source: src/vector.cpp@ 273382

/** \file vector.cpp
 *
 * Function implementations for the class vector.
 *
 */


#include "defs.hpp"
#include "gslmatrix.hpp"
#include "leastsquaremin.hpp"
#include "memoryallocator.hpp"
#include "vector.hpp"
#include "Helpers/fast_functions.hpp"
#include "Helpers/Assert.hpp"
#include "Plane.hpp"
#include "Exceptions/LinearDependenceException.hpp"

#include <gsl/gsl_linalg.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_permutation.h>
#include <gsl/gsl_vector.h>

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

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

/** Constructor of class vector.
 */
Vector::Vector(const double x1, const double x2, const double x3)
{
  x[0] = x1;
  x[1] = x2;
  x[2] = x3;
};

/**
 * Copy constructor
 */
Vector::Vector(const Vector& src)
{
  x[0] = src[0];
  x[1] = src[1];
  x[2] = src[2];
}

/**
 * Assignment operator
 */
Vector& Vector::operator=(const Vector& src){
  // check for self assignment
  if(&src!=this){
    x[0] = src[0];
    x[1] = src[1];
    x[2] = src[2];
  }
  return *this;
}

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

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

/** Calculates distance between this and another vector.
 * \param *y array to second vector
 * \return \f$| x - y |\f$
 */
double Vector::Distance(const Vector &y) const
{
  return (sqrt(DistanceSquared(y)));
};

/** 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$| x - y |\f$
 */
double Vector::PeriodicDistance(const Vector &y, const double * const 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=NDIM;i--;)
          TranslationVector.x[i] = (double)N[i];
        TranslationVector.MatrixMultiplication(matrix);
        // add onto the original vector to compare with
        Shiftedy = y + TranslationVector;
        // get distance and compare with minimum so far
        tmp = Distance(Shiftedy);
        if (tmp < res) res = tmp;
      }
  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::PeriodicDistanceSquared(const Vector &y, const double * const cell_size) const
{
  double res = DistanceSquared(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=NDIM;i--;)
          TranslationVector.x[i] = (double)N[i];
        TranslationVector.MatrixMultiplication(matrix);
        // add onto the original vector to compare with
        Shiftedy = y + TranslationVector;
        // get distance and compare with minimum so far
        tmp = DistanceSquared(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(const double * const matrix)
{
//  int N[NDIM];
//  bool flag = false;
  //vector Shifted, TranslationVector;
  Vector TestVector;
//  Log() << Verbose(1) << "Begin of KeepPeriodic." << endl;
//  Log() << Verbose(2) << "Vector is: ";
//  Output(out);
//  Log() << Verbose(0) << endl;
  TestVector = (*this);
  TestVector.InverseMatrixMultiplication(matrix);
  for(int 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);
  (*this) = TestVector;
//  Log() << Verbose(2) << "New corrected vector is: ";
//  Output(out);
//  Log() << Verbose(0) << endl;
//  Log() << 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=NDIM;i--;)
    res += x[i]*y[i];
  return (res);
};


/** Calculates VectorProduct between this and another vector.
 *  -# returns the Product in place of vector from which it was initiated
 *  -# ATTENTION: Only three dim.
 *  \param *y array to vector with which to calculate crossproduct
 *  \return \f$ x \times y \f&
 */
void Vector::VectorProduct(const Vector &y)
{
  Vector tmp;
  tmp[0] = x[1]* (y[2]) - x[2]* (y[1]);
  tmp[1] = x[2]* (y[0]) - x[0]* (y[2]);
  tmp[2] = x[0]* (y[1]) - x[1]* (y[0]);
  (*this) = tmp;
};


/** projects this vector onto plane defined by \a *y.
 * \param *y normal vector of plane
 * \return \f$\langle x, y \rangle\f$
 */
void Vector::ProjectOntoPlane(const Vector &y)
{
  Vector tmp = y;
  tmp.Normalize();
  tmp *= ScalarProduct(tmp);
  (*this) -= tmp;
};

