source: molecuilder/src/vector.cpp@ 3ce105

/** \file vector.cpp
 * 
 * Function implementations for the class vector.
 * 
 */
 
#include "molecules.hpp"
 

/************************************ Functions for class vector ************************************/

/** Constructor of class vector.
 */
vector::vector() { x[0] = x[1] = x[2] = 0.; };

/** Desctructor of class vector.
 */
vector::~vector() {};

/** Calculates distance between this and another vector.
 * \param *y array to second vector
 * \return \f$| x - y |^2\f$
 */
double vector::Distance(const vector *y) const
{
  double res = 0.;
  for (int i=0;i<NDIM;i++)
    res += (x[i]-y->x[i])*(x[i]-y->x[i]);
  return (res);  
};

/** Calculates distance between this and another vector in a periodic cell.
 * \param *y array to second vector
 * \param *cell_size 6-dimensional array with (xx, xy, yy, xz, yz, zz) entries specifying the periodic cell
 * \return \f$| x - y |^2\f$
 */
double vector::PeriodicDistance(const vector *y, const double *cell_size) const
{
  double res = Distance(y), tmp, matrix[NDIM*NDIM];
  vector Shiftedy, TranslationVector;
  int N[NDIM];
  matrix[0] = cell_size[0];
  matrix[1] = cell_size[1];
  matrix[2] = cell_size[3];
  matrix[3] = cell_size[1];
  matrix[4] = cell_size[2];
  matrix[5] = cell_size[4];
  matrix[6] = cell_size[3];
  matrix[7] = cell_size[4];
  matrix[8] = cell_size[5];
  // in order to check the periodic distance, translate one of the vectors into each of the 27 neighbouring cells
  for (N[0]=-1;N[0]<=1;N[0]++)
    for (N[1]=-1;N[1]<=1;N[1]++)
      for (N[2]=-1;N[2]<=1;N[2]++) {
        // create the translation vector
        TranslationVector.Zero();
        for (int i=0;i<NDIM;i++)
          TranslationVector.x[i] = (double)N[i];
        TranslationVector.MatrixMultiplication(matrix);
        // add onto the original vector to compare with
        Shiftedy.CopyVector(y);
        Shiftedy.AddVector(&TranslationVector);
        // get distance and compare with minimum so far
        tmp = Distance(&Shiftedy);
        if (tmp < res) res = tmp;
      }
  return (res);  
};

/** Keeps the vector in a periodic cell, defined by the symmetric \a *matrix.
 * \param *out ofstream for debugging messages
 * Tries to translate a vector into each adjacent neighbouring cell.
 */
void vector::KeepPeriodic(ofstream *out, double *matrix)
{
//  int N[NDIM];
//  bool flag = false;
  //vector Shifted, TranslationVector;
  vector TestVector;
//  *out << Verbose(1) << "Begin of KeepPeriodic." << endl;
//  *out << Verbose(2) << "Vector is: ";
//  Output(out);
//  *out << endl;
  TestVector.CopyVector(this);
  TestVector.InverseMatrixMultiplication(matrix);
  for(int i=0;i<NDIM;i++) { // correct periodically
    if (TestVector.x[i] < 0) {  // get every coefficient into the interval [0,1)
      TestVector.x[i] += ceil(TestVector.x[i]);
    } else {
      TestVector.x[i] -= floor(TestVector.x[i]);
    }
  }
  TestVector.MatrixMultiplication(matrix);
  CopyVector(&TestVector);
//  *out << Verbose(2) << "New corrected vector is: ";
//  Output(out);
//  *out << endl;
//  *out << Verbose(1) << "End of KeepPeriodic." << endl;
};

/** Calculates scalar product between this and another vector.
 * \param *y array to second vector
 * \return \f$\langle x, y \rangle\f$
 */
double vector::ScalarProduct(const vector *y) const
{
  double res = 0.;
  for (int i=0;i<NDIM;i++)
    res += x[i]*y->x[i];
  return (res);  
};

/** Calculates the projection of a vector onto another \a *y.
 * \param *y array to second vector
 * \return \f$\langle x, y \rangle\f$
 */
double vector::Projection(const vector *y) const
{
  return (ScalarProduct(y)/Norm()/y->Norm());
};

