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colvartypes.cpp
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// -*- c++ -*-
// This file is part of the Collective Variables module (Colvars).
// The original version of Colvars and its updates are located at:
// https://github.com/colvars/colvars
// Please update all Colvars source files before making any changes.
// If you wish to distribute your changes, please submit them to the
// Colvars repository at GitHub.
#include <stdlib.h>
#include <string.h>
#include "colvarmodule.h"
#include "colvartypes.h"
#include "colvarparse.h"
bool colvarmodule::rotation::monitor_crossings = false;
cvm::real colvarmodule::rotation::crossing_threshold = 1.0E-02;
/// Numerical recipes diagonalization
static int jacobi(cvm::real **a, cvm::real *d, cvm::real **v, int *nrot);
/// Eigenvector sort
static int eigsrt(cvm::real *d, cvm::real **v);
/// Transpose the matrix
static int transpose(cvm::real **v);
std::string cvm::rvector::to_simple_string() const
{
std::ostringstream os;
os.setf(std::ios::scientific, std::ios::floatfield);
os.precision(cvm::cv_prec);
os << x << " " << y << " " << z;
return os.str();
}
int cvm::rvector::from_simple_string(std::string const &s)
{
std::stringstream stream(s);
if ( !(stream >> x) ||
!(stream >> y) ||
!(stream >> z) ) {
return COLVARS_ERROR;
}
return COLVARS_OK;
}
std::ostream & operator << (std::ostream &os, colvarmodule::rvector const &v)
{
std::streamsize const w = os.width();
std::streamsize const p = os.precision();
os.width(2);
os << "( ";
os.width(w); os.precision(p);
os << v.x << " , ";
os.width(w); os.precision(p);
os << v.y << " , ";
os.width(w); os.precision(p);
os << v.z << " )";
return os;
}
std::istream & operator >> (std::istream &is, colvarmodule::rvector &v)
{
size_t const start_pos = is.tellg();
char sep;
if ( !(is >> sep) || !(sep == '(') ||
!(is >> v.x) || !(is >> sep) || !(sep == ',') ||
!(is >> v.y) || !(is >> sep) || !(sep == ',') ||
!(is >> v.z) || !(is >> sep) || !(sep == ')') ) {
is.clear();
is.seekg(start_pos, std::ios::beg);
is.setstate(std::ios::failbit);
return is;
}
return is;
}
std::string cvm::quaternion::to_simple_string() const
{
std::ostringstream os;
os.setf(std::ios::scientific, std::ios::floatfield);
os.precision(cvm::cv_prec);
os << q0 << " " << q1 << " " << q2 << " " << q3;
return os.str();
}
int cvm::quaternion::from_simple_string(std::string const &s)
{
std::stringstream stream(s);
if ( !(stream >> q0) ||
!(stream >> q1) ||
!(stream >> q2) ||
!(stream >> q3) ) {
return COLVARS_ERROR;
}
return COLVARS_OK;
