SLAED7 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.  SLAED1 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 occurence the dimension of the


       secular equation problem is reduced by one.  This stage is


       performed by the routine SLAED8.


       The second stage consists of calculating the updated


       eigenvalues. This is done by finding the roots of the secular


       equation via the routine SLAED4 (as called by SLAED9).


       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 REAL array, dimension (N)


         On entry, the eigenvalues of the rank-1-perturbed matrix.


         On exit, the eigenvalues of the repaired matrix.

Q

          Q is REAL 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 REAL


         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 REAL 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 REAL array, dimension (2, N lg N)


         Each number indicates the S value to be used in the


         corresponding Givens rotation.

WORK

          WORK is REAL 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

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

September 2012

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