/** Calculates norm of this vector.
 * \return \f$|x|\f$
 */
double vector::Norm() const
{
  double res = 0.;
  for (int i=0;i<NDIM;i++)
    res += this->x[i]*this->x[i];
  return (sqrt(res));  
};

/** Normalizes this vector.
 */
void vector::Normalize()
{
  double res = 0.;
  for (int i=0;i<NDIM;i++)
    res += this->x[i]*this->x[i];
  res = 1./sqrt(res);
  Scale(&res);
};

/** Zeros all components of this vector.
 */
void vector::Zero()
{
  for (int i=0;i<NDIM;i++)
    this->x[i] = 0.;
};

/** Calculates the angle between this and another vector.
 * \param *y array to second vector
 * \return \f$\frac{\langle x, y \rangle}{|x||y|}\f$
 */
double vector::Angle(vector *y) const
{
  return (this->ScalarProduct(y)/(this->Norm()*y->Norm()));
};

/** Rotates the vector around the axis given by \a *axis by an angle of \a alpha.
 * \param *axis rotation axis
 * \param alpha rotation angle in radian
 */
void vector::RotateVector(const vector *axis, const double alpha)
{
  vector a,y;
  // normalise this vector with respect to axis
  a.CopyVector(this);
  a.Scale(Projection(axis));
  SubtractVector(&a);
  // construct normal vector
  y.MakeNormalVector(axis,this);
  y.Scale(Norm());
  // scale normal vector by sine and this vector by cosine
  y.Scale(sin(alpha));
  Scale(cos(alpha));
  // add scaled normal vector onto this vector
  AddVector(&y);
  // add part in axis direction
  AddVector(&a);
};

/** Sums vector \a to this lhs component-wise.
 * \param a base vector
 * \param b vector components to add
 * \return lhs + a
 */
vector& operator+=(vector& a, const vector& b)
{
  a.AddVector(&b);
  return a;
};
/** factor each component of \a a times a double \a m.
 * \param a base vector
 * \param m factor
 * \return lhs.x[i] * m
 */
vector& operator*=(vector& a, const double m)
{
  a.Scale(m);
  return a;
};

/** Sums two vectors \a  and \b component-wise.
 * \param a first vector 
 * \param b second vector
 * \return a + b
 */
vector& operator+(const vector& a, const vector& b)
{
  vector *x = new vector;
  x->CopyVector(&a);
  x->AddVector(&b);
  return *x;
};

/** Factors given vector \a a times \a m.
 * \param a vector 
 * \param m factor
 * \return a + b
 */
vector& operator*(const vector& a, const double m)
{
  vector *x = new vector;
  x->CopyVector(&a);
  x->Scale(m);
  return *x;
};

/** Prints a 3dim vector.
 * prints no end of line.
 * \param *out output stream
 */
bool vector::Output(ofstream *out) const
{
  if (out != NULL) {
    *out << "(";
    for (int i=0;i<NDIM;i++) {
      *out << x[i];
      if (i != 2)
        *out << ",";
    }
    *out << ")";
    return true;
  } else
    return false;
};

ofstream& operator<<(ofstream& ost,vector& m)
{
        m.Output(&ost);
        return ost;
};

/** Scales each atom coordinate by an individual \a factor.
 * \param *factor pointer to scaling factor
 */
void vector::Scale(double **factor)
{
  for (int i=0;i<NDIM;i++)
    x[i] *= (*factor)[i];
};

void vector::Scale(double *factor)
{
  for (int i=0;i<NDIM;i++)
    x[i] *= *factor;
};

void vector::Scale(double factor)
{
  for (int i=0;i<NDIM;i++)
    x[i] *= factor;
};

/** Translate atom by given vector. 
 * \param trans[] translation vector.
 */
void vector::Translate(const vector *trans)
{
  for (int i=0;i<NDIM;i++)
    x[i] += trans->x[i];
}; 