}
std::ostream & operator << (std::ostream &os, colvarmodule::quaternion const &q)
{
std::streamsize const w = os.width();
std::streamsize const p = os.precision();
os.width(2);
os << "( ";
os.width(w); os.precision(p);
os << q.q0 << " , ";
os.width(w); os.precision(p);
os << q.q1 << " , ";
os.width(w); os.precision(p);
os << q.q2 << " , ";
os.width(w); os.precision(p);
os << q.q3 << " )";
return os;
}
std::istream & operator >> (std::istream &is, colvarmodule::quaternion &q)
{
size_t const start_pos = is.tellg();
std::string euler("");
if ( (is >> euler) && (colvarparse::to_lower_cppstr(euler) ==
std::string("euler")) ) {
// parse the Euler angles
char sep;
cvm::real phi, theta, psi;
if ( !(is >> sep) || !(sep == '(') ||
!(is >> phi) || !(is >> sep) || !(sep == ',') ||
!(is >> theta) || !(is >> sep) || !(sep == ',') ||
!(is >> psi) || !(is >> sep) || !(sep == ')') ) {
is.clear();
is.seekg(start_pos, std::ios::beg);
is.setstate(std::ios::failbit);
return is;
}
q = colvarmodule::quaternion(phi, theta, psi);
} else {
// parse the quaternion components
is.seekg(start_pos, std::ios::beg);
char sep;
if ( !(is >> sep) || !(sep == '(') ||
!(is >> q.q0) || !(is >> sep) || !(sep == ',') ||
!(is >> q.q1) || !(is >> sep) || !(sep == ',') ||
!(is >> q.q2) || !(is >> sep) || !(sep == ',') ||
!(is >> q.q3) || !(is >> sep) || !(sep == ')') ) {
is.clear();
is.seekg(start_pos, std::ios::beg);
is.setstate(std::ios::failbit);
return is;
}
}
return is;
}
cvm::quaternion
cvm::quaternion::position_derivative_inner(cvm::rvector const &pos,
cvm::rvector const &vec) const
{
cvm::quaternion result(0.0, 0.0, 0.0, 0.0);
result.q0 = 2.0 * pos.x * q0 * vec.x
+2.0 * pos.y * q0 * vec.y
+2.0 * pos.z * q0 * vec.z
-2.0 * pos.y * q3 * vec.x
+2.0 * pos.z * q2 * vec.x
+2.0 * pos.x * q3 * vec.y
-2.0 * pos.z * q1 * vec.y
-2.0 * pos.x * q2 * vec.z
+2.0 * pos.y * q1 * vec.z;
result.q1 = +2.0 * pos.x * q1 * vec.x
-2.0 * pos.y * q1 * vec.y
-2.0 * pos.z * q1 * vec.z
+2.0 * pos.y * q2 * vec.x
+2.0 * pos.z * q3 * vec.x
+2.0 * pos.x * q2 * vec.y
-2.0 * pos.z * q0 * vec.y
+2.0 * pos.x * q3 * vec.z
+2.0 * pos.y * q0 * vec.z;
result.q2 = -2.0 * pos.x * q2 * vec.x
+2.0 * pos.y * q2 * vec.y
-2.0 * pos.z * q2 * vec.z
+2.0 * pos.y * q1 * vec.x
+2.0 * pos.z * q0 * vec.x
+2.0 * pos.x * q1 * vec.y
+2.0 * pos.z * q3 * vec.y
-2.0 * pos.x * q0 * vec.z
+2.0 * pos.y * q3 * vec.z;
result.q3 = -2.0 * pos.x * q3 * vec.x
-2.0 * pos.y * q3 * vec.y
+2.0 * pos.z * q3 * vec.z
-2.0 * pos.y * q0 * vec.x
+2.0 * pos.z * q1 * vec.x
+2.0 * pos.x * q0 * vec.y
+2.0 * pos.z * q2 * vec.y
