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lammps/lib/colvars/colvartypes.h
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colvartypes.h
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#ifndef COLVARTYPES_H
#define COLVARTYPES_H
#include <cmath>
#ifndef PI
#define PI 3.14159265358979323846
#endif
// ----------------------------------------------------------------------
/// Linear algebra functions and data types used in the collective
/// variables implemented so far
// ----------------------------------------------------------------------
/// 1-dimensional vector of real numbers with three components
class colvarmodule::rvector {
public:
cvm::real x, y, z;
inline rvector()
: x (0.0), y (0.0), z (0.0)
{}
inline rvector (cvm::real const &x_i,
cvm::real const &y_i,
cvm::real const &z_i)
: x (x_i), y (y_i), z (z_i)
{}
inline rvector (cvm::real v)
: x (v), y (v), z (v)
{}
/// \brief Set all components to a scalar value
inline void set (cvm::real const value = 0.0) {
x = y = z = value;
}
/// \brief Set all components to zero
inline void reset() {
x = y = z = 0.0;
}
/// \brief Access cartesian components by index
inline cvm::real & operator [] (int const &i) {
return (i == 0) ? x : (i == 1) ? y : (i == 2) ? z : x;
}
/// \brief Access cartesian components by index
inline cvm::real const & operator [] (int const &i) const {
return (i == 0) ? x : (i == 1) ? y : (i == 2) ? z : x;
}
inline cvm::rvector & operator = (cvm::real const &v)
{
x = v;
y = v;
z = v;
return *this;
}
inline void operator += (cvm::rvector const &v)
{
x += v.x;
y += v.y;
z += v.z;
}
inline void operator -= (cvm::rvector const &v)
{
x -= v.x;
y -= v.y;
z -= v.z;
}
inline void operator *= (cvm::real const &v)
{
x *= v;
y *= v;
z *= v;
}
inline void operator /= (cvm::real const& v)
{
x /= v;
y /= v;
z /= v;
}
inline cvm::real norm2() const
{
return (x*x + y*y + z*z);
}
inline cvm::real norm() const
{
return std::sqrt (this->norm2());
}
inline cvm::rvector unit() const
{
const cvm::real n = this->norm();
return (n > 0. ? cvm::rvector (x, y, z)/n : cvm::rvector (1., 0., 0.));
}
static inline size_t output_width (size_t const &real_width)
{
return 3*real_width + 10;
}
static inline cvm::rvector outer (cvm::rvector const &v1, cvm::rvector const &v2)
{
return cvm::rvector ( v1.y*v2.z - v2.y*v1.z,
-v1.x*v2.z + v2.x*v1.z,
v1.x*v2.y - v2.x*v1.y);
}
friend inline cvm::rvector operator - (cvm::rvector const &v)
{
return cvm::rvector (-v.x, -v.y, -v.z);
}
friend inline int operator == (cvm::rvector const &v1, cvm::rvector const &v2)
{
return (v1.x == v2.x) && (v1.y == v2.y) && (v1.z == v2.z);
}
friend inline int operator != (cvm::rvector const &v1, cvm::rvector const &v2)
{
return (v1.x != v2.x) || (v1.y != v2.y) || (v1.z != v2.z);
}
friend inline cvm::rvector operator + (cvm::rvector const &v1, cvm::rvector const &v2)
{
return cvm::rvector (v1.x + v2.x, v1.y + v2.y, v1.z + v2.z);
}
friend inline cvm::rvector operator - (cvm::rvector const &v1, cvm::rvector const &v2)
{
return cvm::rvector (v1.x - v2.x, v1.y - v2.y, v1.z - v2.z);
}
friend inline cvm::real operator * (cvm::rvector const &v1, cvm::rvector const &v2)
{
return v1.x * v2.x + v1.y * v2.y + v1.z * v2.z;
}
friend inline cvm::rvector operator * (cvm::real const &a, cvm::rvector const &v)
{
return cvm::rvector (a*v.x, a*v.y, a*v.z);
}
friend inline cvm::rvector operator * (cvm::rvector const &v, cvm::real const &a)
{
return cvm::rvector (a*v.x, a*v.y, a*v.z);
}
friend inline cvm::rvector operator / (cvm::rvector const &v, cvm::real const &a)