/** Calculates the minimum distance of this vector to the plane.
 * \param *out output stream for debugging
 * \param *PlaneNormal normal of plane
 * \param *PlaneOffset offset of plane
 * \return distance to plane
 */
double Vector::DistanceToPlane(const Vector &PlaneNormal, const Vector &PlaneOffset) const
{
  // first create part that is orthonormal to PlaneNormal with withdraw
  Vector temp = (*this) - PlaneOffset;
  temp.MakeNormalTo(PlaneNormal);
  temp *= -1.;
  // then add connecting vector from plane to point
  temp += (*this);
  temp -= PlaneOffset;
  double sign = temp.ScalarProduct(PlaneNormal);
  if (fabs(sign) > MYEPSILON)
    sign /= fabs(sign);
  else
    sign = 0.;

  return (temp.Norm()*sign);
};

/** Calculates the projection of a vector onto another \a *y.
 * \param *y array to second vector
 */
void Vector::ProjectIt(const Vector &y)
{
  Vector helper = y;
  helper.Scale(-(ScalarProduct(y)));
  AddVector(helper);
};

/** Calculates the projection of a vector onto another \a *y.
 * \param *y array to second vector
 * \return Vector
 */
Vector Vector::Projection(const Vector &y) const
{
  Vector helper = y;
  helper.Scale((ScalarProduct(y)/y.NormSquared()));

  return helper;
};

/** Calculates norm of this vector.
 * \return \f$|x|\f$
 */
double Vector::Norm() const
{
  return (sqrt(NormSquared()));
};

/** Calculates squared norm of this vector.
 * \return \f$|x|^2\f$
 */
double Vector::NormSquared() const
{
  return (ScalarProduct(*this));
};

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

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

/** Zeros all components of this vector.
 */
void Vector::One(const double one)
{
  for (int i=NDIM;i--;)
    this->x[i] = one;
};

/** Initialises all components of this vector.
 */
void Vector::Init(const double x1, const double x2, const double x3)
{
  x[0] = x1;
  x[1] = x2;
  x[2] = x3;
};

/** Checks whether vector has all components zero.
 * @return true - vector is zero, false - vector is not
 */
bool Vector::IsZero() const
{
  return (fabs(x[0])+fabs(x[1])+fabs(x[2]) < MYEPSILON);
};

/** Checks whether vector has length of 1.
 * @return true - vector is normalized, false - vector is not
 */
bool Vector::IsOne() const
{
  return (fabs(Norm() - 1.) < MYEPSILON);
};

/** Checks whether vector is normal to \a *normal.
 * @return true - vector is normalized, false - vector is not
 */
bool Vector::IsNormalTo(const Vector &normal) const
{
  if (ScalarProduct(normal) < MYEPSILON)
    return true;
  else
    return false;
};

/** Checks whether vector is normal to \a *normal.
 * @return true - vector is normalized, false - vector is not
 */
bool Vector::IsEqualTo(const Vector &a) const
{
  bool status = true;
  for (int i=0;i<NDIM;i++) {
    if (fabs(x[i] - a[i]) > MYEPSILON)
      status = false;
  }
  return status;
};

/** Calculates the angle between this and another vector.
 * \param *y array to second vector
 * \return \f$\acos\bigl(frac{\langle x, y \rangle}{|x||y|}\bigr)\f$
 */
double Vector::Angle(const Vector &y) const
{
  double norm1 = Norm(), norm2 = y.Norm();
  double angle = -1;
  if ((fabs(norm1) > MYEPSILON) && (fabs(norm2) > MYEPSILON))
    angle = this->ScalarProduct(y)/norm1/norm2;
  // -1-MYEPSILON occured due to numerical imprecision, catch ...
  //Log() << Verbose(2) << "INFO: acos(-1) = " << acos(-1) << ", acos(-1+MYEPSILON) = " << acos(-1+MYEPSILON) << ", acos(-1-MYEPSILON) = " << acos(-1-MYEPSILON) << "." << endl;
  if (angle < -1)
    angle = -1;
  if (angle > 1)
    angle = 1;
  return acos(angle);
};


double& Vector::operator[](size_t i){
  ASSERT(i<=NDIM && i>=0,"Vector Index out of Range");
  return x[i];
}

const double& Vector::operator[](size_t i) const{
  ASSERT(i<=NDIM && i>=0,"Vector Index out of Range");
  return x[i];
}

double& Vector::at(size_t i){
  return (*this)[i];
}

const double& Vector::at(size_t i) const{
  return (*this)[i];
}

double* Vector::get(){
  return x;
}

/** Compares vector \a to vector \a b component-wise.
 * \param a base vector
 * \param b vector components to add
 * \return a == b
 */
bool Vector::operator==(const Vector& b) const
{
  bool status = true;
  for (int i=0;i<NDIM;i++)
    status = status && (fabs((*this)[i] - b[i]) < MYEPSILON);
  return status;
};

