SLAED8 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.

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.

K

          K is INTEGER


         The number of non-deflated eigenvalues, and 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 orthogonal matrix used to reduce


         the full matrix to tridiagonal form.  QSIZ >= N if ICOMPQ = 1.

D

          D is REAL array, dimension (N)


         On entry, the eigenvalues of the two submatrices to be


         combined.  On exit, the trailing (N-K) updated eigenvalues


         (those which were deflated) sorted into increasing order.

Q

          Q is REAL array, dimension (LDQ,N)


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


         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).

INDXQ

          INDXQ is INTEGER array, dimension (N)


         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.

RHO

          RHO is REAL


         On entry, the off-diagonal element associated with the rank-1


         cut which originally split the two submatrices which are now


         being recombined.


         On exit, RHO has been modified to the value required by


         SLAED3.

CUTPNT

          CUTPNT is INTEGER


         The location of the last eigenvalue in the leading


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

Z

          Z is REAL array, dimension (N)


         On entry, Z contains the updating vector (the last row of


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


         second sub-eigenvector matrix).


         On exit, the contents of Z are destroyed by the updating


         process.

DLAMDA

          DLAMDA is REAL array, dimension (N)


         A copy of the first K eigenvalues which will be used by


         SLAED3 to form the secular equation.

Q2

          Q2 is REAL array, dimension (LDQ2,N)


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


         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)


         The first k values of the final deflation-altered z-vector and


         will be passed to SLAED3.

PERM

          PERM is INTEGER array, dimension (N)


         The permutations (from deflation and sorting) to be applied


         to each eigenblock.

GIVPTR

          GIVPTR is INTEGER


         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.

INDXP

          INDXP is INTEGER array, dimension (N)


         The permutation used to place deflated values of D at the end


         of the array.  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)


         The permutation used to sort the contents of D into ascending


         order.

INFO

          INFO is INTEGER


          = 0:  successful exit.


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

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

December 2016

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