{
return cvm::rvector (v.x/a, v.y/a, v.z/a);
}
};
/// \brief Arbitrary size array (one dimensions) suitable for linear
/// algebra operations (i.e. for floating point numbers it can be used
/// with library functions)
template <class T, size_t const length> class colvarmodule::vector1d
{
protected:
/// Underlying C-array
T *array;
public:
/// Length of the array
inline size_t size()
{
return length;
}
/// Default constructor
inline vector1d (T const &t = T())
{
array = new T[length];
reset();
}
/// Set all elements to zero
inline void reset()
{
for (size_t i = 0; i < length; i++) {
array[i] = T (0.0);
}
}
/// Constructor from a 1-d C array
inline vector1d (T const *v)
{
array = new T[length];
for (size_t i = 0; i < length; i++) {
array[i] = v[i];
}
}
/// Copy constructor
inline vector1d (vector1d<T, length> const &v)
{
array = new T[length];
for (size_t i = 0; i < length; i++) {
array[i] = v.array[i];
}
}
/// Assignment
inline vector1d<T, length> & operator = (vector1d<T, length> const &v)
{
for (size_t i = 0; i < length; i++) {
this->array[i] = v.array[i];
}
return *this;
}
/// Destructor
inline ~vector1d() {
delete [] array;
}
/// Return the 1-d C array
inline T *c_array() { return array; }
/// Return the 1-d C array
inline operator T *() { return array; }
/// Inner product
inline friend T operator * (vector1d<T, length> const &v1,
vector1d<T, length> const &v2)
{
T prod (0.0);
for (size_t i = 0; i < length; i++) {
prod += v1.array[i] * v2.array[i];
}
return prod;
}
/// Formatted output
friend std::ostream & operator << (std::ostream &os,
vector1d<T, length> const &v)
{
std::streamsize const w = os.width();
std::streamsize const p = os.precision();
os << "( ";
for (size_t i = 0; i < length-1; i++) {
os.width (w); os.precision (p);
os << v.array[i] << " , ";
}
os.width (w); os.precision (p);
os << v.array[length-1] << " )";
return os;
}
};
/// \brief Arbitrary size array (two dimensions) suitable for linear
/// algebra operations (i.e. for floating point numbers it can be used
/// with library functions)
template <class T,
size_t const outer_length,
size_t const inner_length> class colvarmodule::matrix2d
{
protected:
/// Underlying C array
T **array;
public:
/// Allocation routine, used by all constructors
inline void alloc() {
array = new T * [outer_length];
for (size_t i = 0; i < outer_length; i++) {
array[i] = new T [inner_length];
}
}
/// Set all elements to zero
inline void reset()
{
for (size_t i = 0; i < outer_length; i++) {
for (size_t j = 0; j < inner_length; j++) {
array[i][j] = T (0.0);
}
}
}
/// Default constructor
inline matrix2d()
{
this->alloc();
reset();
}
/// Constructor from a 2-d C array
inline matrix2d (T const **m)
{
this->alloc();
for (size_t i = 0; i < outer_length; i++) {
for (size_t j = 0; j < inner_length; j++)
array[i][j] = m[i][j];
}
}
/// Copy constructor
inline matrix2d (matrix2d<T, outer_length, inner_length> const &m)
{
this->alloc();
for (size_t i = 0; i < outer_length; i++) {
for (size_t j = 0; j < inner_length; j++)
this->array[i][j] = m.array[i][j];
}
}
/// Assignment
inline matrix2d<T, outer_length, inner_length> &
operator = (matrix2d<T, outer_length, inner_length> const &m)
{
for (size_t i = 0; i < outer_length; i++) {
for (size_t j = 0; j < inner_length; j++)
this->array[i][j] = m.array[i][j];
}
return *this;
}
/// Destructor
inline ~matrix2d() {
for (size_t i = 0; i < outer_length; i++) {
delete [] array[i];
}
delete [] array;
}
/// Return the 2-d C array
