ESMF_DLAHR2 Subroutine

subroutine ESMF_DLAHR2(N, K, NB, A, LDA, TAU, T, LDT, Y, LDY)

\brief \b ESMF_DLAHR2 reduces the specified number of first columns of a general \htmlonly Download ESMF_DLAHR2 + dependencies [TGZ] [ZIP] [TXT] \endhtmlonly \par Purpose:

\verbatim

ESMF_DLAHR2 reduces the first NB columns of A real general n-BY-(n-k+1) matrix A so that elements below the k-th subdiagonal are zero. The reduction is performed by an orthogonal similarity transformation QT * A * Q. The routine returns the matrices V and T which determine Q as a block reflector I - VTVT, and also the matrix Y = A * V * T.

This is an auxiliary routine called by ESMF_DGEHRD. \endverbatim \param[in] N \verbatim N is INTEGER The order of the matrix A. \endverbatim

\param[in] K \verbatim K is INTEGER The offset for the reduction. Elements below the k-th subdiagonal in the first NB columns are reduced to zero. K < N. \endverbatim

\param[in] NB \verbatim NB is INTEGER The number of columns to be reduced. \endverbatim

\param[in,out] A \verbatim A is DOUBLE PRECISION array, dimension (LDA,N-K+1) On entry, the n-by-(n-k+1) general matrix A. On exit, the elements on and above the k-th subdiagonal in the first NB columns are overwritten with the corresponding elements of the reduced matrix; the elements below the k-th subdiagonal, with the array TAU, represent the matrix Q as a product of elementary reflectors. The other columns of A are unchanged. See Further Details. \endverbatim

\param[in] LDA \verbatim LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). \endverbatim

\param[out] TAU \verbatim TAU is DOUBLE PRECISION array, dimension (NB) The scalar factors of the elementary reflectors. See Further Details. \endverbatim

\param[out] T \verbatim T is DOUBLE PRECISION array, dimension (LDT,NB) The upper triangular matrix T. \endverbatim

\param[in] LDT \verbatim LDT is INTEGER The leading dimension of the array T. LDT >= NB. \endverbatim

\param[out] Y \verbatim Y is DOUBLE PRECISION array, dimension (LDY,NB) The n-by-nb matrix Y. \endverbatim

\param[in] LDY \verbatim LDY is INTEGER The leading dimension of the array Y. LDY >= N. \endverbatim \author Univ. of Tennessee \author Univ. of California Berkeley \author Univ. of Colorado Denver \author NAG Ltd. \date December 2016 \ingroup doubleOTHERauxiliary \par Further Details:

\verbatim

The matrix Q is represented as a product of nb elementary reflectors

Q = H(1) H(2) . . . H(nb).

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in A(i+k+1:n,i), and tau in TAU(i).

The elements of the vectors v together form the (n-k+1)-by-nb matrix V which is needed, with T and Y, to apply the transformation to the unreduced part of the matrix, using an update of the form: A := (I - VTVT) * (A - Y*VT).

The contents of A on exit are illustrated by the following example with n = 7, k = 3 and nb = 2:

( a   a   a   a   a )
( a   a   a   a   a )
( a   a   a   a   a )
( h   h   a   a   a )
( v1  h   a   a   a )
( v1  v2  a   a   a )
( v1  v2  a   a   a )

where a denotes an element of the original matrix A, h denotes a modified element of the upper Hessenberg matrix H, and vi denotes an element of the vector defining H(i).

This subroutine is a slight modification of LAPACK-3.0’s DLAHRD incorporating improvements proposed by Quintana-Orti and Van de Gejin. Note that the entries of A(1:K,2:NB) differ from those returned by the original LAPACK-3.0’s DLAHRD routine. (This subroutine is not backward compatible with LAPACK-3.0’s DLAHRD.) \endverbatim \par References:

Gregorio Quintana-Orti and Robert van de Geijn, “Improving the performance of reduction to Hessenberg form,” ACM Transactions on Mathematical Software, 32(2):180-194, June 2006.

Arguments