inline
void Mp_M(DMatrixSlice* gms_lhs, const M& M_rhs, BLAS_Cpp::Transp trans_rhs) {
Mp_StM(gms_lhs,1.0,M_rhs,trans_rhs);
}
void MatrixSymPosDefLBFGS::update_Q() const
{
using DenseLinAlgPack::tri;
using DenseLinAlgPack::tri_ele;
using DenseLinAlgPack::Mp_StM;
//
// We need update the factorizations to solve for:
//
// x = inv(Q) * y
//
// [ y1 ] = [ (1/gk)*S'S L ] * [ x1 ]
// [ y2 ] [ L' -D ] [ x2 ]
//
// We will solve the above system using the schur complement:
//
// C = (1/gk)*S'S + L*inv(D)*L'
//
// According to the referenced paper, C is p.d. so:
//
// C = J*J'
//
// We then compute the solution as:
//
// x1 = inv(C) * ( y1 + L*inv(D)*y2 )
// x2 = - inv(D) * ( y2 - L'*x1 )
//
// Therefore we will just update the factorization C = J*J'
//
// Form the upper triangular part of C which will become J
// which we are using storage of QJ
if( QJ_.rows() < m_ )
QJ_.resize( m_, m_ );
const size_type
mb = m_bar_;
DMatrixSlice
C = QJ_(1,mb,1,mb);
// C = L * inv(D) * L'
//
// Here L is a strictly lower triangular (zero diagonal) matrix where:
//
// L = [ 0 0 ]
// [ Lb 0 ]
//
// Lb is lower triangular (nonzero diagonal)
//
// Therefore we compute L*inv(D)*L' as:
//
// C = [ 0 0 ] * [ Db 0 ] * [ 0 Lb' ]
// [ Lb 0 ] [ 0 db ] [ 0 0 ]
//
// = [ 0 0 ] = [ 0 0 ]
// [ 0 Cb ] [ 0 Lb*Db*Lb' ]
//
// We need to compute the upper triangular part of Cb.
C.row(1) = 0.0;
if( mb > 1 )
comp_Cb( STY_(2,mb,1,mb-1), STY_.diag(0)(1,mb-1), &C(2,mb,2,mb) );
// C += (1/gk)*S'S
const DMatrixSliceSym &STS = this->STS();
Mp_StM( &C, (1/gamma_k_), tri( STS.gms(), STS.uplo(), BLAS_Cpp::nonunit )
, BLAS_Cpp::trans );
// Now perform a cholesky factorization of C
// After this factorization the upper triangular part of QJ
// (through C) will contain the cholesky factor.
DMatrixSliceTriEle C_upper = tri_ele( C, BLAS_Cpp::upper );
try {
DenseLinAlgLAPack::potrf( &C_upper );
}
catch( const DenseLinAlgLAPack::FactorizationException &fe ) {
TEUCHOS_TEST_FOR_EXCEPTION(
true, UpdateFailedException
,"Error, the factorization of Q which should be s.p.d. failed with"
" the error message: {" << fe.what() << "}";
);
}
