void Ricatti( Matrix<F>& W, Matrix<F>& X, SignCtrl<Base<F>> ctrl )
{
DEBUG_ONLY(CallStackEntry cse("Ricatti"))
Sign( W, ctrl );
const Int n = W.Height()/2;
Matrix<F> WTL, WTR,
WBL, WBR;
PartitionDownDiagonal
( W, WTL, WTR,
WBL, WBR, n );
// (ML, MR) = sgn(W) - I
UpdateDiagonal( W, F(-1) );
// Solve for X in ML X = -MR
Matrix<F> ML, MR;
PartitionRight( W, ML, MR, n );
Scale( F(-1), MR );
LeastSquares( NORMAL, ML, MR, X );
}
inline ValueInt<BASE(F)>
QDWHDivide
( UpperOrLower uplo, DistMatrix<F>& A, DistMatrix<F>& G, bool returnQ=false )
{
DEBUG_ONLY(CallStackEntry cse("herm_eig::QDWHDivide"))
// G := sgn(G)
// G := 1/2 ( G + I )
herm_polar::QDWH( uplo, G );
UpdateDiagonal( G, F(1) );
Scale( F(1)/F(2), G );
// Compute the pivoted QR decomposition of the spectral projection
const Grid& g = A.Grid();
DistMatrix<F,MD,STAR> t(g);
DistMatrix<Int,VR,STAR> p(g);
elem::QR( G, t, p );
// A := Q^H A Q
MakeHermitian( uplo, A );
const Base<F> oneA = OneNorm( A );
if( returnQ )
{
ExpandPackedReflectors( LOWER, VERTICAL, CONJUGATED, 0, G, t );
DistMatrix<F> B(g);
Gemm( ADJOINT, NORMAL, F(1), G, A, B );
Gemm( NORMAL, NORMAL, F(1), B, G, A );
}
else
{
qr::ApplyQ( LEFT, ADJOINT, G, t, A );
qr::ApplyQ( RIGHT, NORMAL, G, t, A );
}
// Return || E21 ||1 / || A ||1 and the chosen rank
auto part = ComputePartition( A );
part.value /= oneA;
return part;
}