inline T **c_array() { return array; }
/// Return the 2-d C array
inline operator T **() { return array; }
// /// Matrix addi(c)tion
// inline friend matrix2d<T, outer_length, inner_length>
// operator + (matrix2d<T, outer_length, inner_length> const &mat1,
// matrix2d<T, outer_length, inner_length> const &mat2) {
// matrix2d<T, outer_length, inner_length> sum;
// for (size_t i = 0; i < outer_length; i++) {
// for (size_t j = 0; j < inner_length; j++) {
// sum[i][j] = mat1[i][j] + mat2[i][j];
// }
// }
// }
// /// Matrix subtraction
// inline friend matrix2d<T, outer_length, inner_length>
// operator - (matrix2d<T, outer_length, inner_length> const &mat1,
// matrix2d<T, outer_length, inner_length> const &mat2) {
// matrix2d<T, outer_length, inner_length> sum;
// for (size_t i = 0; i < outer_length; i++) {
// for (size_t j = 0; j < inner_length; j++) {
// sum[i][j] = mat1[i][j] - mat2[i][j];
// }
// }
// }
/// Formatted output
friend std::ostream & operator << (std::ostream &os,
matrix2d<T, outer_length, inner_length> const &m)
{
std::streamsize const w = os.width();
std::streamsize const p = os.precision();
os << "(";
for (size_t i = 0; i < outer_length; i++) {
os << " ( ";
for (size_t j = 0; j < inner_length-1; j++) {
os.width (w);
os.precision (p);
os << m.array[i][j] << " , ";
}
os.width (w);
os.precision (p);
os << m.array[i][inner_length-1] << " )";
}
os << " )";
return os;
}
};
/// \brief 2-dimensional array of real numbers with three components
/// along each dimension (works with colvarmodule::rvector)
class colvarmodule::rmatrix
: public colvarmodule::matrix2d<colvarmodule::real, 3, 3> {
private:
public:
/// Return the xx element
inline cvm::real & xx() { return array[0][0]; }
/// Return the xy element
inline cvm::real & xy() { return array[0][1]; }
/// Return the xz element
inline cvm::real & xz() { return array[0][2]; }
/// Return the yx element
inline cvm::real & yx() { return array[1][0]; }
/// Return the yy element
inline cvm::real & yy() { return array[1][1]; }
/// Return the yz element
inline cvm::real & yz() { return array[1][2]; }
/// Return the zx element
inline cvm::real & zx() { return array[2][0]; }
/// Return the zy element
inline cvm::real & zy() { return array[2][1]; }
/// Return the zz element
inline cvm::real & zz() { return array[2][2]; }
/// Return the xx element
inline cvm::real xx() const { return array[0][0]; }
/// Return the xy element
inline cvm::real xy() const { return array[0][1]; }
/// Return the xz element
inline cvm::real xz() const { return array[0][2]; }
/// Return the yx element
inline cvm::real yx() const { return array[1][0]; }
/// Return the yy element
inline cvm::real yy() const { return array[1][1]; }
/// Return the yz element
inline cvm::real yz() const { return array[1][2]; }
/// Return the zx element
inline cvm::real zx() const { return array[2][0]; }
/// Return the zy element
inline cvm::real zy() const { return array[2][1]; }
/// Return the zz element
inline cvm::real zz() const { return array[2][2]; }
/// Constructor from a 2-d C array
inline rmatrix (cvm::real const **m)
: cvm::matrix2d<cvm::real, 3, 3> (m)
{}
/// Default constructor
inline rmatrix()
: cvm::matrix2d<cvm::real, 3, 3>()
{}
/// Constructor component by component
inline rmatrix (cvm::real const &xxi,
cvm::real const &xyi,
cvm::real const &xzi,
cvm::real const &yxi,
cvm::real const &yyi,
cvm::real const &yzi,
cvm::real const &zxi,
cvm::real const &zyi,
cvm::real const &zzi)
: cvm::matrix2d<cvm::real, 3, 3>()
{
this->xx() = xxi;
