dVector ortogonalizacao(const dVector u, const dVector v)
{
double escalar = produtoEscalar(u, v) / produtoEscalar(v, v);
dVector vetor(u.size());
for(int i = 0;i< u.size();i++)vetor[i]=u[i] - escalar*v[i];
return vetor;
}
// Adds the dVector b to the dVector a. a is modified in-place.
void RKF45::sum_in_place(dVector& a, const dVector& b) const {
assert ( a.size() == b.size()
&& "To sum two vectors, they must be the same size.");
for (unsigned int i = 0; i < a.size(); i++) {
a[i] += b[i];
}
}
// Adds k*b to the dVector a. k is a scalar. b is a dVector.
void RKF45::linear_combination_in_place(dVector& a,
double k, const dVector& b) const {
assert ( a.size() == b.size()
&& "To sum two vectors, they must be the same size.");
for (unsigned int i = 0; i < a.size(); i++) {
a[i] += k*b[i];
}
}
// Takes the norm of the difference between two dVectors.
double RKF45::norm_difference(const dVector& a, const dVector& b) const {
assert ( a.size() == b.size()
&& "To sum two vectors, they must be the same size.");
double output = 0;
for (unsigned int i = 0; i < a.size(); i++) {
output += (a[i] - b[i]) * (a[i] - b[i]);
}
return sqrt(output);
}
// Adds two dVectors a and b and outputs a new dVector that is their
// sum. Assumes that the vectors are the same size. If they're not,
// raises an error.
dVector RKF45::sum(const dVector& a, const dVector& b) const {
assert ( a.size() == b.size()
&& "To sum two vectors, they must be the same size.");
dVector output;
int size = a.size();
output.resize(size);
for (int i = 0; i < size; i++) {
output[i] = a[i] + b[i];
}
return output;
}
// ----------------------------------------------------------------------
double derivative(int i, const dVector& v, double lattice_spacing) {
// A local "num points" variable. Used for simplicity.
int num_points = get_num_points(v);
assert (0 <= i && i < (int)v.size()
&& "The ith elemennt must be in the vector");
if ( i % num_points == 0 ) { // We're at the lower boundary of the grid.
if (DEBUGGING) {
cout << "\tUsing forward difference." << endl;
}
return forward_difference(i,v,lattice_spacing);
}
else if (i % num_points == num_points - 1) {
// We're at the upper boundary of the grid.
if (DEBUGGING) {
cout << "\tUsing backward difference." << endl;
}
return backward_difference(i,v,lattice_spacing);
}
else { // We're in the middle of the grid.
if (DEBUGGING) {
cout << "\tUsing centered difference." << endl;
}
return centered_difference(i,v,lattice_spacing);
}
}
void Quaternion::set_v(const dVector & v)
{
if(v.size() == 3) {
v_ = v;
}
else {
assert (0 && "Quaternion::set_v: input has a wrong size.");
}
}
Quaternion::Quaternion(const double angle, const dVector & axis)
{
if(axis.size() != 3) {
assert (0 && "Quaternion::Quaternion, size of axis != 3");
return;
}
// make sure axis is a unit vector
v_ = sin(angle/2) * axis/Norm2(axis);
s_ = cos(angle/2);
}
// ----------------------------------------------------------------------
// Iterative Function
// ----------------------------------------------------------------------
dVector f(double t, const dVector& y, const dVector& optional_args) {
// t is not used. But it is required by the integrator.
// The optional arguments vector contains the lattice spacing and c^2.
double h = optional_args[0]; // h = lattice spacing
double c2 = optional_args[1]; // c2 = c^2.
// For convenience, get the number of points in the lattice
int num_points = get_num_points(y);
// The output dVector. Starts empty.
dVector output(y.size(),0);
// The order of elements in v is s,r,u. So we have the following system:
// (d/dt) [s,r,u] = [c^2 (dr/dx), (ds/dx), s]
// Fill the output array
for (int i = 0; i < num_points; i++) {
// Fill the (du/dt) terms with s.
// ES: A nicer notation would allow you to write e.g.
// "output[i].*s" or "output.*s[i]" (see "member pointers), or
// "output[i][s]", or maybe even "s[i]" where "s" is a local
// variable defined before the loop.
output[get_linear_index_u(i,output)] = get_ith_s(i,y);
// Fill the (dr/dt) vectors with (ds/dx).
// ES: A nicer notation would allow you to write e.g.
// "diff(h,y,i,s)" or "diff(h,y,s,i)" instead.
output[get_linear_index_r(i,output)] = derivative_for_s(i,y,h);
// Fill the (ds/dt) vectors with c^2(dr/dx).
output[get_linear_index_s(i,output)] = c2 * derivative_for_r(i,y,h);
}
// (du/dt) at the boundary is forced to be zero. Impose this.
// ES: What about the boundary conditions for the other fields?
output[get_linear_index_s(0,output)] = 0;
output[get_linear_index_s(num_points-1,output)] = 0;
// That's it. Return the array!
return output;
}
dVector subtracao(dVector u, dVector v){
dVector retorno(v.size());
for (int i = 0; i < v.size(); i++)
retorno[i] = u[i] - v[i];
return retorno;
}
dVector soma(dVector u, dVector v){
dVector retorno(v.size());
for (int i = 0; i < v.size(); i++)
retorno[i] = u[i] + v[i];
return retorno;
}
dVector normalize(dVector v)
{
double norma = getNorma(v);
for(int i = 0;i< v.size();i++) v[i]=v[i]/norma;
return v;
}
double getNorma(const dVector v)
{
double retorno = 0;
for(int i = 0;i< v.size();i++) retorno+=pow(v[i], 2);
return (sqrt(retorno));
}
double produtoEscalar(const dVector u, const dVector v)
{
double retorno =0;
for(int i = 0;i< u.size();i++) retorno+=u[i]*v[i];
return retorno;
}
// ----------------------------------------------------------------------
int get_num_points(const dVector& v) {
assert ( v.size() % NUM_VAR_TYPES == 0
&& "The vector must be for the variables of the system.");
return v.size() / 3;
}