/** Do a matrix multiplication.
 * \param *matrix NDIM_NDIM array
 */
void vector::MatrixMultiplication(double *M)
{
  vector C;
  // do the matrix multiplication
  C.x[0] = M[0]*x[0]+M[3]*x[1]+M[6]*x[2];
  C.x[1] = M[1]*x[0]+M[4]*x[1]+M[7]*x[2];
  C.x[2] = M[2]*x[0]+M[5]*x[1]+M[8]*x[2];
  // transfer the result into this
  for (int i=0;i<NDIM;i++)
    x[i] = C.x[i];
};

/** Do a matrix multiplication with \a *matrix' inverse.
 * \param *matrix NDIM_NDIM array
 */
void vector::InverseMatrixMultiplication(double *A)
{
  vector C;
  double B[NDIM*NDIM];
  double detA = RDET3(A);
  double detAReci;

  // calculate the inverse B
  if (fabs(detA) > MYEPSILON) {;  // RDET3(A) yields precisely zero if A irregular
    detAReci = 1./detA;
    B[0] =  detAReci*RDET2(A[4],A[5],A[7],A[8]);    // A_11
    B[1] = -detAReci*RDET2(A[1],A[2],A[7],A[8]);    // A_12
    B[2] =  detAReci*RDET2(A[1],A[2],A[4],A[5]);    // A_13
    B[3] = -detAReci*RDET2(A[3],A[5],A[6],A[8]);    // A_21
    B[4] =  detAReci*RDET2(A[0],A[2],A[6],A[8]);    // A_22
    B[5] = -detAReci*RDET2(A[0],A[2],A[3],A[5]);    // A_23
    B[6] =  detAReci*RDET2(A[3],A[4],A[6],A[7]);    // A_31
    B[7] = -detAReci*RDET2(A[0],A[1],A[6],A[7]);    // A_32
    B[8] =  detAReci*RDET2(A[0],A[1],A[3],A[4]);    // A_33
  
    // do the matrix multiplication
    C.x[0] = B[0]*x[0]+B[3]*x[1]+B[6]*x[2];
    C.x[1] = B[1]*x[0]+B[4]*x[1]+B[7]*x[2];
    C.x[2] = B[2]*x[0]+B[5]*x[1]+B[8]*x[2];
    // transfer the result into this
    for (int i=0;i<NDIM;i++)
      x[i] = C.x[i];
  } else {
    cerr << "ERROR: inverse of matrix does not exists!" << endl;
  }
};


/** Creates this vector as the b y *factors' components scaled linear combination of the given three.
 * this vector = x1*factors[0] + x2* factors[1] + x3*factors[2]
 * \param *x1 first vector
 * \param *x2 second vector
 * \param *x3 third vector
 * \param *factors three-component vector with the factor for each given vector
 */
void vector::LinearCombinationOfVectors(const vector *x1, const vector *x2, const vector *x3, double *factors)
{
  for(int i=0;i<NDIM;i++)
    x[i] = factors[0]*x1->x[i] + factors[1]*x2->x[i] + factors[2]*x3->x[i]; 
};

/** Mirrors atom against a given plane. 
 * \param n[] normal vector of mirror plane.
 */
void vector::Mirror(const vector *n)
{
  double projection;
  projection = ScalarProduct(n)/((vector *)n)->ScalarProduct(n);    // remove constancy from n (keep as logical one)
  // withdraw projected vector twice from original one
  cout << Verbose(1) << "Vector: ";
  Output((ofstream *)&cout);
  cout << "\t";
  for (int i=0;i<NDIM;i++)
    x[i] -= 2.*projection*n->x[i];
  cout << "Projected vector: ";
  Output((ofstream *)&cout);
  cout << endl;
}; 

