DLAED1 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 eigenvectors of a tridiagonal matrix.  DLAED7 handles


 the case in which eigenvalues only or eigenvalues and eigenvectors


 of a full symmetric matrix (which was reduced to tridiagonal form)


 are desired.


   T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out)


    where Z = Q**T*u, 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 DLAED2.


       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 DLAED3).


       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.

N

          N is INTEGER


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

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)


         On entry, the permutation which separately sorts the two


         subproblems in D into ascending order.


         On exit, the permutation which will reintegrate the


         subproblems back into sorted order,


         i.e. D( INDXQ( I = 1, N ) ) will be in ascending order.

RHO

          RHO is DOUBLE PRECISION


         The subdiagonal entry used to create the rank-1 modification.

CUTPNT

          CUTPNT is INTEGER


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


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

WORK

          WORK is DOUBLE PRECISION array, dimension (4*N + N**2)

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.

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

 SLAED1 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 eigenvectors of a tridiagonal matrix.  SLAED7 handles


 the case in which eigenvalues only or eigenvalues and eigenvectors


 of a full symmetric matrix (which was reduced to tridiagonal form)


 are desired.


   T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out)


    where Z = Q**T*u, 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 SLAED2.


       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 SLAED3).


       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.

N

          N is INTEGER


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

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)


         On entry, the permutation which separately sorts the two


         subproblems in D into ascending order.


         On exit, the permutation which will reintegrate the


         subproblems back into sorted order,


         i.e. D( INDXQ( I = 1, N ) ) will be in ascending order.

RHO

          RHO is REAL


         The subdiagonal entry used to create the rank-1 modification.

CUTPNT

          CUTPNT is INTEGER


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


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

WORK

          WORK is REAL array, dimension (4*N + N**2)

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.

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