void inline cuppen_inplace(Matrix& A, Matrix& Q, Vector& lambda)
{
using std::abs; using mtl::irange; using mtl::imax; using mtl::iall; using vector::iota;
typedef typename Collection<Matrix>::value_type value_type;
typedef typename Collection<Matrix>::size_type size_type;
typedef mtl::vector::dense_vector<size_type, vector::parameters<> > size_vector; // todo: with type trait
size_type nrows= num_rows(A);
MTL_THROW_IF(nrows != num_cols(A), matrix_not_square());
const value_type zero= 0, one= 1;
if (nrows == 1){
lambda[0]= A[0][0];
Q= one;
} else {
size_type m= size_type(nrows/2);
irange till_m(m), from_m(m, imax);
size_vector perm(nrows);
Matrix T1(A[till_m][till_m]), T2(A[from_m][from_m]), // sub-matrices of A
Q0(nrows, nrows), Q1(Q0[till_m][till_m]), Q2(Q0[from_m][from_m]); // Q0 and sub-matrices
Vector v(nrows, zero), diag(nrows), lambda1(diag[till_m]), lambda2(diag[from_m]); // sub-vectors of diag
//DIVIDE
value_type b= A[m-1][m];
T1[m-1][m-1]-= abs(b);
T2[0][0]-= abs(b);
v[m-1]= b > zero ? one : -one;
v[m]= one;
cuppen_inplace(T1, Q1, lambda1);
cuppen_inplace(T2, Q2, lambda2);
Q0[till_m][from_m]= zero; Q0[from_m][till_m]= zero; // zero out non-diagonal blocks
T1[m-1][m-1]+= abs(b);
T2[0][0]+= abs(b);
iota(perm);
sort(diag, perm);
// CONQUER, start with eq. (3.0.2) using rows (not columns)
v[till_m]= b < zero ? Vector(-trans(Q1[m-1][iall])) : trans(Q1[m-1][iall]);
v[from_m]= trans(Q2[0][iall]);
// permutation on v
mtl::matrix::traits::permutation<>::type P= mtl::matrix::permutation(perm);
Vector v1(P * v);
lambda= secular(v1, diag, abs(b)); // solve secular equation
// std::cout << "lambda is " << lambda << "\ndiag is " << diag << '\n';
//Lemma 3.0.2 ... calculate eigenvectors
Matrix Q_tilde(nrows, nrows);
for (size_type i = 0; i < nrows; i++) {
for (size_type j= 0; j < size(diag); ++j)
MTL_DEBUG_THROW_IF (diag[j] == lambda[i], logic_error("Can't compute eigenvector, probably due to double eigenvalue."));
Vector li(nrows, lambda[i]), lambda_i(ele_quot(v1, diag - li));
Q_tilde[iall][i]= lambda_i / two_norm(lambda_i); // normalized eigenvector in Matrix Q
}
Q= Q0 * P * Q_tilde;
#if 0
for (size_type i = 0; i < nrows; i++) {
// Vector qi(Q[iall][i]); // Todo: find out valgrind complains about the memory of qi for reasons inexplicable
Vector qi(nrows);
for (size_type j= 0; j < nrows; j++)
qi[j]= Q[j][i];
std::cout << "q[" << i << "] = " << qi << ", lambda[i] = " << lambda[i]
<< ", diff = " << two_norm(Vector(A*qi - lambda[i]*qi)) << ", A*qi = " << Vector(A*qi) << ", li*qi = " << Vector(lambda[i]*qi) << '\n';
itl::basic_iteration<double> iter(1.0, 20, 1e-5, 1e-5);
fsm(A, qi, lambda[i], 0.1, iter);
std::cout << "q[" << i << "] = " << qi << ", diff = " << two_norm(Vector(A*qi - lambda[i]*qi)) << '\n';
Q[iall][i]= qi;
}
#endif
}
}