this->xy() = xyi;
this->xz() = xzi;
this->yx() = yxi;
this->yy() = yyi;
this->yz() = yzi;
this->zx() = zxi;
this->zy() = zyi;
this->zz() = zzi;
}
/// Destructor
inline ~rmatrix()
{}
/// Return the determinant
inline cvm::real determinant() const
{
return
( xx() * (yy()*zz() - zy()*yz()))
- (yx() * (xy()*zz() - zy()*xz()))
+ (zx() * (xy()*yz() - yy()*xz()));
}
inline cvm::rmatrix transpose() const
{
return cvm::rmatrix (this->xx(),
this->yx(),
this->zx(),
this->xy(),
this->yy(),
this->zy(),
this->xz(),
this->yz(),
this->zz());
}
friend cvm::rvector operator * (cvm::rmatrix const &m, cvm::rvector const &r);
// friend inline cvm::rmatrix const operator * (cvm::rmatrix const &m1, cvm::rmatrix const &m2) {
// return cvm::rmatrix (m1.xx()*m2.xx() + m1.xy()*m2.yx() + m1.xz()*m2.yz(),
// m1.xx()*m2.xy() + m1.xy()*m2.yy() + m1.xz()*m2.zy(),
// m1.xx()*m2.xz() + m1.xy()*m2.yz() + m1.xz()*m2.zz(),
// m1.yx()*m2.xx() + m1.yy()*m2.yx() + m1.yz()*m2.yz(),
// m1.yx()*m2.xy() + m1.yy()*m2.yy() + m1.yz()*m2.yy(),
// m1.yx()*m2.xz() + m1.yy()*m2.yz() + m1.yz()*m2.yz(),
// m1.zx()*m2.xx() + m1.zy()*m2.yx() + m1.zz()*m2.yz(),
// m1.zx()*m2.xy() + m1.zy()*m2.yy() + m1.zz()*m2.yy(),
// m1.zx()*m2.xz() + m1.zy()*m2.yz() + m1.zz()*m2.yz());
// }
};
inline cvm::rvector operator * (cvm::rmatrix const &m,
cvm::rvector const &r)
{
return cvm::rvector (m.xx()*r.x + m.xy()*r.y + m.xz()*r.z,
m.yx()*r.x + m.yy()*r.y + m.yz()*r.z,
m.zx()*r.x + m.zy()*r.y + m.zz()*r.z);
}
/// Numerical recipes diagonalization
void jacobi (cvm::real **a, int n, cvm::real d[], cvm::real **v, int *nrot);
/// Eigenvector sort
void eigsrt (cvm::real d[], cvm::real **v, int n);
/// Transpose the matrix
void transpose (cvm::real **v, int n);
/// \brief 1-dimensional vector of real numbers with four components and
/// a quaternion algebra
class colvarmodule::quaternion {
public:
cvm::real q0, q1, q2, q3;
/// Constructor from a 3-d vector
inline quaternion (cvm::real const &x, cvm::real const &y, cvm::real const &z)
: q0 (0.0), q1 (x), q2 (y), q3 (z)
{}
/// Constructor component by component
inline quaternion (cvm::real const qv[4])
: q0 (qv[0]), q1 (qv[1]), q2 (qv[2]), q3 (qv[3])
{}
/// Constructor component by component
inline quaternion (cvm::real const &q0i,
cvm::real const &q1i,
cvm::real const &q2i,
cvm::real const &q3i)
: q0 (q0i), q1 (q1i), q2 (q2i), q3 (q3i)
{}
/// "Constructor" after Euler angles (in radians)
///
/// http://en.wikipedia.org/wiki/Conversion_between_quaternions_and_Euler_angles
inline void set_from_euler_angles (cvm::real const &phi_in,
cvm::real const &theta_in,
cvm::real const &psi_in)
{
q0 = ( (std::cos (phi_in/2.0)) * (std::cos (theta_in/2.0)) * (std::cos (psi_in/2.0)) +
(std::sin (phi_in/2.0)) * (std::sin (theta_in/2.0)) * (std::sin (psi_in/2.0)) );
q1 = ( (std::sin (phi_in/2.0)) * (std::cos (theta_in/2.0)) * (std::cos (psi_in/2.0)) -
(std::cos (phi_in/2.0)) * (std::sin (theta_in/2.0)) * (std::sin (psi_in/2.0)) );
q2 = ( (std::cos (phi_in/2.0)) * (std::sin (theta_in/2.0)) * (std::cos (psi_in/2.0)) +
(std::sin (phi_in/2.0)) * (std::cos (theta_in/2.0)) * (std::sin (psi_in/2.0)) );
q3 = ( (std::cos (phi_in/2.0)) * (std::cos (theta_in/2.0)) * (std::sin (psi_in/2.0)) -
(std::sin (phi_in/2.0)) * (std::sin (theta_in/2.0)) * (std::cos (psi_in/2.0)) );
}
/// \brief Default constructor
inline quaternion()
{
reset();
}
/// \brief Set all components to a scalar
inline void set (cvm::real const value = 0.0)
{
q0 = q1 = q2 = q3 = value;
}
/// \brief Set all components to zero (null quaternion)
