CLAED8 merges the two sets of eigenvalues together into a single


 sorted set.  Then it tries to deflate the size of the problem.


 There are two ways in which deflation can occur:  when two or more


 eigenvalues are close together or if there is a tiny element in the


 Z vector.  For each such occurrence the order of the related secular


 equation problem is reduced by one.

K

          K is INTEGER


         Contains the number of non-deflated eigenvalues.


         This is the order of the related secular equation.

N

          N is INTEGER


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

QSIZ

          QSIZ is INTEGER


         The dimension of the unitary matrix used to reduce


         the dense or band matrix to tridiagonal form.


         QSIZ >= N if ICOMPQ = 1.

Q

          Q is COMPLEX array, dimension (LDQ,N)


         On entry, Q contains the eigenvectors of the partially solved


         system which has been previously updated in matrix


         multiplies with other partially solved eigensystems.


         On exit, Q contains the trailing (N-K) updated eigenvectors


         (those which were deflated) in its last N-K columns.

LDQ

          LDQ is INTEGER


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

D

          D is REAL array, dimension (N)


         On entry, D contains the eigenvalues of the two submatrices to


         be combined.  On exit, D contains the trailing (N-K) updated


         eigenvalues (those which were deflated) sorted into increasing


         order.

RHO

          RHO is REAL


         Contains the off diagonal element associated with the rank-1


         cut which originally split the two submatrices which are now


         being recombined. RHO is modified during the computation to


         the value required by SLAED3.

CUTPNT

          CUTPNT is INTEGER


         Contains the location of the last eigenvalue in the leading


         sub-matrix.  MIN(1,N) <= CUTPNT <= N.

Z

          Z is REAL array, dimension (N)


         On input this vector contains the updating vector (the last


         row of the first sub-eigenvector matrix and the first row of


         the second sub-eigenvector matrix).  The contents of Z are


         destroyed during the updating process.

DLAMDA

          DLAMDA is REAL array, dimension (N)


         Contains a copy of the first K eigenvalues which will be used


         by SLAED3 to form the secular equation.

Q2

          Q2 is COMPLEX array, dimension (LDQ2,N)


         If ICOMPQ = 0, Q2 is not referenced.  Otherwise,


         Contains a copy of the first K eigenvectors which will be used


         by SLAED7 in a matrix multiply (SGEMM) to update the new


         eigenvectors.

LDQ2

          LDQ2 is INTEGER


         The leading dimension of the array Q2.  LDQ2 >= max( 1, N ).

W

          W is REAL array, dimension (N)


         This will hold the first k values of the final


         deflation-altered z-vector and will be passed to SLAED3.

INDXP

          INDXP is INTEGER array, dimension (N)


         This will contain the permutation used to place deflated


         values of D at the end of the array. On output INDXP(1:K)


         points to the nondeflated D-values and INDXP(K+1:N)


         points to the deflated eigenvalues.

INDX

          INDX is INTEGER array, dimension (N)


         This will contain the permutation used to sort the contents of


         D into ascending order.

INDXQ

          INDXQ is INTEGER array, dimension (N)


         This contains the permutation which separately sorts the two


         sub-problems in D into ascending order.  Note that elements in


         the second half of this permutation must first have CUTPNT


         added to their values in order to be accurate.

PERM

          PERM is INTEGER array, dimension (N)


         Contains the permutations (from deflation and sorting) to be


         applied to each eigenblock.

GIVPTR

          GIVPTR is INTEGER


         Contains the number of Givens rotations which took place in


         this subproblem.

GIVCOL

          GIVCOL is INTEGER array, dimension (2, 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)


         Each number indicates the S value to be used in the


         corresponding Givens rotation.

INFO

          INFO is INTEGER


          = 0:  successful exit.


          < 0:  if INFO = -i, the i-th argument had an illegal value.

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December 2016