static void NonBlockHessenberg(
MatrixView<T> A, VectorView<T> Ubeta)
{
#ifdef XDEBUG
cout<<"Start NonBlock Hessenberg Reduction: A = "<<A<<endl;
Matrix<T> A0(A);
#endif
// Decompose A into U H Ut
// H is a Hessenberg Matrix
// U is a Unitary Matrix
// On output, H is stored in the upper-Hessenberg part of A
// U is stored in compact form in the rest of A along with
// the vector Ubeta.
const ptrdiff_t N = A.rowsize();
TMVAssert(A.colsize() == A.rowsize());
TMVAssert(N > 0);
TMVAssert(Ubeta.size() == N-1);
TMVAssert(A.iscm() || A.isrm());
TMVAssert(!Ubeta.isconj());
TMVAssert(Ubeta.step()==1);
// We use Householder reflections to reduce A to the Hessenberg form:
T* Uj = Ubeta.ptr();
T det = 0; // Ignore Householder det calculations
for(ptrdiff_t j=0;j<N-1;++j,++Uj) {
#ifdef TMVFLDEBUG
TMVAssert(Uj >= Ubeta._first);
TMVAssert(Uj < Ubeta._last);
#endif
*Uj = Householder_Reflect(A.subMatrix(j+1,N,j,N),det);
if (*Uj != T(0))
Householder_LMult(A.col(j+2,N),*Uj,A.subMatrix(0,N,j+1,N).adjoint());
}
#ifdef XDEBUG
Matrix<T> U(N,N,T(0));
U.subMatrix(1,N,1,N) = A.subMatrix(1,N,0,N-1);
U.upperTri().setZero();
Vector<T> Ubeta2(N);
Ubeta2.subVector(1,N) = Ubeta;
Ubeta2(0) = T(0);
GetQFromQR(U.view(),Ubeta2);
Matrix<T> H = A;
if (N>2) LowerTriMatrixViewOf(H).offDiag(2).setZero();
Matrix<T> AA = U*H*U.adjoint();
if (Norm(A0-AA) > 0.001*Norm(A0)) {
cerr<<"NonBlock Hessenberg: A = "<<Type(A)<<" "<<A0<<endl;
cerr<<"A = "<<A<<endl;
cerr<<"Ubeta = "<<Ubeta<<endl;
cerr<<"U = "<<U<<endl;
cerr<<"H = "<<H<<endl;
cerr<<"UHUt = "<<AA<<endl;
abort();
}
#endif
}
static void BlockHessenberg(
MatrixView<T> A, VectorView<T> Ubeta)
{
// Much like the block version of Bidiagonalize, we try to maintain
// the operation of several successive Householder matrices in
// a block form, where the net Block Householder is I - YZYt.
//
// But as with the bidiagonlization algorithm (and unlike a simple
// block QR decomposition), we update the matrix from both the left
// and the right, so we also need to keep track of the product
// ZYtm in addition.
//
// The block update at the end of the block loop is
// m' = (I-YZYt) m (I-YZtYt)
//
// The Y matrix is stored in the first K columns of m,
// and the Hessenberg portion of these columns is updated as we go.
// For the right-hand-side update, m -= mYZtYt, the m on the right
// needs to be the full original matrix m, including the original
// versions of these K columns. Therefore, we can't wait until
// the end for this calculation.
//
// Instead, we keep track of mYZt as we progress, so the final update
// is:
//
// m' = (I-YZYt) (m - mYZt Y)
//
// We also need to do this same calculation for each column as we
// progress through the block.