/** Calculates normal vector for three given vectors (being three points in space).
 * Makes this vector orthonormal to the three given points, making up a place in 3d space.
 * \param *y1 first vector
 * \param *y2 second vector
 * \param *y3 third vector
 * \return true - success, vectors are linear independent, false - failure due to linear dependency
 */
bool vector::MakeNormalVector(const vector *y1, const vector *y2, const vector *y3)
{
  vector x1, x2;

  x1.CopyVector(y1);
  x1.SubtractVector(y2);
  x2.CopyVector(y3);
  x2.SubtractVector(y2);
  if ((x1.Norm()==0) || (x2.Norm()==0)) {
    cout << Verbose(4) << "Given vectors are linear dependent." << endl;
    return false;
  }
  cout << Verbose(4) << "relative, first plane coordinates:";
  x1.Output((ofstream *)&cout);
  cout << endl;
  cout << Verbose(4) << "second plane coordinates:";
  x2.Output((ofstream *)&cout);
  cout << endl;

  this->x[0] = (x1.x[1]*x2.x[2] - x1.x[2]*x2.x[1]);
  this->x[1] = (x1.x[2]*x2.x[0] - x1.x[0]*x2.x[2]);
  this->x[2] = (x1.x[0]*x2.x[1] - x1.x[1]*x2.x[0]);
  Normalize();
  
  return true;
};


/** Calculates orthonormal vector to two given vectors.
 * Makes this vector orthonormal to two given vectors. This is very similar to the other
 * vector::MakeNormalVector(), only there three points whereas here two difference
 * vectors are given.
 * \param *x1 first vector
 * \param *x2 second vector
 * \return true - success, vectors are linear independent, false - failure due to linear dependency
 */
bool vector::MakeNormalVector(const vector *y1, const vector *y2)
{
  vector x1,x2;
  x1.CopyVector(y1);
  x2.CopyVector(y2);
  Zero();
  if ((x1.Norm()==0) || (x2.Norm()==0)) {
    cout << Verbose(4) << "Given vectors are linear dependent." << endl;
    return false;
  }
  cout << Verbose(4) << "relative, first plane coordinates:";
  x1.Output((ofstream *)&cout);
  cout << endl;
  cout << Verbose(4) << "second plane coordinates:";
  x2.Output((ofstream *)&cout);
  cout << endl;

  this->x[0] = (x1.x[1]*x2.x[2] - x1.x[2]*x2.x[1]);
  this->x[1] = (x1.x[2]*x2.x[0] - x1.x[0]*x2.x[2]);
  this->x[2] = (x1.x[0]*x2.x[1] - x1.x[1]*x2.x[0]);
  Normalize();

  return true;
};

/** Calculates orthonormal vector to one given vectors.
 * Just subtracts the projection onto the given vector from this vector.
 * \param *x1 vector
 * \return true - success, false - vector is zero
 */
bool vector::MakeNormalVector(const vector *y1)
{
        bool result = false;
  vector x1;
  x1.CopyVector(y1);
  x1.Scale(x1.Projection(this));
  SubtractVector(&x1);
  for (int i=0;i<NDIM;i++)
          result = result || (fabs(x[i]) > MYEPSILON);

  return result;
};

/** Creates this vector as one of the possible orthonormal ones to the given one.
 * Just scan how many components of given *vector are unequal to zero and
 * try to get the skp of both to be zero accordingly.  
 * \param *vector given vector
 * \return true - success, false - failure (null vector given)
 */
bool vector::GetOneNormalVector(const vector *vector)
{
  int Components[NDIM]; // contains indices of non-zero components
  int Last = 0;   // count the number of non-zero entries in vector
  int j;  // loop variables
  double norm;

  cout << Verbose(4);
  vector->Output((ofstream *)&cout);
  cout << endl;
  for (j=0;j<NDIM;j++)
    Components[j] = -1;
  // find two components != 0
  for (j=0;j<NDIM;j++)
    if (fabs(vector->x[j]) > MYEPSILON)
      Components[Last++] = j;
  cout << Verbose(4) << Last << " Components != 0: (" << Components[0] << "," << Components[1] << "," << Components[2] << ")" << endl;
        
  switch(Last) {
    case 3:  // threecomponent system
    case 2:  // two component system
      norm = sqrt(1./(vector->x[Components[1]]*vector->x[Components[1]]) + 1./(vector->x[Components[0]]*vector->x[Components[0]]));
      x[Components[2]] = 0.;
      // in skp both remaining parts shall become zero but with opposite sign and third is zero
      x[Components[1]] = -1./vector->x[Components[1]] / norm;
      x[Components[0]] = 1./vector->x[Components[0]] / norm;
      return true;
      break;
    case 1: // one component system
      // set sole non-zero component to 0, and one of the other zero component pendants to 1
      x[(Components[0]+2)%NDIM] = 0.;   
      x[(Components[0]+1)%NDIM] = 1.;   
      x[Components[0]] = 0.;   
      return true;
      break;
    default:
      return false;
  }
};