+2.0 * pos.x * q1 * vec.z
+2.0 * pos.y * q2 * vec.z;
return result;
}
// Calculate the optimal rotation between two groups, and implement it
// as a quaternion. Uses the method documented 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 colvarmodule::rotation::build_matrix(std::vector<cvm::atom_pos> const &pos1,
std::vector<cvm::atom_pos> const &pos2,
cvm::matrix2d<cvm::real> &S)
{
// build the correlation matrix
C.resize(3, 3);
C.reset();
size_t i;
for (i = 0; i < pos1.size(); i++) {
C.xx() += pos1[i].x * pos2[i].x;
C.xy() += pos1[i].x * pos2[i].y;
C.xz() += pos1[i].x * pos2[i].z;
C.yx() += pos1[i].y * pos2[i].x;
C.yy() += pos1[i].y * pos2[i].y;
C.yz() += pos1[i].y * pos2[i].z;
C.zx() += pos1[i].z * pos2[i].x;
C.zy() += pos1[i].z * pos2[i].y;
C.zz() += pos1[i].z * pos2[i].z;
}
// build the "overlap" matrix, whose eigenvectors are stationary
// points of the RMSD in the space of rotations
S[0][0] = C.xx() + C.yy() + C.zz();
S[1][0] = C.yz() - C.zy();
S[0][1] = S[1][0];
S[2][0] = - C.xz() + C.zx() ;
S[0][2] = S[2][0];
S[3][0] = C.xy() - C.yx();
S[0][3] = S[3][0];
S[1][1] = C.xx() - C.yy() - C.zz();
S[2][1] = C.xy() + C.yx();
S[1][2] = S[2][1];
S[3][1] = C.xz() + C.zx();
S[1][3] = S[3][1];
S[2][2] = - C.xx() + C.yy() - C.zz();
S[3][2] = C.yz() + C.zy();
S[2][3] = S[3][2];
S[3][3] = - C.xx() - C.yy() + C.zz();
}
void colvarmodule::rotation::diagonalize_matrix(cvm::matrix2d<cvm::real> &S,
cvm::vector1d<cvm::real> &S_eigval,
cvm::matrix2d<cvm::real> &S_eigvec)
{
S_eigval.resize(4);
S_eigval.reset();
S_eigvec.resize(4,4);
S_eigvec.reset();
// diagonalize
int jac_nrot = 0;
if (jacobi(S.c_array(), S_eigval.c_array(), S_eigvec.c_array(), &jac_nrot) !=
COLVARS_OK) {
cvm::error("Too many iterations in routine jacobi.\n"
"This is usually the result of an ill-defined set of atoms for "
"rotational alignment (RMSD, rotateReference, etc).\n");
}
eigsrt(S_eigval.c_array(), S_eigvec.c_array());
// jacobi saves eigenvectors by columns
transpose(S_eigvec.c_array());
// normalize eigenvectors
for (size_t ie = 0; ie < 4; ie++) {
cvm::real norm2 = 0.0;
size_t i;
for (i = 0; i < 4; i++) {
norm2 += std::pow(S_eigvec[ie][i], int(2));
}
cvm::real const norm = std::sqrt(norm2);
for (i = 0; i < 4; i++) {
S_eigvec[ie][i] /= norm;
}
}
}
// Calculate the rotation, plus its derivatives
void colvarmodule::rotation::calc_optimal_rotation(std::vector<cvm::atom_pos> const &pos1,
std::vector<cvm::atom_pos> const &pos2)
{
S.resize(4,4);
S.reset();
build_matrix(pos1, pos2, S);
S_backup.resize(4,4);
S_backup = S;
if (b_debug_gradients) {