//
const ptrdiff_t N = A.rowsize();
#ifdef XDEBUG
Matrix<T> A0(A);
#endif
TMVAssert(A.rowsize() == A.colsize());
TMVAssert(N > 0);
TMVAssert(Ubeta.size() == N-1);
TMVAssert(!Ubeta.isconj());
TMVAssert(Ubeta.step()==1);
ptrdiff_t ncolmax = MIN(HESS_BLOCKSIZE,N-1);
Matrix<T,RowMajor> mYZt_full(N,ncolmax);
UpperTriMatrix<T,NonUnitDiag|ColMajor> Z_full(ncolmax);
T det(0); // Ignore Householder Determinant calculations
T* Uj = Ubeta.ptr();
for(ptrdiff_t j1=0;j1<N-1;) {
ptrdiff_t j2 = MIN(N-1,j1+HESS_BLOCKSIZE);
ptrdiff_t ncols = j2-j1;
MatrixView<T> mYZt = mYZt_full.subMatrix(0,N-j1,0,ncols);
UpperTriMatrixView<T> Z = Z_full.subTriMatrix(0,ncols);
for(ptrdiff_t j=j1,jj=0;j<j2;++j,++jj,++Uj) { // jj = j-j1
// Update current column of A
//
// m' = (I - YZYt) (m - mYZt Yt)
// A(0:N,j)' = A(0:N,j) - mYZt(0:N,0:j) Y(j,0:j)t
A.col(j,j1+1,N) -= mYZt.Cols(0,j) * A.row(j,0,j).Conjugate();
//
// A(0:N,j)'' = A(0:N,j) - Y Z Yt A(0:N,j)'
//
// Let Y = (L) where L is unit-diagonal, lower-triangular,
// (M) and M is rectangular
//
LowerTriMatrixView<T> L =
LowerTriMatrixViewOf(A.subMatrix(j1+1,j+1,j1,j),UnitDiag);
MatrixView<T> M = A.subMatrix(j+1,N,j1,j);
// Use the last column of Z as temporary storage for Yt A(0:N,j)'
VectorView<T> YtAj = Z.col(jj,0,jj);
YtAj = L.adjoint() * A.col(j,j1+1,j+1);
YtAj += M.adjoint() * A.col(j,j+1,N);
YtAj = Z.subTriMatrix(0,jj) * YtAj;
A.col(j,j1+1,j+1) -= L * YtAj;
A.col(j,j+1,N) -= M * YtAj;
// Do the Householder reflection
VectorView<T> u = A.col(j,j+1,N);
T bu = Householder_Reflect(u,det);
#ifdef TMVFLDEBUG
TMVAssert(Uj >= Ubeta._first);
TMVAssert(Uj < Ubeta._last);
#endif
*Uj = bu;
// Save the top of the u vector, which isn't actually part of u
T& Atemp = *u.cptr();
TMVAssert(IMAG(Atemp) == RealType(T)(0));
RealType(T) Aorig = REAL(Atemp);
Atemp = RealType(T)(1);
// Update Z
VectorView<T> Zj = Z.col(jj,0,jj);
Zj = -bu * M.adjoint() * u;
Zj = Z * Zj;
Z(jj,jj) = -bu;
// Update mYtZt:
//
// mYZt(0:N,j) = m(0:N,0:N) Y(0:N,0:j) Zt(0:j,j)
// = m(0:N,j+1:N) Y(j+1:N,j) Zt(j,j)
// = bu* m(0:N,j+1:N) u
//
mYZt.col(jj) = CONJ(bu) * A.subMatrix(j1,N,j+1,N) * u;
// Restore Aorig, which is actually part of the Hessenberg matrix.
Atemp = Aorig;
}
// Update the rest of the matrix:
// A(j2,j2-1) needs to be temporarily changed to 1 for use in Y
T& Atemp = *(A.ptr() + j2*A.stepi() + (j2-1)*A.stepj());
TMVAssert(IMAG(Atemp) == RealType(T)(0));
RealType(T) Aorig = Atemp;
Atemp = RealType(T)(1);
// m' = (I-YZYt) (m - mYZt Y)
MatrixView<T> m = A.subMatrix(j1,N,j2,N);
ConstMatrixView<T> Y = A.subMatrix(j2+1,N,j1,j2);
m -= mYZt * Y.adjoint();
BlockHouseholder_LMult(Y,Z,m);
// Restore A(j2,j2-1)
Atemp = Aorig;
j1 = j2;
}
#ifdef XDEBUG
Matrix<T> U(N,N,T(0));
U.subMatrix(1,N,1,N) = A.subMatrix(1,N,0,N-1);
U.upperTri().setZero();
U(0,0) = T(1);
Vector<T> Ubeta2(N);
Ubeta2.subVector(1,N) = Ubeta;
Ubeta2(0) = T(0);
GetQFromQR(U.view(),Ubeta2);
Matrix<T> H = A;
if (N>2) LowerTriMatrixViewOf(H).offDiag(2).setZero();
Matrix<T> AA = U*H*U.adjoint();
if (Norm(A0-AA) > 0.001*Norm(A0)) {
cerr<<"NonBlock Hessenberg: A = "<<Type(A)<<" "<<A0<<endl;
cerr<<"A = "<<A<<endl;
cerr<<"Ubeta = "<<Ubeta<<endl;
cerr<<"U = "<<U<<endl;
cerr<<"H = "<<H<<endl;
cerr<<"UHUt = "<<AA<<endl;
Matrix<T,ColMajor> A2 = A0;
Vector<T> Ub2(Ubeta.size());
NonBlockHessenberg(A2.view(),Ub2.view());
cerr<<"cf NonBlock: A -> "<<A2<<endl;
cerr<<"Ubeta = "<<Ub2<<endl;
abort();
}
#endif
}