/** Sums vector \a to this lhs component-wise.
 * \param a base vector
 * \param b vector components to add
 * \return lhs + a
 */
const Vector& Vector::operator+=(const Vector& b)
{
  this->AddVector(b);
  return *this;
};

/** Subtracts vector \a from this lhs component-wise.
 * \param a base vector
 * \param b vector components to add
 * \return lhs - a
 */
const Vector& Vector::operator-=(const Vector& b)
{
  this->SubtractVector(b);
  return *this;
};

/** factor each component of \a a times a double \a m.
 * \param a base vector
 * \param m factor
 * \return lhs.x[i] * m
 */
const 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 const Vector::operator+(const Vector& b) const
{
  Vector x = *this;
  x.AddVector(b);
  return x;
};

/** Subtracts vector \a from \b component-wise.
 * \param a first vector
 * \param b second vector
 * \return a - b
 */
Vector const Vector::operator-(const Vector& b) const
{
  Vector x = *this;
  x.SubtractVector(b);
  return x;
};

/** Factors given vector \a a times \a m.
 * \param a vector
 * \param m factor
 * \return m * a
 */
Vector const operator*(const Vector& a, const double m)
{
  Vector x(a);
  x.Scale(m);
  return x;
};

/** Factors given vector \a a times \a m.
 * \param m factor
 * \param a vector
 * \return m * a
 */
Vector const operator*(const double m, const Vector& a )
{
  Vector x(a);
  x.Scale(m);
  return x;
};

ostream& operator<<(ostream& ost, const Vector& m)
{
  ost << "(";
  for (int i=0;i<NDIM;i++) {
    ost << m[i];
    if (i != 2)
      ost << ",";
  }
  ost << ")";
  return ost;
};

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

void Vector::Scale(const double * const factor)
{
  for (int i=NDIM;i--;)
    x[i] *= *factor;
};

void Vector::Scale(const double factor)
{
  for (int i=NDIM;i--;)
    x[i] *= factor;
};

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

/** Given a box by its matrix \a *M and its inverse *Minv the vector is made to point within that box.
 * \param *M matrix of box
 * \param *Minv inverse matrix
 */
void Vector::WrapPeriodically(const double * const M, const double * const Minv)
{
  MatrixMultiplication(Minv);
  // truncate to [0,1] for each axis
  for (int i=0;i<NDIM;i++) {
    x[i] += 0.5;  // set to center of box
    while (x[i] >= 1.)
      x[i] -= 1.;
    while (x[i] < 0.)
      x[i] += 1.;
  }
  MatrixMultiplication(M);
};

/** Do a matrix multiplication.
 * \param *matrix NDIM_NDIM array
 */
void Vector::MatrixMultiplication(const double * const 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=NDIM;i--;)
    x[i] = C.x[i];
};

/** Do a matrix multiplication with the \a *A' inverse.
 * \param *matrix NDIM_NDIM array
 */
bool Vector::InverseMatrixMultiplication(const double * const 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=NDIM;i--;)
      x[i] = C.x[i];
    return true;
  } else {
    return false;
  }
};


/** 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, const double * const factors)
{
  (*this) = (factors[0]*x1) +
            (factors[1]*x2) +
            (factors[2]*x3);
};

/** Mirrors atom against a given plane.
 * \param n[] normal vector of mirror plane.
 */
void Vector::Mirror(const Vector &n)
{
  double projection;
  projection = ScalarProduct(n)/n.NormSquared();    // remove constancy from n (keep as logical one)
  // withdraw projected vector twice from original one
  for (int i=NDIM;i--;)
    x[i] -= 2.*projection*n[i];
};


