ESMF_DLAED7 Subroutine

subroutine ESMF_DLAED7(ICOMPQ, N, QSIZ, TLVLS, CURLVL, CURPBM, D, Q, LDQ, INDXQ, RHO, CUTPNT, QSTORE, QPTR, PRMPTR, PERM, GIVPTR, GIVCOL, GIVNUM, WORK, IWORK, INFO)

\brief \b ESMF_DLAED7 used by sstedc. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense. \htmlonly Download ESMF_DLAED7 + dependencies [TGZ] [ZIP] [TXT] \endhtmlonly \par Purpose:

\verbatim

ESMF_DLAED7 computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. This routine is used only for the eigenproblem which requires all eigenvalues and optionally eigenvectors of a dense symmetric matrix that has been reduced to tridiagonal form. ESMF_DLAED1 handles the case in which all eigenvalues and eigenvectors of a symmetric tridiagonal matrix are desired.

T = Q(in) ( D(in) + RHO * ZZT ) QT(in) = Q(out) * D(out) * Q*T(out)

where Z = Q**Tu, u is a vector of length N with ones in the CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.

The eigenvectors of the original matrix are stored in Q, and the eigenvalues are in D. The algorithm consists of three stages:

  The first stage consists of deflating the size of the problem
  when there are multiple eigenvalues or if there is a zero in
  the Z vector.  For each such occurrence the dimension of the
  secular equation problem is reduced by one.  This stage is
  performed by the routine ESMF_DLAED8.

  The second stage consists of calculating the updated
  eigenvalues. This is done by finding the roots of the secular
  equation via the routine ESMF_DLAED4 (as called by ESMF_DLAED9).
  This routine also calculates the eigenvectors of the current
  problem.

  The final stage consists of computing the updated eigenvectors
  directly using the updated eigenvalues.  The eigenvectors for
  the current problem are multiplied with the eigenvectors from
  the overall problem.

\endverbatim \param[in] ICOMPQ \verbatim ICOMPQ is INTEGER = 0: Compute eigenvalues only. = 1: Compute eigenvectors of original dense symmetric matrix also. On entry, Q contains the orthogonal matrix used to reduce the original matrix to tridiagonal form. \endverbatim

\param[in] N \verbatim N is INTEGER The dimension of the symmetric tridiagonal matrix. N >= 0. \endverbatim

\param[in] QSIZ \verbatim QSIZ is INTEGER The dimension of the orthogonal matrix used to reduce the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1. \endverbatim

\param[in] TLVLS \verbatim TLVLS is INTEGER The total number of merging levels in the overall divide and conquer tree. \endverbatim

\param[in] CURLVL \verbatim CURLVL is INTEGER The current level in the overall merge routine, 0 <= CURLVL <= TLVLS. \endverbatim

\param[in] CURPBM \verbatim CURPBM is INTEGER The current problem in the current level in the overall merge routine (counting from upper left to lower right). \endverbatim

\param[in,out] D \verbatim D is DOUBLE PRECISION array, dimension (N) On entry, the eigenvalues of the rank-1-perturbed matrix. On exit, the eigenvalues of the repaired matrix. \endverbatim

\param[in,out] Q \verbatim Q is DOUBLE PRECISION array, dimension (LDQ, N) On entry, the eigenvectors of the rank-1-perturbed matrix. On exit, the eigenvectors of the repaired tridiagonal matrix. \endverbatim

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

\param[out] INDXQ \verbatim INDXQ is INTEGER array, dimension (N) The permutation which will reintegrate the subproblem just solved back into sorted order, i.e., D( INDXQ( I = 1, N ) ) will be in ascending order. \endverbatim

\param[in] RHO \verbatim RHO is DOUBLE PRECISION The subdiagonal element used to create the rank-1 modification. \endverbatim

\param[in] CUTPNT \verbatim CUTPNT is INTEGER Contains the location of the last eigenvalue in the leading sub-matrix. min(1,N) <= CUTPNT <= N. \endverbatim

\param[in,out] QSTORE \verbatim QSTORE is DOUBLE PRECISION array, dimension (N**2+1) Stores eigenvectors of submatrices encountered during divide and conquer, packed together. QPTR points to beginning of the submatrices. \endverbatim

\param[in,out] QPTR \verbatim QPTR is INTEGER array, dimension (N+2) List of indices pointing to beginning of submatrices stored in QSTORE. The submatrices are numbered starting at the bottom left of the divide and conquer tree, from left to right and bottom to top. \endverbatim

\param[in] PRMPTR \verbatim PRMPTR is INTEGER array, dimension (N lg N) Contains a list of pointers which indicate where in PERM a level’s permutation is stored. PRMPTR(i+1) - PRMPTR(i) indicates the size of the permutation and also the size of the full, non-deflated problem. \endverbatim

\param[in] PERM \verbatim PERM is INTEGER array, dimension (N lg N) Contains the permutations (from deflation and sorting) to be applied to each eigenblock. \endverbatim

\param[in] GIVPTR \verbatim GIVPTR is INTEGER array, dimension (N lg N) Contains a list of pointers which indicate where in GIVCOL a level’s Givens rotations are stored. GIVPTR(i+1) - GIVPTR(i) indicates the number of Givens rotations. \endverbatim

\param[in] GIVCOL \verbatim GIVCOL is INTEGER array, dimension (2, N lg N) Each pair of numbers indicates a pair of columns to take place in a Givens rotation. \endverbatim

\param[in] GIVNUM \verbatim GIVNUM is DOUBLE PRECISION array, dimension (2, N lg N) Each number indicates the S value to be used in the corresponding Givens rotation. \endverbatim

\param[out] WORK \verbatim WORK is DOUBLE PRECISION array, dimension (3N+2QSIZ*N) \endverbatim

\param[out] IWORK \verbatim IWORK is INTEGER array, dimension (4*N) \endverbatim

\param[out] INFO \verbatim INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = 1, an eigenvalue did not converge \endverbatim \author Univ. of Tennessee \author Univ. of California Berkeley \author Univ. of Colorado Denver \author NAG Ltd. \date June 2016 \ingroup auxOTHERcomputational \par Contributors:

Jeff Rutter, Computer Science Division, University of California at Berkeley, USA

Arguments