inline void reset()
{
q0 = q1 = q2 = q3 = 0.0;
}
/// \brief Set the q0 component to 1 and the others to 0 (quaternion
/// representing no rotation)
inline void reset_rotation()
{
q0 = 1.0;
q1 = q2 = q3 = 0.0;
}
/// Tell the number of characters required to print a quaternion, given that of a real number
static inline size_t output_width (size_t const &real_width)
{
return 4*real_width + 13;
}
/// \brief Formatted output operator
friend std::ostream & operator << (std::ostream &os, cvm::quaternion const &q);
/// \brief Formatted input operator
friend std::istream & operator >> (std::istream &is, cvm::quaternion &q);
/// Access the quaternion as a 4-d array (return a reference)
inline cvm::real & operator [] (int const &i) {
switch (i) {
case 0:
return this->q0;
case 1:
return this->q1;
case 2:
return this->q2;
case 3:
return this->q3;
default:
cvm::fatal_error ("Error: incorrect quaternion component.\n");
return q0;
}
}
/// Access the quaternion as a 4-d array (return a value)
inline cvm::real operator [] (int const &i) const {
switch (i) {
case 0:
return this->q0;
case 1:
return this->q1;
case 2:
return this->q2;
case 3:
return this->q3;
default:
cvm::fatal_error ("Error: trying to access a quaternion "
"component which is not between 0 and 3.\n");
return this->q0;
}
}
/// Square norm of the quaternion
inline cvm::real norm2() const
{
return q0*q0 + q1*q1 + q2*q2 + q3*q3;
}
/// Norm of the quaternion
inline cvm::real norm() const
{
return std::sqrt (this->norm2());
}
/// Return the conjugate quaternion
inline cvm::quaternion conjugate() const
{
return cvm::quaternion (q0, -q1, -q2, -q3);
}
inline void operator *= (cvm::real const &a)
{
q0 *= a; q1 *= a; q2 *= a; q3 *= a;
}
inline void operator /= (cvm::real const &a)
{
q0 /= a; q1 /= a; q2 /= a; q3 /= a;
}
inline void set_positive()
{
if (q0 > 0.0) return;
q0 = -q0;
q1 = -q1;
q2 = -q2;
q3 = -q3;
}
inline void operator += (cvm::quaternion const &h)
{
q0+=h.q0; q1+=h.q1; q2+=h.q2; q3+=h.q3;
}
inline void operator -= (cvm::quaternion const &h)
{
q0-=h.q0; q1-=h.q1; q2-=h.q2; q3-=h.q3;
}
/// Promote a 3-vector to a quaternion
static inline cvm::quaternion promote (cvm::rvector const &v)
{
return cvm::quaternion (0.0, v.x, v.y, v.z);
}
/// Return the vector component
inline cvm::rvector get_vector() const
{
return cvm::rvector (q1, q2, q3);
}
friend inline cvm::quaternion operator + (cvm::quaternion const &h, cvm::quaternion const &q)
{
return cvm::quaternion (h.q0+q.q0, h.q1+q.q1, h.q2+q.q2, h.q3+q.q3);
}
friend inline cvm::quaternion operator - (cvm::quaternion const &h, cvm::quaternion const &q)
{
return cvm::quaternion (h.q0-q.q0, h.q1-q.q1, h.q2-q.q2, h.q3-q.q3);
}
/// \brief Provides the quaternion product. \b NOTE: for the inner
/// product use: \code h.inner (q); \endcode
friend inline cvm::quaternion operator * (cvm::quaternion const &h, cvm::quaternion const &q)
{
return cvm::quaternion (h.q0*q.q0 - h.q1*q.q1 - h.q2*q.q2 - h.q3*q.q3,
h.q0*q.q1 + h.q1*q.q0 + h.q2*q.q3 - h.q3*q.q2,
h.q0*q.q2 + h.q2*q.q0 + h.q3*q.q1 - h.q1*q.q3,
h.q0*q.q3 + h.q3*q.q0 + h.q1*q.q2 - h.q2*q.q1);
}
friend inline cvm::quaternion operator * (cvm::real const &c,
cvm::quaternion const &q)
{
return cvm::quaternion (c*q.q0, c*q.q1, c*q.q2, c*q.q3);
}
friend inline cvm::quaternion operator * (cvm::quaternion const &q,
cvm::real const &c)
{
return cvm::quaternion (q.q0*c, q.q1*c, q.q2*c, q.q3*c);
}
friend inline cvm::quaternion operator / (cvm::quaternion const &q,
cvm::real const &c)
{