/** Creates a new vector as the one with least square distance to a given set of \a vectors.
 * \param *vectors set of vectors
 * \param num number of vectors
 * \return true if success, false if failed due to linear dependency
 */
bool vector::LSQdistance(vector **vectors, int num) 
{
        int j;
                                
        for (j=0;j<num;j++) {
                cout << Verbose(1) << j << "th atom's vector: ";
                (vectors[j])->Output((ofstream *)&cout);
                cout << endl;
        }

  int np = 3;
        struct LSQ_params par;
    
   const gsl_multimin_fminimizer_type *T =
     gsl_multimin_fminimizer_nmsimplex;
   gsl_multimin_fminimizer *s = NULL;
   gsl_vector *ss, *x;
   gsl_multimin_function minex_func;
 
   size_t iter = 0, i;
   int status;
   double size;
 
   /* Initial vertex size vector */
   ss = gsl_vector_alloc (np);
   x = gsl_vector_alloc (np);
 
   /* Set all step sizes to 1 */
   gsl_vector_set_all (ss, 1.0);
 
   /* Starting point */
   par.vectors = vectors;
         par.num = num;
        
         for (i=0;i<NDIM;i++)
                gsl_vector_set(x, i, (vectors[0]->x[i] - vectors[1]->x[i])/2.); 
         
   /* Initialize method and iterate */
   minex_func.f = &LSQ;
   minex_func.n = np;
   minex_func.params = (void *)&par;
 
   s = gsl_multimin_fminimizer_alloc (T, np);
   gsl_multimin_fminimizer_set (s, &minex_func, x, ss);
 
   do
     {
       iter++;
       status = gsl_multimin_fminimizer_iterate(s);
 
       if (status)
         break;
 
       size = gsl_multimin_fminimizer_size (s);
       status = gsl_multimin_test_size (size, 1e-2);
 
       if (status == GSL_SUCCESS)
         {
           printf ("converged to minimum at\n");
         }
 
       printf ("%5d ", (int)iter);
       for (i = 0; i < (size_t)np; i++)
         {
           printf ("%10.3e ", gsl_vector_get (s->x, i));
         }
       printf ("f() = %7.3f size = %.3f\n", s->fval, size);
     }
   while (status == GSL_CONTINUE && iter < 100);
 
  for (i=0;i<(size_t)np;i++)
    this->x[i] = gsl_vector_get(s->x, i);
   gsl_vector_free(x);
   gsl_vector_free(ss);
   gsl_multimin_fminimizer_free (s);

        return true;
};

/** Adds vector \a *y componentwise.
 * \param *y vector
 */
void vector::AddVector(const vector *y)
{
  for (int i=0;i<NDIM;i++)
    this->x[i] += y->x[i];
}

/** Adds vector \a *y componentwise.
 * \param *y vector
 */
void vector::SubtractVector(const vector *y)
{
  for (int i=0;i<NDIM;i++)
    this->x[i] -= y->x[i];
}

/** Copy vector \a *y componentwise.
 * \param *y vector
 */
void vector::CopyVector(const vector *y)
{
  for (int i=0;i<NDIM;i++)
    this->x[i] = y->x[i];
}


/** Asks for position, checks for boundary.
 * \param cell_size unitary size of cubic cell, coordinates must be within 0...cell_size
 * \param check whether bounds shall be checked (true) or not (false)
 */
void vector::AskPosition(double *cell_size, bool check)
{
  char coords[3] = {'x','y','z'};
  int j = -1;
  for (int i=0;i<3;i++) {
    j += i+1;
    do {
      cout << Verbose(0) << coords[i] << "[0.." << cell_size[j] << "]: ";
      cin >> x[i];
    } while (((x[i] < 0) || (x[i] >= cell_size[j])) && (check));
  }
};