cvm::log("S = "+cvm::to_str(cvm::to_str(S_backup), cvm::cv_width, cvm::cv_prec)+"\n");
}
diagonalize_matrix(S, S_eigval, S_eigvec);
// eigenvalues and eigenvectors
cvm::real const L0 = S_eigval[0];
cvm::real const L1 = S_eigval[1];
cvm::real const L2 = S_eigval[2];
cvm::real const L3 = S_eigval[3];
cvm::quaternion const Q0(S_eigvec[0]);
cvm::quaternion const Q1(S_eigvec[1]);
cvm::quaternion const Q2(S_eigvec[2]);
cvm::quaternion const Q3(S_eigvec[3]);
lambda = L0;
q = Q0;
if (cvm::rotation::monitor_crossings) {
if (q_old.norm2() > 0.0) {
q.match(q_old);
if (q_old.inner(q) < (1.0 - crossing_threshold)) {
cvm::log("Warning: one molecular orientation has changed by more than "+
cvm::to_str(crossing_threshold)+": discontinuous rotation ?\n");
}
}
q_old = q;
}
if (b_debug_gradients) {
cvm::log("L0 = "+cvm::to_str(L0, cvm::cv_width, cvm::cv_prec)+
", Q0 = "+cvm::to_str(Q0, cvm::cv_width, cvm::cv_prec)+
", Q0*Q0 = "+cvm::to_str(Q0.inner(Q0), cvm::cv_width, cvm::cv_prec)+
"\n");
cvm::log("L1 = "+cvm::to_str(L1, cvm::cv_width, cvm::cv_prec)+
", Q1 = "+cvm::to_str(Q1, cvm::cv_width, cvm::cv_prec)+
", Q0*Q1 = "+cvm::to_str(Q0.inner(Q1), cvm::cv_width, cvm::cv_prec)+
"\n");
cvm::log("L2 = "+cvm::to_str(L2, cvm::cv_width, cvm::cv_prec)+
", Q2 = "+cvm::to_str(Q2, cvm::cv_width, cvm::cv_prec)+
", Q0*Q2 = "+cvm::to_str(Q0.inner(Q2), cvm::cv_width, cvm::cv_prec)+
"\n");
cvm::log("L3 = "+cvm::to_str(L3, cvm::cv_width, cvm::cv_prec)+
", Q3 = "+cvm::to_str(Q3, cvm::cv_width, cvm::cv_prec)+
", Q0*Q3 = "+cvm::to_str(Q0.inner(Q3), cvm::cv_width, cvm::cv_prec)+
"\n");
}
// calculate derivatives of L0 and Q0 with respect to each atom in
// either group; note: if dS_1 is a null vector, nothing will be
// calculated
size_t ia;
for (ia = 0; ia < dS_1.size(); ia++) {
cvm::real const &a2x = pos2[ia].x;
cvm::real const &a2y = pos2[ia].y;
cvm::real const &a2z = pos2[ia].z;
cvm::matrix2d<cvm::rvector> &ds_1 = dS_1[ia];
// derivative of the S matrix
ds_1.reset();
ds_1[0][0].set( a2x, a2y, a2z);
ds_1[1][0].set( 0.0, a2z, -a2y);
ds_1[0][1] = ds_1[1][0];
ds_1[2][0].set(-a2z, 0.0, a2x);
ds_1[0][2] = ds_1[2][0];
ds_1[3][0].set( a2y, -a2x, 0.0);
ds_1[0][3] = ds_1[3][0];
ds_1[1][1].set( a2x, -a2y, -a2z);
ds_1[2][1].set( a2y, a2x, 0.0);
ds_1[1][2] = ds_1[2][1];
ds_1[3][1].set( a2z, 0.0, a2x);
ds_1[1][3] = ds_1[3][1];
ds_1[2][2].set(-a2x, a2y, -a2z);
ds_1[3][2].set( 0.0, a2z, a2y);
ds_1[2][3] = ds_1[3][2];
ds_1[3][3].set(-a2x, -a2y, a2z);
cvm::rvector &dl0_1 = dL0_1[ia];
cvm::vector1d<cvm::rvector> &dq0_1 = dQ0_1[ia];
// matrix multiplications; derivatives of L_0 and Q_0 are