return cvm::quaternion (q.q0/c, q.q1/c, q.q2/c, q.q3/c);
}
/// \brief Rotate v through this quaternion (put it in the rotated
/// reference frame)
inline cvm::rvector rotate (cvm::rvector const &v) const
{
return ((*this) * promote (v) * ((*this).conjugate())).get_vector();
}
/// \brief Rotate Q2 through this quaternion (put it in the rotated
/// reference frame)
inline cvm::quaternion rotate (cvm::quaternion const &Q2) const
{
cvm::rvector const vq_rot = this->rotate (Q2.get_vector());
return cvm::quaternion (Q2.q0, vq_rot.x, vq_rot.y, vq_rot.z);
}
/// Return the 3x3 matrix associated to this quaternion
inline cvm::rmatrix rotation_matrix() const
{
cvm::rmatrix R;
R.xx() = q0*q0 + q1*q1 - q2*q2 - q3*q3;
R.yy() = q0*q0 - q1*q1 + q2*q2 - q3*q3;
R.zz() = q0*q0 - q1*q1 - q2*q2 + q3*q3;
R.xy() = 2.0 * (q1*q2 - q0*q3);
R.xz() = 2.0 * (q0*q2 + q1*q3);
R.yx() = 2.0 * (q0*q3 + q1*q2);
R.yz() = 2.0 * (q2*q3 - q0*q1);
R.zx() = 2.0 * (q1*q3 - q0*q2);
R.zy() = 2.0 * (q0*q1 + q2*q3);
return R;
}
/// \brief Multiply the given vector by the derivative of the given
/// (rotated) position with respect to the quaternion
cvm::quaternion position_derivative_inner (cvm::rvector const &pos,
cvm::rvector const &vec) const;
/// \brief Return the cosine between the orientation frame
/// associated to this quaternion and another
inline cvm::real cosine (cvm::quaternion const &q) const
{
if (q0*q0 >= (1.0-1.0E-10)) {
// null quaternion, return the square of the cosine of the other
// quaternion's rotation angle
return (2.0*q.q0*q.q0 - 1.0);
} else if (q.q0*q.q0 >= (1.0-1.0E-10)) {
return (2.0*q0*q0 - 1.0);
} else {
// get a vector orthogonal to both axes of rotation
cvm::rvector const op = (cvm::rvector::outer (this->get_vector(), q.get_vector()));
cvm::real const opl2 = op.norm2();
// rotate it with both quaternions and get the normalized inner product
return ( (opl2 > 0.0) ?
((this->rotate (op)) * (q.rotate (op))) / opl2 :
1.0 );
}
}
/// \brief Square distance from another quaternion on the
/// 4-dimensional unit sphere: returns the square of the angle along
/// the shorter of the two geodesics
inline cvm::real dist2 (cvm::quaternion const &Q2) const
{
cvm::real const cos_omega = this->q0*Q2.q0 + this->q1*Q2.q1 +
this->q2*Q2.q2 + this->q3*Q2.q3;
cvm::real const omega = std::acos ( (cos_omega > 1.0) ? 1.0 :
( (cos_omega < -1.0) ? -1.0 : cos_omega) );
// get the minimum distance: x and -x are the same quaternion
if (cos_omega > 0.0)
return omega * omega;
else
return (PI-omega) * (PI-omega);
}
/// Gradient of the square distance: returns a 4-vector equivalent
/// to the one provided by slerp
inline cvm::quaternion dist2_grad (cvm::quaternion const &Q2) const
{
cvm::real const cos_omega = this->q0*Q2.q0 + this->q1*Q2.q1 + this->q2*Q2.q2 + this->q3*Q2.q3;
cvm::real const omega = std::acos ( (cos_omega > 1.0) ? 1.0 :
( (cos_omega < -1.0) ? -1.0 : cos_omega) );
cvm::real const sin_omega = std::sin (omega);
if (std::fabs (sin_omega) < 1.0E-14) {
// return a null 4d vector
return cvm::quaternion (0.0, 0.0, 0.0, 0.0);
}
cvm::quaternion const
grad1 ((-1.0)*sin_omega*Q2.q0 + cos_omega*(this->q0-cos_omega*Q2.q0)/sin_omega,
(-1.0)*sin_omega*Q2.q1 + cos_omega*(this->q1-cos_omega*Q2.q1)/sin_omega,
(-1.0)*sin_omega*Q2.q2 + cos_omega*(this->q2-cos_omega*Q2.q2)/sin_omega,
(-1.0)*sin_omega*Q2.q3 + cos_omega*(this->q3-cos_omega*Q2.q3)/sin_omega);
if (cos_omega > 0.0) {
return 2.0*omega*grad1;
}
else {
return -2.0*(PI-omega)*grad1;
}
}
/// \brief Choose the closest between Q2 and -Q2 and save it back.