/** Solves a vectorial system consisting of two orthogonal statements and a norm statement.
 * This is linear system of equations to be solved, however of the three given (skp of this vector\
 * with either of the three hast to be zero) only two are linear independent. The third equation
 * is that the vector should be of magnitude 1 (orthonormal). This all leads to a case-based solution
 * where very often it has to be checked whether a certain value is zero or not and thus forked into
 * another case.
 * \param *x1 first vector
 * \param *x2 second vector
 * \param *y third vector
 * \param alpha first angle
 * \param beta second angle
 * \param c norm of final vector 
 * \return a vector with \f$\langle x1,x2 \rangle=A\f$, \f$\langle x1,y \rangle = B\f$ and with norm \a c.
 * \bug this is not yet working properly 
 */
bool vector::SolveSystem(vector *x1, vector *x2, vector *y, double alpha, double beta, double c)
{
  double D1,D2,D3,E1,E2,F1,F2,F3,p,q=0., A, B1, B2, C;
  double ang; // angle on testing
  double sign[3];
  int i,j,k;
  A = cos(alpha) * x1->Norm() * c;
  B1 = cos(beta + M_PI/2.) * y->Norm() * c;
  B2 = cos(beta) * x2->Norm() * c;
  C = c * c;
  cout << Verbose(2) << "A " << A << "\tB " << B1 << "\tC " << C << endl;
  int flag = 0;
  if (fabs(x1->x[0]) < MYEPSILON) { // check for zero components for the later flipping and back-flipping
    if (fabs(x1->x[1]) > MYEPSILON) {
      flag = 1;    
    } else if (fabs(x1->x[2]) > MYEPSILON) {
       flag = 2;
    } else {
      return false;
    }
  }
  switch (flag) {
    default:
    case 0:
      break;
    case 2:
      flip(&x1->x[0],&x1->x[1]);
      flip(&x2->x[0],&x2->x[1]);
      flip(&y->x[0],&y->x[1]);
      //flip(&x[0],&x[1]);
      flip(&x1->x[1],&x1->x[2]);
      flip(&x2->x[1],&x2->x[2]);
      flip(&y->x[1],&y->x[2]);
      //flip(&x[1],&x[2]);
    case 1:
      flip(&x1->x[0],&x1->x[1]);
      flip(&x2->x[0],&x2->x[1]);
      flip(&y->x[0],&y->x[1]);
      //flip(&x[0],&x[1]);
      flip(&x1->x[1],&x1->x[2]);
      flip(&x2->x[1],&x2->x[2]);
      flip(&y->x[1],&y->x[2]);
      //flip(&x[1],&x[2]);
      break;
  }
  // now comes the case system
  D1 = -y->x[0]/x1->x[0]*x1->x[1]+y->x[1];
  D2 = -y->x[0]/x1->x[0]*x1->x[2]+y->x[2]; 
  D3 = y->x[0]/x1->x[0]*A-B1;
  cout << Verbose(2) << "D1 " << D1 << "\tD2 " << D2 << "\tD3 " << D3 << "\n";
  if (fabs(D1) < MYEPSILON) {
    cout << Verbose(2) << "D1 == 0!\n"; 
    if (fabs(D2) > MYEPSILON) {
      cout << Verbose(3) << "D2 != 0!\n"; 
      x[2] = -D3/D2;
      E1 = A/x1->x[0] + x1->x[2]/x1->x[0]*D3/D2;
      E2 = -x1->x[1]/x1->x[0];
      cout << Verbose(3) << "E1 " << E1 << "\tE2 " << E2 << "\n";
      F1 = E1*E1 + 1.;
      F2 = -E1*E2;
      F3 = E1*E1 + D3*D3/(D2*D2) - C;
      cout << Verbose(3) << "F1 " << F1 << "\tF2 " << F2 << "\tF3 " << F3 << "\n";
      if (fabs(F1) < MYEPSILON) {
        cout << Verbose(4) << "F1 == 0!\n"; 