// calculated using Hellmann-Feynman theorem (i.e. exploiting the
// fact that the eigenvectors Q_i form an orthonormal basis)
dl0_1.reset();
for (size_t i = 0; i < 4; i++) {
for (size_t j = 0; j < 4; j++) {
dl0_1 += Q0[i] * ds_1[i][j] * Q0[j];
}
}
dq0_1.reset();
for (size_t p = 0; p < 4; p++) {
for (size_t i = 0; i < 4; i++) {
for (size_t j = 0; j < 4; j++) {
dq0_1[p] +=
(Q1[i] * ds_1[i][j] * Q0[j]) / (L0-L1) * Q1[p] +
(Q2[i] * ds_1[i][j] * Q0[j]) / (L0-L2) * Q2[p] +
(Q3[i] * ds_1[i][j] * Q0[j]) / (L0-L3) * Q3[p];
}
}
}
}
// do the same for the second group
for (ia = 0; ia < dS_2.size(); ia++) {
cvm::real const &a1x = pos1[ia].x;
cvm::real const &a1y = pos1[ia].y;
cvm::real const &a1z = pos1[ia].z;
cvm::matrix2d<cvm::rvector> &ds_2 = dS_2[ia];
ds_2.reset();
ds_2[0][0].set( a1x, a1y, a1z);
ds_2[1][0].set( 0.0, -a1z, a1y);
ds_2[0][1] = ds_2[1][0];
ds_2[2][0].set( a1z, 0.0, -a1x);
ds_2[0][2] = ds_2[2][0];
ds_2[3][0].set(-a1y, a1x, 0.0);
ds_2[0][3] = ds_2[3][0];
ds_2[1][1].set( a1x, -a1y, -a1z);
ds_2[2][1].set( a1y, a1x, 0.0);
ds_2[1][2] = ds_2[2][1];
ds_2[3][1].set( a1z, 0.0, a1x);
ds_2[1][3] = ds_2[3][1];
ds_2[2][2].set(-a1x, a1y, -a1z);
ds_2[3][2].set( 0.0, a1z, a1y);
ds_2[2][3] = ds_2[3][2];
ds_2[3][3].set(-a1x, -a1y, a1z);
cvm::rvector &dl0_2 = dL0_2[ia];
cvm::vector1d<cvm::rvector> &dq0_2 = dQ0_2[ia];
dl0_2.reset();
for (size_t i = 0; i < 4; i++) {
for (size_t j = 0; j < 4; j++) {
dl0_2 += Q0[i] * ds_2[i][j] * Q0[j];
}
}
dq0_2.reset();
for (size_t p = 0; p < 4; p++) {
for (size_t i = 0; i < 4; i++) {
for (size_t j = 0; j < 4; j++) {
dq0_2[p] +=
(Q1[i] * ds_2[i][j] * Q0[j]) / (L0-L1) * Q1[p] +
(Q2[i] * ds_2[i][j] * Q0[j]) / (L0-L2) * Q2[p] +
(Q3[i] * ds_2[i][j] * Q0[j]) / (L0-L3) * Q3[p];
}
}
}
if (b_debug_gradients) {
cvm::matrix2d<cvm::real> S_new(4, 4);
cvm::vector1d<cvm::real> S_new_eigval(4);
cvm::matrix2d<cvm::real> S_new_eigvec(4, 4);
// make an infitesimal move along each cartesian coordinate of
// this atom, and solve again the eigenvector problem
for (size_t comp = 0; comp < 3; comp++) {
S_new = S_backup;
// diagonalize the new overlap matrix
for (size_t i = 0; i < 4; i++) {
for (size_t j = 0; j < 4; j++) {
S_new[i][j] +=
colvarmodule::debug_gradients_step_size * ds_2[i][j][comp];
}
}
// cvm::log("S_new = "+cvm::to_str(cvm::to_str (S_new), cvm::cv_width, cvm::cv_prec)+"\n");
diagonalize_matrix(S_new, S_new_eigval, S_new_eigvec);
cvm::real const &L0_new = S_new_eigval[0];
cvm::quaternion const Q0_new(S_new_eigvec[0]);
cvm::real const DL0 = (dl0_2[comp]) * colvarmodule::debug_gradients_step_size;
