ESMF_DGEHRD Subroutine

subroutine ESMF_DGEHRD(N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO)

\brief \b ESMF_DGEHRD \htmlonly Download ESMF_DGEHRD + dependencies [TGZ] [ZIP] [TXT] \endhtmlonly \par Purpose:

\verbatim

ESMF_DGEHRD reduces a real general matrix A to upper Hessenberg form H by an orthogonal similarity transformation: Q**T * A * Q = H . \endverbatim \param[in] N \verbatim N is INTEGER The order of the matrix A. N >= 0. \endverbatim

\param[in] ILO \verbatim ILO is INTEGER \endverbatim

\param[in] IHI \verbatim IHI is INTEGER

     It is assumed that A is already upper triangular in rows
     and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
     set by a previous call to ESMF_DGEBAL; otherwise they should be
     set to 1 and N respectively. See Further Details.
     1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

\endverbatim

\param[in,out] A \verbatim A is DOUBLE PRECISION array, dimension (LDA,N) On entry, the N-by-N general matrix to be reduced. On exit, the upper triangle and the first subdiagonal of A are overwritten with the upper Hessenberg matrix H, and the elements below the first subdiagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors. 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 (N-1) The scalar factors of the elementary reflectors (see Further Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to zero. \endverbatim

\param[out] WORK \verbatim WORK is DOUBLE PRECISION array, dimension (LWORK) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. \endverbatim

\param[in] LWORK \verbatim LWORK is INTEGER The length of the array WORK. LWORK >= max(1,N). For good performance, LWORK should generally be larger.

     If LWORK = -1, then a workspace query is assumed; the routine
     only calculates the optimal size of the WORK array, returns
     this value as the first entry of the WORK array, and no error
     message related to LWORK is issued by ESMF_XERBLA.

\endverbatim

\param[out] INFO \verbatim INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value. \endverbatim \author Univ. of Tennessee \author Univ. of California Berkeley \author Univ. of Colorado Denver \author NAG Ltd. \date December 2016 \ingroup doubleGEcomputational \par Further Details:

\verbatim

The matrix Q is represented as a product of (ihi-ilo) elementary reflectors

Q = H(ilo) H(ilo+1) . . . H(ihi-1).

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) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on exit in A(i+2:ihi,i), and tau in TAU(i).

The contents of A are illustrated by the following example, with n = 7, ilo = 2 and ihi = 6:

on entry, on exit,

( a a a a a a a ) ( a a h h h h a ) ( a a a a a a ) ( a h h h h a ) ( a a a a a a ) ( h h h h h h ) ( a a a a a a ) ( v2 h h h h h ) ( a a a a a a ) ( v2 v3 h h h h ) ( a a a a a a ) ( v2 v3 v4 h h h ) ( 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 file is a slight modification of LAPACK-3.0’s ESMF_DGEHRD subroutine incorporating improvements proposed by Quintana-Orti and Van de Geijn (2006). (See ESMF_DLAHR2.) \endverbatim

Arguments