void Riccati
( ElementalMatrix<F>& WPre,
ElementalMatrix<F>& X,
SignCtrl<Base<F>> ctrl )
{
DEBUG_CSE
DistMatrixReadProxy<F,F,MC,MR> WProx( WPre );
auto& W = WProx.Get();
const Grid& g = W.Grid();
Sign( W, ctrl );
const Int n = W.Height()/2;
DistMatrix<F> WTL(g), WTR(g),
WBL(g), WBR(g);
PartitionDownDiagonal
( W, WTL, WTR,
WBL, WBR, n );
// (ML, MR) = sgn(W) - I
ShiftDiagonal( W, F(-1) );
// Solve for X in ML X = -MR
DistMatrix<F> ML(g), MR(g);
PartitionRight( W, ML, MR, n );
MR *= -1;
ls::Overwrite( NORMAL, ML, MR, X );
}
void Sylvester
( Int m,
ElementalMatrix<F>& WPre,
ElementalMatrix<F>& X,
SignCtrl<Base<F>> ctrl )
{
EL_DEBUG_CSE
DistMatrixReadProxy<F,F,MC,MR> WProx( WPre );
auto& W = WProx.Get();
const Grid& g = W.Grid();
Sign( W, ctrl );
DistMatrix<F> WTL(g), WTR(g),
WBL(g), WBR(g);
PartitionDownDiagonal
( W, WTL, WTR,
WBL, WBR, m );
// WTL and WBR should be the positive and negative identity, WBL should be
// zero, and WTR should be -2 X
Copy( WTR, X );
X *= -F(1)/F(2);
// TODO: Think of how to probe for checks on other quadrants.
// Add UpdateDiagonal routine to avoid explicit identity Axpy?
/*
typedef Base<F> Real;
UpdateDiagonal( WTL, F(-1) );
const Real errorWTL = FrobeniusNorm( WTL );
const Int n = W.Height() - m;
UpdateDiagonal( WBR, F(1) );
const Real errorWBR = FrobeniusNorm( WBR );
const Real errorWBL = FrobeniusNorm( WBL );
*/
}
inline int
Lyapunov( const DistMatrix<F>& A, const DistMatrix<F>& C, DistMatrix<F>& X )
{
#ifndef RELEASE
CallStackEntry cse("Sylvester");
if( A.Height() != A.Width() )
LogicError("A must be square");
if( C.Height() != A.Height() || C.Width() != A.Height() )
LogicError("C must conform with A");
if( A.Grid() != C.Grid() )
LogicError("A and C must have the same grid");
#endif
const Grid& g = A.Grid();
const Int m = A.Height();
DistMatrix<F> W(g), WTL(g), WTR(g),
WBL(g), WBR(g);
Zeros( W, 2*m, 2*m );
PartitionDownDiagonal
( W, WTL, WTR,
WBL, WBR, m );
WTL = A;
Adjoint( A, WBR ); Scale( F(-1), WBR );
WTR = C; Scale( F(-1), WTR );
return Sylvester( m, W, X );
}
