!>
!> 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.  DLAED1 handles
!> the case in which all eigenvalues and eigenvectors of a symmetric
!> tridiagonal matrix are desired.
!>
!>   T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(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 DLAED8.
!>
!>       The second stage consists of calculating the updated
!>       eigenvalues. This is done by finding the roots of the secular
!>       equation via the routine DLAED4 (as called by 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.
!> 

ICOMPQ

!>          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.
!> 

N

!>          N is INTEGER
!>         The dimension of the symmetric tridiagonal matrix.  N >= 0.
!> 

QSIZ

!>          QSIZ is INTEGER
!>         The dimension of the orthogonal matrix used to reduce
!>         the full matrix to tridiagonal form.  QSIZ >= N if ICOMPQ = 1.
!> 

TLVLS

!>          TLVLS is INTEGER
!>         The total number of merging levels in the overall divide and
!>         conquer tree.
!> 

CURLVL

!>          CURLVL is INTEGER
!>         The current level in the overall merge routine,
!>         0 <= CURLVL <= TLVLS.
!> 

CURPBM

!>          CURPBM is INTEGER
!>         The current problem in the current level in the overall
!>         merge routine (counting from upper left to lower right).
!> 

D

!>          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.
!> 

Q

!>          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.
!> 

LDQ

!>          LDQ is INTEGER
!>         The leading dimension of the array Q.  LDQ >= max(1,N).
!> 

INDXQ

!>          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.
!> 

RHO

!>          RHO is DOUBLE PRECISION
!>         The subdiagonal element used to create the rank-1
!>         modification.
!> 

CUTPNT

!>          CUTPNT is INTEGER
!>         Contains the location of the last eigenvalue in the leading
!>         sub-matrix.  min(1,N) <= CUTPNT <= N.
!> 

QSTORE

!>          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.
!> 

QPTR

!>          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.
!> 

PRMPTR

!>          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.
!> 

PERM

!>          PERM is INTEGER array, dimension (N lg N)
!>         Contains the permutations (from deflation and sorting) to be
!>         applied to each eigenblock.
!> 

GIVPTR

!>          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.
!> 

GIVCOL

!>          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.
!> 

GIVNUM

!>          GIVNUM is DOUBLE PRECISION array, dimension (2, N lg N)
!>         Each number indicates the S value to be used in the
!>         corresponding Givens rotation.
!> 

WORK

!>          WORK is DOUBLE PRECISION array, dimension (3*N+2*QSIZ*N)
!> 

IWORK

!>          IWORK is INTEGER array, dimension (4*N)
!> 

INFO

!>          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
!> 

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Univ. of California Berkeley

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