cvm::quaternion const DQ0(dq0_2[0][comp] * colvarmodule::debug_gradients_step_size,
dq0_2[1][comp] * colvarmodule::debug_gradients_step_size,
dq0_2[2][comp] * colvarmodule::debug_gradients_step_size,
dq0_2[3][comp] * colvarmodule::debug_gradients_step_size);
cvm::log( "|(l_0+dl_0) - l_0^new|/l_0 = "+
cvm::to_str(std::fabs(L0+DL0 - L0_new)/L0, cvm::cv_width, cvm::cv_prec)+
", |(q_0+dq_0) - q_0^new| = "+
cvm::to_str((Q0+DQ0 - Q0_new).norm(), cvm::cv_width, cvm::cv_prec)+
"\n");
}
}
}
}
// Numerical Recipes routine for diagonalization
#define ROTATE(a,i,j,k,l) g=a[i][j]; \
h=a[k][l]; \
a[i][j]=g-s*(h+g*tau); \
a[k][l]=h+s*(g-h*tau);
#define n 4
int jacobi(cvm::real **a, cvm::real *d, cvm::real **v, int *nrot)
{
int j,iq,ip,i;
cvm::real tresh,theta,tau,t,sm,s,h,g,c;
cvm::vector1d<cvm::real> b(n);
cvm::vector1d<cvm::real> z(n);
for (ip=0;ip<n;ip++) {
for (iq=0;iq<n;iq++) {
v[ip][iq]=0.0;
}
v[ip][ip]=1.0;
}
for (ip=0;ip<n;ip++) {
b[ip]=d[ip]=a[ip][ip];
z[ip]=0.0;
}
*nrot=0;
for (i=0;i<=50;i++) {
sm=0.0;
for (ip=0;ip<n-1;ip++) {
for (iq=ip+1;iq<n;iq++)
sm += std::fabs(a[ip][iq]);
}
if (sm == 0.0) {
return COLVARS_OK;
}
if (i < 4)
tresh=0.2*sm/(n*n);
else
tresh=0.0;
for (ip=0;ip<n-1;ip++) {
for (iq=ip+1;iq<n;iq++) {
g=100.0*std::fabs(a[ip][iq]);
if (i > 4 && (cvm::real)(std::fabs(d[ip])+g) == (cvm::real)std::fabs(d[ip])
&& (cvm::real)(std::fabs(d[iq])+g) == (cvm::real)std::fabs(d[iq]))
a[ip][iq]=0.0;
else if (std::fabs(a[ip][iq]) > tresh) {
h=d[iq]-d[ip];
if ((cvm::real)(std::fabs(h)+g) == (cvm::real)std::fabs(h))
t=(a[ip][iq])/h;
else {
theta=0.5*h/(a[ip][iq]);
t=1.0/(std::fabs(theta)+std::sqrt(1.0+theta*theta));
if (theta < 0.0) t = -t;
}
c=1.0/std::sqrt(1+t*t);
s=t*c;
tau=s/(1.0+c);
h=t*a[ip][iq];
z[ip] -= h;
z[iq] += h;
d[ip] -= h;
d[iq] += h;
a[ip][iq]=0.0;
for (j=0;j<=ip-1;j++) {
ROTATE(a,j,ip,j,iq)
}
for (j=ip+1;j<=iq-1;j++) {
ROTATE(a,ip,j,j,iq)
}
for (j=iq+1;j<n;j++) {
ROTATE(a,ip,j,iq,j)
}
for (j=0;j<n;j++) {
ROTATE(v,j,ip,j,iq)
}
++(*nrot);
}
}
}
for (ip=0;ip<n;ip++) {
b[ip] += z[ip];
d[ip]=b[ip];
z[ip]=0.0;
}
}
return COLVARS_ERROR;
}
int eigsrt(cvm::real *d, cvm::real **v)
{
int k,j,i;
cvm::real p;
for (i=0;i<n;i++) {
p=d[k=i];
for (j=i+1;j<n;j++)
if (d[j] >= p) p=d[k=j];
if (k != i) {
d[k]=d[i];
d[i]=p;
for (j=0;j<n;j++) {
p=v[j][i];
v[j][i]=v[j][k];
v[j][k]=p;
}
}
}
return COLVARS_OK;
}
int transpose(cvm::real **v)
{
cvm::real p;
int i,j;
for (i=0;i<n;i++) {
for (j=i+1;j<n;j++) {
p=v[i][j];
v[i][j]=v[j][i];
v[j][i]=p;
}
}
return COLVARS_OK;
}
#undef n
#undef ROTATE

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