/// Not required for dist2() and dist2_grad()
inline void match (cvm::quaternion &Q2) const
{
cvm::real const cos_omega = this->q0*Q2.q0 + this->q1*Q2.q1 +
this->q2*Q2.q2 + this->q3*Q2.q3;
if (cos_omega < 0.0) Q2 *= -1.0;
}
/// \brief Inner product (as a 4-d vector) with Q2; requires match()
/// if the largest overlap is looked for
inline cvm::real inner (cvm::quaternion const &Q2) const
{
cvm::real const prod = this->q0*Q2.q0 + this->q1*Q2.q1 +
this->q2*Q2.q2 + this->q3*Q2.q3;
return prod;
}
};
/// \brief A rotation between two sets of coordinates (for the moment
/// a wrapper for colvarmodule::quaternion)
class colvarmodule::rotation
{
public:
/// \brief The rotation itself (implemented as a quaternion)
cvm::quaternion q;
/// \brief Eigenvalue corresponding to the optimal rotation
cvm::real lambda;
/// \brief Perform gradient tests
bool b_debug_gradients;
/// \brief Positions to superimpose: the rotation should brings pos1
/// into pos2
std::vector< cvm::atom_pos > pos1, pos2;
/// Derivatives of S
std::vector< cvm::matrix2d<cvm::rvector, 4, 4> > dS_1, dS_2;
/// Derivatives of leading eigenvalue
std::vector< cvm::rvector > dL0_1, dL0_2;
/// Derivatives of leading eigenvector
std::vector< cvm::vector1d<cvm::rvector, 4> > dQ0_1, dQ0_2;
/// Allocate space for the derivatives of the rotation
inline void request_group1_gradients (size_t const &n)
{
dS_1.resize (n, cvm::matrix2d<cvm::rvector, 4, 4>());
dL0_1.resize (n, cvm::rvector (0.0, 0.0, 0.0));
dQ0_1.resize (n, cvm::vector1d<cvm::rvector, 4>());
}
inline void request_group2_gradients (size_t const &n)
{
dS_2.resize (n, cvm::matrix2d<cvm::rvector, 4, 4>());
dL0_2.resize (n, cvm::rvector (0.0, 0.0, 0.0));
dQ0_2.resize (n, cvm::vector1d<cvm::rvector, 4>());
}
/// \brief Calculate the optimal rotation and store the
/// corresponding eigenvalue and eigenvector in the arguments l0 and
/// q0; if the gradients have been previously requested, calculate
/// them as well
///
/// The method to derive the optimal rotation is defined in:
/// Coutsias EA, Seok C, Dill KA.
/// Using quaternions to calculate RMSD.
/// J Comput Chem. 25(15):1849-57 (2004)
/// DOI: 10.1002/jcc.20110 PubMed: 15376254
void calc_optimal_rotation (std::vector<atom_pos> const &pos1,
std::vector<atom_pos> const &pos2);
/// Default constructor
inline rotation()
: b_debug_gradients (false)
{}
/// Constructor after a quaternion
inline rotation (cvm::quaternion const &qi)
: b_debug_gradients (false)
{
q = qi;
}
/// Constructor after an axis of rotation and an angle (in radians)
inline rotation (cvm::real const &angle, cvm::rvector const &axis)
: b_debug_gradients (false)
{
cvm::rvector const axis_n = axis.unit();
cvm::real const sina = std::sin (angle/2.0);
q = cvm::quaternion (std::cos (angle/2.0),
sina * axis_n.x, sina * axis_n.y, sina * axis_n.z);
}
/// Destructor
inline ~rotation()
{}
/// Return the rotated vector
inline cvm::rvector rotate (cvm::rvector const &v) const
{
return q.rotate (v);
}
/// Return the inverse of this rotation
inline cvm::rotation inverse() const
{
return cvm::rotation (this->q.conjugate());
}
/// Return the associated 3x3 matrix
inline cvm::rmatrix matrix() const
{
return q.rotation_matrix();
}
