/**
* Computes the eigenvalues and eigenvectors of a symmetric matrix, using
* the jacobi eigenvalue algorithm.
*
* @pre matrix must be square
* @pre matrix must be symmetric
* @pre matrix must be at least 2x2
* @param input symmetric n x n matrix
* @return tuple containing the n-vector of eigenvalues, a matrix of eigenvectors and the number of rotations performed.
*/
std::tuple<vec, mat, uint32_t> eigenvalues(mat input){
// @pre matrix must be square
assert(input.n_cols == input.n_rows);
// @pre matrix must be symmetric
assert(util::check_symmetric(input));
// @pre matrix must be at least 2x2
assert(input.n_cols >= 2);
mat current = input;
mat eigenvectors = arma::eye(input.n_cols, input.n_cols);
uint32_t iteration = 1;
while(true){
auto largest = util::find_largest(current);
if(std::abs(current(std::get<0>(largest), std::get<1>(largest))) < GLOBAL_PRECISION_THRESHOLD){
//std::cout << "Desired precision reached, exiting." << std::endl;
break;
}
//std::cout << "Iteration #" << iteration << ": iterating about " << std::get<0>(largest)
// << ", " << std::get<1>(largest) << std::endl;
std::tie(current, eigenvectors) = givens_rotation(current, largest, eigenvectors);
//std::cout << "Matrix after iteration: " << std::endl << current << std::endl;
iteration++;
}
return std::make_tuple(current.diag(0), eigenvectors, iteration);
}
void tridiagonalQRStep(std::vector<double> &diag, std::vector<double> &subdiag, int start, int end, e::Matrix<double, 3, 3> &Q)
{
double td = (diag[end-1] - diag[end])*0.5;
double e2 = subdiag[end-1]*subdiag[end-1];
double mu = diag[end] - e2/(td + (td>0? 1: -1)*std::sqrt(td*td + e2));
double x = diag[start] - mu;
double z = subdiag[start];
for(int k = start; k < end; ++k)
{
double c, s;
givens_rotation(x, z, c, s);
//do T = G' T G
double sdk = s * diag[k] + c * subdiag[k];
double dkp1 = s*subdiag[k] + c*diag[k + 1];
diag[k] = c * (c * diag[k] - s * subdiag[k]) - s * (c * subdiag[k] - s * diag[k+1]);
diag[k+1] = s * sdk + c * dkp1;
subdiag[k] = c * sdk - s * dkp1;
if (k > start)
subdiag[k - 1] = c * subdiag[k-1] - s * z;
x = subdiag[k];
if (k < end - 1)
{
z = -s * subdiag[k+1];
subdiag[k + 1] = c * subdiag[k+1];
}
//apply givens rotation
//this only modifies two columns k and k+1
//0
double m_i_k = Q(0, k);
double m_i_k1 = Q(0, k+1);
Q(0, k) = c*m_i_k - s*m_i_k1;
Q(0, k+1) = s*m_i_k + c*m_i_k1;
//1
m_i_k = Q(1, k);
m_i_k1 = Q(1, k+1);
Q(1, k) = c*m_i_k - s*m_i_k1;
Q(1, k+1) = s*m_i_k + c*m_i_k1;
//2
m_i_k = Q(2, k);
m_i_k1 = Q(2, k+1);
Q(2, k) = c*m_i_k - s*m_i_k1;
Q(2, k+1) = s*m_i_k + c*m_i_k1;
}
}