DLAED2 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 entry in the


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


 equation problem is reduced by one.

K

          K is INTEGER


         The number of non-deflated eigenvalues, and the order of the


         related secular equation. 0 <= K <=N.

N

          N is INTEGER


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

N1

          N1 is INTEGER


         The location of the last eigenvalue in the leading sub-matrix.


         min(1,N) <= N1 <= N/2.

D

          D is DOUBLE PRECISION 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.

Q

          Q is DOUBLE PRECISION array, dimension (LDQ, N)


         On entry, Q contains the eigenvectors of two submatrices in


         the two square blocks with corners at (1,1), (N1,N1)


         and (N1+1, N1+1), (N,N).


         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 N1 added to their


         values. Destroyed on exit.

RHO

          RHO is DOUBLE PRECISION


         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


         DLAED3.

Z

          Z is DOUBLE PRECISION 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 have been destroyed by the updating


         process.

DLAMBDA

          DLAMBDA is DOUBLE PRECISION array, dimension (N)


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


         DLAED3 to form the secular equation.

W

          W is DOUBLE PRECISION array, dimension (N)


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


         which will be passed to DLAED3.

Q2

          Q2 is DOUBLE PRECISION array, dimension (N1**2+(N-N1)**2)


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


         DLAED3 in a matrix multiply (DGEMM) to solve for the new


         eigenvectors.

INDX

          INDX is INTEGER array, dimension (N)


         The permutation used to sort the contents of DLAMBDA into


         ascending order.

INDXC

          INDXC is INTEGER array, dimension (N)


         The permutation used to arrange the columns of the deflated


         Q matrix into three groups:  the first group contains non-zero


         elements only at and above N1, the second contains


         non-zero elements only below N1, and the third is dense.

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.

COLTYP

          COLTYP is INTEGER array, dimension (N)


         During execution, a label which will indicate which of the


         following types a column in the Q2 matrix is:


         1 : non-zero in the upper half only;


         2 : dense;


         3 : non-zero in the lower half only;


         4 : deflated.


         On exit, COLTYP(i) is the number of columns of type i,


         for i=1 to 4 only.

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.

Jeff Rutter, Computer Science Division, University of California at Berkeley, USA
Modified by Francoise Tisseur, University of Tennessee

 SLAED2 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 entry in the


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


 equation problem is reduced by one.

K

          K is INTEGER


         The number of non-deflated eigenvalues, and the order of the


         related secular equation. 0 <= K <=N.

N

          N is INTEGER


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

N1

          N1 is INTEGER


         The location of the last eigenvalue in the leading sub-matrix.


         min(1,N) <= N1 <= N/2.

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.

Q

          Q is REAL array, dimension (LDQ, N)


         On entry, Q contains the eigenvectors of two submatrices in


         the two square blocks with corners at (1,1), (N1,N1)


         and (N1+1, N1+1), (N,N).


         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 N1 added to their


         values. Destroyed on exit.

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.

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 have been destroyed by the updating


         process.

DLAMBDA

          DLAMBDA is REAL array, dimension (N)


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


         SLAED3 to form the secular equation.

W

          W is REAL array, dimension (N)


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


         which will be passed to SLAED3.

Q2

          Q2 is REAL array, dimension (N1**2+(N-N1)**2)


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


         SLAED3 in a matrix multiply (SGEMM) to solve for the new


         eigenvectors.

INDX

          INDX is INTEGER array, dimension (N)


         The permutation used to sort the contents of DLAMBDA into


         ascending order.

INDXC

          INDXC is INTEGER array, dimension (N)


         The permutation used to arrange the columns of the deflated


         Q matrix into three groups:  the first group contains non-zero


         elements only at and above N1, the second contains


         non-zero elements only below N1, and the third is dense.

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.

COLTYP

          COLTYP is INTEGER array, dimension (N)


         During execution, a label which will indicate which of the


         following types a column in the Q2 matrix is:


         1 : non-zero in the upper half only;


         2 : dense;


         3 : non-zero in the lower half only;


         4 : deflated.


         On exit, COLTYP(i) is the number of columns of type i,


         for i=1 to 4 only.

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.

Jeff Rutter, Computer Science Division, University of California at Berkeley, USA
Modified by Francoise Tisseur, University of Tennessee