/// \brief Return the spin angle (in degrees) with respect to the
/// provided axis (which MUST be normalized)
inline cvm::real spin_angle (cvm::rvector const &axis) const
{
cvm::rvector const q_vec = q.get_vector();
cvm::real alpha = (180.0/PI) * 2.0 * std::atan2 (axis * q_vec, q.q0);
while (alpha > 180.0) alpha -= 360;
while (alpha < -180.0) alpha += 360;
return alpha;
}
/// \brief Return the derivative of the spin angle with respect to
/// the quaternion
inline cvm::quaternion dspin_angle_dq (cvm::rvector const &axis) const
{
cvm::rvector const q_vec = q.get_vector();
cvm::real const iprod = axis * q_vec;
if (q.q0 != 0.0) {
// cvm::real const x = iprod/q.q0;
cvm::real const dspindx = (180.0/PI) * 2.0 * (1.0 / (1.0 + (iprod*iprod)/(q.q0*q.q0)));
return
cvm::quaternion ( dspindx * (iprod * (-1.0) / (q.q0*q.q0)),
dspindx * ((1.0/q.q0) * axis.x),
dspindx * ((1.0/q.q0) * axis.y),
dspindx * ((1.0/q.q0) * axis.z));
} else {
// (1/(1+x^2)) ~ (1/x)^2
return
cvm::quaternion ((180.0/PI) * 2.0 * ((-1.0)/iprod), 0.0, 0.0, 0.0);
// XX TODO: What if iprod == 0? XX
}
}
/// \brief Return the projection of the orientation vector onto a
/// predefined axis
inline cvm::real cos_theta (cvm::rvector const &axis) const
{
cvm::rvector const q_vec = q.get_vector();
cvm::real const alpha =
(180.0/PI) * 2.0 * std::atan2 (axis * q_vec, q.q0);
cvm::real const cos_spin_2 = std::cos (alpha * (PI/180.0) * 0.5);
cvm::real const cos_theta_2 = ( (cos_spin_2 != 0.0) ?
(q.q0 / cos_spin_2) :
(0.0) );
// cos(2t) = 2*cos(t)^2 - 1
return 2.0 * (cos_theta_2*cos_theta_2) - 1.0;
}
/// Return the derivative of the tilt wrt the quaternion
inline cvm::quaternion dcos_theta_dq (cvm::rvector const &axis) const
{
cvm::rvector const q_vec = q.get_vector();
cvm::real const iprod = axis * q_vec;
cvm::real const cos_spin_2 = std::cos (std::atan2 (iprod, q.q0));
if (q.q0 != 0.0) {
cvm::real const d_cos_theta_dq0 =
(4.0 * q.q0 / (cos_spin_2*cos_spin_2)) *
(1.0 - (iprod*iprod)/(q.q0*q.q0) / (1.0 + (iprod*iprod)/(q.q0*q.q0)));
cvm::real const d_cos_theta_dqn =
(4.0 * q.q0 / (cos_spin_2*cos_spin_2) *
(iprod/q.q0) / (1.0 + (iprod*iprod)/(q.q0*q.q0)));
return cvm::quaternion (d_cos_theta_dq0,
d_cos_theta_dqn * axis.x,
d_cos_theta_dqn * axis.y,
d_cos_theta_dqn * axis.z);
} else {
cvm::real const d_cos_theta_dqn =
(4.0 / (cos_spin_2*cos_spin_2 * iprod));
return cvm::quaternion (0.0,
d_cos_theta_dqn * axis.x,
d_cos_theta_dqn * axis.y,
d_cos_theta_dqn * axis.z);
}
}
/// \brief Threshold for the eigenvalue crossing test
static cvm::real crossing_threshold;
protected:
/// \brief Previous value of the rotation (used to warn the user
/// when the structure changes too much, and there may be an
/// eigenvalue crossing)
cvm::quaternion q_old;
/// Build the overlap matrix S (used by calc_optimal_rotation())
void build_matrix (std::vector<cvm::atom_pos> const &pos1,
std::vector<cvm::atom_pos> const &pos2,
cvm::matrix2d<real, 4, 4> &S);
/// Diagonalize the overlap matrix S (used by calc_optimal_rotation())
void diagonalize_matrix (cvm::matrix2d<cvm::real, 4, 4> &S,
cvm::real S_eigval[4],
cvm::matrix2d<cvm::real, 4, 4> &S_eigvec);
};
#endif
// Emacs
// Local Variables:
// mode: C++
// End:
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