        cout << Verbose(4) << "Gleichungssystem linear\n";
        x[1] = F3/(2.*F2); 
      } else {
        p = F2/F1;
        q = p*p - F3/F1;
        cout << Verbose(4) << "p " << p << "\tq " << q << endl;  
        if (q < 0) {
          cout << Verbose(4) << "q < 0" << endl;
          return false;
        }
        x[1] = p + sqrt(q);
      }
      x[0] =  A/x1->x[0] - x1->x[1]/x1->x[0]*x[1] + x1->x[2]/x1->x[0]*x[2];
    } else {
      cout << Verbose(2) << "Gleichungssystem unterbestimmt\n";
      return false;
    }
  } else {
    E1 = A/x1->x[0]+x1->x[1]/x1->x[0]*D3/D1;
    E2 = x1->x[1]/x1->x[0]*D2/D1 - x1->x[2];
    cout << Verbose(2) << "E1 " << E1 << "\tE2 " << E2 << "\n";
    F1 = E2*E2 + D2*D2/(D1*D1) + 1.;
    F2 = -(E1*E2 + D2*D3/(D1*D1));
    F3 = E1*E1 + D3*D3/(D1*D1) - C;
    cout << Verbose(2) << "F1 " << F1 << "\tF2 " << F2 << "\tF3 " << F3 << "\n";
    if (fabs(F1) < MYEPSILON) {
      cout << Verbose(3) << "F1 == 0!\n"; 
      cout << Verbose(3) << "Gleichungssystem linear\n";
      x[2] = F3/(2.*F2);     
    } else {
      p = F2/F1;
      q = p*p - F3/F1;
      cout << Verbose(3) << "p " << p << "\tq " << q << endl;  
      if (q < 0) {
        cout << Verbose(3) << "q < 0" << endl;
        return false;
      }
      x[2] = p + sqrt(q);
    }
    x[1] = (-D2 * x[2] - D3)/D1;
    x[0] = A/x1->x[0] - x1->x[1]/x1->x[0]*x[1] + x1->x[2]/x1->x[0]*x[2];
  }
  switch (flag) { // back-flipping
    default:
    case 0:
      break;
    case 2:
      flip(&x1->x[0],&x1->x[1]);
      flip(&x2->x[0],&x2->x[1]);
      flip(&y->x[0],&y->x[1]);
      flip(&x[0],&x[1]);
      flip(&x1->x[1],&x1->x[2]);
      flip(&x2->x[1],&x2->x[2]);
      flip(&y->x[1],&y->x[2]);
      flip(&x[1],&x[2]);
    case 1:
      flip(&x1->x[0],&x1->x[1]);
      flip(&x2->x[0],&x2->x[1]);
      flip(&y->x[0],&y->x[1]);
      //flip(&x[0],&x[1]);
      flip(&x1->x[1],&x1->x[2]);
      flip(&x2->x[1],&x2->x[2]);
      flip(&y->x[1],&y->x[2]);
      flip(&x[1],&x[2]);
      break;
  }
  // one z component is only determined by its radius (without sign)
  // thus check eight possible sign flips and determine by checking angle with second vector
  for (i=0;i<8;i++) {
    // set sign vector accordingly
    for (j=2;j>=0;j--) {
      k = (i & pot(2,j)) << j;
      cout << Verbose(2) << "k " << k << "\tpot(2,j) " << pot(2,j) << endl;
      sign[j] = (k == 0) ? 1. : -1.;
    }
    cout << Verbose(2) << i << ": sign matrix is " << sign[0] << "\t" << sign[1] << "\t" << sign[2] << "\n";
    // apply sign matrix 
    for (j=0;j<NDIM;j++)
      x[j] *= sign[j];
    // calculate angle and check
    ang = x2->Angle (this);
    cout << Verbose(1) << i << "th angle " << ang << "\tbeta " << cos(beta) << " :\t";
    if (fabs(ang - cos(beta)) < MYEPSILON) {  
      break;
    }
    // unapply sign matrix (is its own inverse)
    for (j=0;j<NDIM;j++)
      x[j] *= sign[j];
  }
  return true;
};