source: src/vector.cpp@ 09af1b

/** \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.; };

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

/** 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->x[i])*(x[i]-y->x[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
{
        double res = 0.;
        for (int i=NDIM;i--;)
                res += (x[i]-y->x[i])*(x[i]-y->x[i]);
        return (sqrt(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$| x - y |\f$
 */
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=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);
};

/** 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 *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.CopyVector(y);
                                Shiftedy.AddVector(&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(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=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=NDIM;i--;)
                res += x[i]*y->x[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.x[0] = x[1]* (y->x[2]) - x[2]* (y->x[1]);
        tmp.x[1] = x[2]* (y->x[0]) - x[0]* (y->x[2]);
        tmp.x[2] = x[0]* (y->x[1]) - x[1]* (y->x[0]);
        this->CopyVector(&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;
        tmp.CopyVector(y);
        tmp.Normalize();
        tmp.Scale(ScalarProduct(&tmp));
        this->SubtractVector(&tmp);
};

/** 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));
};

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

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

/** 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
{
        return acos(this->ScalarProduct(y)/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;
};

ostream& operator<<(ostream& ost,Vector& m)
{
        ost << "(";
        for (int i=0;i<NDIM;i++) {
                ost << m.x[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(double **factor)
{
        for (int i=NDIM;i--;)
                x[i] *= (*factor)[i];
};

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

void Vector::Scale(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->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=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=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=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)/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=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 ((fabs(x1.Norm()) < MYEPSILON) || (fabs(x2.Norm()) < MYEPSILON) || (fabs(x1.Angle(&x2)) < MYEPSILON)) {
                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 ((fabs(x1.Norm()) < MYEPSILON) || (fabs(x2.Norm()) < MYEPSILON) || (fabs(x1.Angle(&x2)) < MYEPSILON)) {
                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=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;

        cout << Verbose(4);
        GivenVector->Output((ofstream *)&cout);
        cout << endl;
        for (j=NDIM;j--;)
                Components[j] = -1;
        // find two components != 0
        for (j=0;j<NDIM;j++)
                if (fabs(GivenVector->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./(GivenVector->x[Components[1]]*GivenVector->x[Components[1]]) + 1./(GivenVector->x[Components[0]]*GivenVector->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./GivenVector->x[Components[1]] / norm;
                        x[Components[0]] = 1./GivenVector->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;
        }
};

/** Determines paramter needed to multiply this vector to obtain intersection point with plane defined by \a *A, \a *B and \a *C.
 * \param *A first plane vector
 * \param *B second plane vector
 * \param *C third plane vector
 * \return scaling parameter for this vector
 */
double Vector::CutsPlaneAt(Vector *A, Vector *B, Vector *C)
{
//      cout << Verbose(3) << "For comparison: ";
//      cout << "A " << A->Projection(this) << "\t";
//      cout << "B " << B->Projection(this) << "\t";
//      cout << "C " << C->Projection(this) << "\t";
//      cout << endl;
        return A->Projection(this);
};

/** 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, *y;
         gsl_multimin_function minex_func;

         size_t iter = 0, i;
         int status;
         double size;

         /* Initial vertex size vector */
         ss = gsl_vector_alloc (np);
         y = 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=NDIM;i--;)
                gsl_vector_set(y, 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, y, 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=(size_t)np;i--;)
                this->x[i] = gsl_vector_get(s->x, i);
         gsl_vector_free(y);
         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=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=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=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=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=NDIM;j--;)
                        x[j] *= sign[j];
        }
        return true;
};