DMatrix orthonormalize(const matrix_T& mat) {
// after the method of Augustin A. Dubrulle:
// "An Optimum Iteration for the Matrix Polar Decomposition", Elect. Trans. Num. Anal, 1999
BOOST_ASSERT(mat.size1() == mat.size2());
// std::cout << "fixing " << ublas::prod(mat,ublas::trans(mat)) << " with det " << determinant(mat) << std::flush;
DMatrix A = mat;
DMatrix W = A;
double eps = 1e-15;
double limit = (1+eps)*sqrt(mat.size1());
A = inv(ublas::trans(W));
double g = sqrt(norm_frobenius(A)/norm_frobenius(W)); // frobenius norm
W = 0.5*(g*W+(1/g)*A);
double f = norm_frobenius(W);
double pf = DBL_MAX;
while ((f > limit) && (f < pf)) {
// std::cout << "." << std::flush;
pf = f;
A = inv(ublas::trans(W));
g = sqrt(norm_frobenius(A)/f);
W = 0.5*(g*W+(1/g)*A);
f = norm_frobenius(W);
}
// now, W^T*W=I, i.e. W^T = inv(W)
//W /= determinant(W); ?? Necessary??
// std::cout << "to " << ublas::prod(mat,ublas::trans(W)) << " with det " << determinant(W) << std::flush;
return W;
}
int do_memory_type(int n, W workspace) {
typedef typename boost::numeric::bindings::traits::type_traits<T>::real_type real_type ;
typedef std::complex< real_type > complex_type ;
typedef ublas::matrix<T, ublas::column_major> matrix_type ;
typedef ublas::vector<T> vector_type ;
// Set matrix
matrix_type a( n, n );
vector_type tau( n );
randomize( a );
matrix_type a2( a );
matrix_type a3( a );
// Compute QR factorization.
lapack::geqrf( a, tau, workspace ) ;
// Apply the orthogonal transformations to a2
lapack::ormqr( 'L', transpose<T>::value, a, tau, a2, workspace );
// The upper triangular parts of a and a2 must be equal.
if (norm_frobenius( upper_part( a - a2 ) )
> std::numeric_limits<real_type>::epsilon() * 10.0 * norm_frobenius( upper_part( a ) ) ) return 255 ;
// Generate orthogonal matrix
lapack::orgqr( a, tau, workspace );
// The result of lapack::ormqr and the equivalent matrix product must be equal.
if (norm_frobenius( a2 - prod(herm(a), a3) )
> std::numeric_limits<real_type>::epsilon() * 10.0 * norm_frobenius( a2 ) ) return 255 ;
return 0 ;
} // do_value_type()
void cofi::NormEvaluator::eval(cofi::Problem& p, std::map<std::string, double>& results){
results[NU] = norm_frobenius(p.getU());
results[NM] = norm_frobenius(p.getM());
if(p.usingGraphKernel()){
results[NA] = norm_frobenius(p.getA());
}
else{
results[NA] = -1.0;
}
}
