new 'X'-axis in reaction plane
// we use the phi of the major axis (largest eigenvector)
fAziReacPlane.SetToIdentity();
fAziReacPlane.RotateZ(e(1).Phi());
fFlowReacPlane.SetToIdentity();
fFlowReacPlane.RotateY(e(1).Theta());
fFlowReacPlane.RotateZ(e(1).Phi());
// calculate Gutbrod squeeze angle (PRC42(1990)640
// defined as the angle by which the middle eigenvector e2
// needs to be rotated around the e1 flow axis
// in order to be brought in to the reaction plane
TVector3 normReac = e(1).Cross(TVector3(0, 0, 1)); //normal to reaction plane: e1 x z
Double_t angle = TMath::RadToDeg() * e(2).Angle(normReac);
// on the contrary to Gutbrod et al, we define this angle between 0 and 90 degrees.
// see Fig. 3 of paper: only 0-90 angles can be defined
fSqueezeAngle = (angle < 90 ? 90 - angle : angle - 90);
// now calculate the in-plane and out-of-plane flow according to Fig. 5 of Gutbrod et al
Double_t a = pow(f(2), 0.5); //semi-axes of ellipsoid given by square roots of eigenvalues
Double_t b = pow(f(3), 0.5);
Double_t tan_t = -a / b * TMath::Tan(fSqueezeAngle * TMath::DegToRad());
Double_t t0 = TMath::ATan(tan_t);
Double_t inPlane = a * cos(t0) * cos(fSqueezeAngle * TMath::DegToRad()) - b * sin(t0) * sin(fSqueezeAngle * TMath::DegToRad());
tan_t = a / (b * tan(fSqueezeAngle * TMath::DegToRad()));
t0 = atan(tan_t);
Double_t outOfPlane = a * cos(t0) * sin(fSqueezeAngle * TMath::DegToRad()) + b * sin(t0) * cos(fSqueezeAngle * TMath::DegToRad());
if (inPlane <= 0.) fSqOutRatio = -1.;
else fSqOutRatio = TMath::Min(1.e+03, outOfPlane / inPlane);
fCalculated = kTRUE;
}
void solveLinear(Double_t eps = 1.e-12)
{
cout << "Perform the fit y = c0 + c1 * x in four different ways" << endl;
const Int_t nrVar = 2;
const Int_t nrPnts = 4;
Double_t ax[] = {0.0,1.0,2.0,3.0};
Double_t ay[] = {1.4,1.5,3.7,4.1};
Double_t ae[] = {0.5,0.2,1.0,0.5};
// Make the vectors 'Use" the data : they are not copied, the vector data
// pointer is just set appropriately
TVectorD x; x.Use(nrPnts,ax);
TVectorD y; y.Use(nrPnts,ay);
TVectorD e; e.Use(nrPnts,ae);
TMatrixD A(nrPnts,nrVar);
TMatrixDColumn(A,0) = 1.0;
TMatrixDColumn(A,1) = x;
cout << " - 1. solve through Normal Equations" << endl;
const TVectorD c_norm = NormalEqn(A,y,e);
cout << " - 2. solve through SVD" << endl;
// numerically preferred method
// first bring the weights in place
TMatrixD Aw = A;
TVectorD yw = y;
for (Int_t irow = 0; irow < A.GetNrows(); irow++) {
TMatrixDRow(Aw,irow) *= 1/e(irow);
yw(irow) /= e(irow);
}
TDecompSVD svd(Aw);
Bool_t ok;
const TVectorD c_svd = svd.Solve(yw,ok);
cout << " - 3. solve with pseudo inverse" << endl;
const TMatrixD pseudo1 = svd.Invert();
TVectorD c_pseudo1 = yw;
c_pseudo1 *= pseudo1;
cout << " - 4. solve with pseudo inverse, calculated brute force" << endl;
TMatrixDSym AtA(TMatrixDSym::kAtA,Aw);
const TMatrixD pseudo2 = AtA.Invert() * Aw.T();
TVectorD c_pseudo2 = yw;
c_pseudo2 *= pseudo2;
cout << " - 5. Minuit through TGraph" << endl;
TGraphErrors *gr = new TGraphErrors(nrPnts,ax,ay,0,ae);
TF1 *f1 = new TF1("f1","pol1",0,5);
gr->Fit("f1","Q");
TVectorD c_graph(nrVar);
c_graph(0) = f1->GetParameter(0);
c_graph(1) = f1->GetParameter(1);
// Check that all 4 answers are identical within a certain
// tolerance . The 1e-12 is somewhat arbitrary . It turns out that
// the TGraph fit is different by a few times 1e-13.
Bool_t same = kTRUE;
same &= VerifyVectorIdentity(c_norm,c_svd,0,eps);
same &= VerifyVectorIdentity(c_norm,c_pseudo1,0,eps);
same &= VerifyVectorIdentity(c_norm,c_pseudo2,0,eps);
same &= VerifyVectorIdentity(c_norm,c_graph,0,eps);
if (same)
cout << " All solutions are the same within tolerance of " << eps << endl;
else
cout << " Some solutions differ more than the allowed tolerance of " << eps << endl;
}
