ESMF_DLAED1 Subroutine

subroutine ESMF_DLAED1(N, D, Q, LDQ, INDXQ, RHO, CUTPNT, WORK, IWORK, INFO)

\brief \b ESMF_DLAED1 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 tridiagonal. \htmlonly Download ESMF_DLAED1 + dependencies [TGZ] [ZIP] [TXT] \endhtmlonly \par Purpose:

\verbatim

ESMF_DLAED1 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 eigenvectors of a tridiagonal matrix. ESMF_DLAED7 handles the case in which eigenvalues only or eigenvalues and eigenvectors of a full symmetric matrix (which was reduced to tridiagonal form) 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_DLAED2.

  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_DLAED3).
  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] N \verbatim N is INTEGER The dimension of the symmetric tridiagonal matrix. N >= 0. \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[in,out] INDXQ \verbatim INDXQ is INTEGER array, dimension (N) On entry, the permutation which separately sorts the two subproblems in D into ascending order. On exit, the permutation which will reintegrate the subproblems 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 entry used to create the rank-1 modification. \endverbatim

\param[in] CUTPNT \verbatim CUTPNT is INTEGER The location of the last eigenvalue in the leading sub-matrix. min(1,N) <= CUTPNT <= N/2. \endverbatim

\param[out] WORK \verbatim WORK is DOUBLE PRECISION array, dimension (4N + N*2) \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 \n Modified by Francoise Tisseur, University of Tennessee

Arguments