/** Calculates orthonormal vector to one given vector.
 * Just subtracts the projection onto the given vector from this vector.
 * The removed part of the vector is Vector::Projection()
 * \param *x1 vector
 * \return true - success, false - vector is zero
 */
bool Vector::MakeNormalTo(const Vector &y1)
{
  bool result = false;
  double factor = y1.ScalarProduct(*this)/y1.NormSquared();
  Vector x1 = factor * y1 ;
  SubtractVector(x1);
  for (int 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 &GivenVector)
{
  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;

  for (j=NDIM;j--;)
    Components[j] = -1;
  // find two components != 0
  for (j=0;j<NDIM;j++)
    if (fabs(GivenVector[j]) > MYEPSILON)
      Components[Last++] = j;

  switch(Last) {
    case 3:  // threecomponent system
    case 2:  // two component system
      norm = sqrt(1./(GivenVector[Components[1]]*GivenVector[Components[1]]) + 1./(GivenVector[Components[0]]*GivenVector[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./GivenVector[Components[1]] / norm;
      x[Components[0]] = 1./GivenVector[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;
  }
};

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

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

/** Copy vector \a y componentwise.
 * \param y vector
 */
void Vector::CopyVector(const Vector &y)
{
  // check for self assignment
  if(&y!=this) {
    for (int i=NDIM;i--;)
      this->x[i] = y.x[i];
  }
}

// this function is completely unused so it is deactivated until new uses arrive and a new
// place can be found
#if 0
/** 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, const double alpha, const double beta, const 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;
  Log() << 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;
  Log() << Verbose(2) << "D1 " << D1 << "\tD2 " << D2 << "\tD3 " << D3 << "\n";
  if (fabs(D1) < MYEPSILON) {
    Log() << Verbose(2) << "D1 == 0!\n";
    if (fabs(D2) > MYEPSILON) {
      Log() << 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];
      Log() << Verbose(3) << "E1 " << E1 << "\tE2 " << E2 << "\n";
      F1 = E1*E1 + 1.;
      F2 = -E1*E2;
      F3 = E1*E1 + D3*D3/(D2*D2) - C;
      Log() << Verbose(3) << "F1 " << F1 << "\tF2 " << F2 << "\tF3 " << F3 << "\n";
      if (fabs(F1) < MYEPSILON) {
        Log() << Verbose(4) << "F1 == 0!\n";
        Log() << Verbose(4) << "Gleichungssystem linear\n";
        x[1] = F3/(2.*F2);
      } else {
        p = F2/F1;
        q = p*p - F3/F1;
        Log() << Verbose(4) << "p " << p << "\tq " << q << endl;
        if (q < 0) {
          Log() << 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 {
      Log() << 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];
    Log() << 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;
    Log() << Verbose(2) << "F1 " << F1 << "\tF2 " << F2 << "\tF3 " << F3 << "\n";
    if (fabs(F1) < MYEPSILON) {
      Log() << Verbose(3) << "F1 == 0!\n";
      Log() << Verbose(3) << "Gleichungssystem linear\n";
      x[2] = F3/(2.*F2);
    } else {
      p = F2/F1;
      q = p*p - F3/F1;
      Log() << Verbose(3) << "p " << p << "\tq " << q << endl;
      if (q < 0) {
        Log() << 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;
      Log() << Verbose(2) << "k " << k << "\tpot(2,j) " << pot(2,j) << endl;
      sign[j] = (k == 0) ? 1. : -1.;
    }
    Log() << Verbose(2) << i << ": sign matrix is " << sign[0] << "\t" << sign[1] << "\t" << sign[2] << "\n";
    // apply sign matrix
    for (j=NDIM;j--;)
      x[j] *= sign[j];
    // calculate angle and check
    ang = x2->Angle (this);
    Log() << 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=NDIM;j--;)
      x[j] *= sign[j];
  }
  return true;
};

#endif

/**
 * Checks whether this vector is within the parallelepiped defined by the given three vectors and
 * their offset.
 *
 * @param offest for the origin of the parallelepiped
 * @param three vectors forming the matrix that defines the shape of the parallelpiped
 */
bool Vector::IsInParallelepiped(const Vector &offset, const double * const parallelepiped) const
{
  Vector a = (*this) - offset;
  a.InverseMatrixMultiplication(parallelepiped);
  bool isInside = true;

  for (int i=NDIM;i--;)
    isInside = isInside && ((a.x[i] <= 1) && (a.x[i] >= 0));

  return isInside;
}