LAED7 - 5.2 English - 68552

LAED7 used by sstedc. Computes the updated eigensystem of a diagonal

matrix after modification by a rank-one symmetric matrix. Used when the

Purpose:

 LAED7 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 occurrence 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.

Parameters:

• ICOMPQ – [in]

  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 – [in]

  N is INTEGER

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

• QSIZ – [in]

  QSIZ is INTEGER

  The dimension of the orthogonal matrix used to reduce the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1.

• TLVLS – [in]

  TLVLS is INTEGER

  The total number of merging levels in the overall divide and conquer tree.

• CURLVL – [in]

  CURLVL is INTEGER

  The current level in the overall merge routine, 0 <= CURLVL <= TLVLS.

• CURPBM – [in]

  CURPBM is INTEGER

  The current problem in the current level in the overall merge routine (counting from upper left to lower right).

• D – [inout]

  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 – [inout]

  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 – [in]

  LDQ is INTEGER

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

• INDXQ – [out]

  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 – [in]

  RHO is REAL

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

• CUTPNT – [in]

  CUTPNT is INTEGER

  Contains the location of the last eigenvalue in the leading sub-matrix. min(1,N) <= CUTPNT <= N.

• QSTORE – [inout]

  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 – [inout]

  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 – [in]

  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 – [in]

  PERM is INTEGER array, dimension (N lg N)

  Contains the permutations (from deflation and sorting) to be applied to each eigenblock.

• GIVPTR – [in]

  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 – [in]

  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 – [in]

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

  Each number indicates the S value to be used in the corresponding Givens rotation.

• WORK – [out] WORK is REAL array, dimension (3*N+2*QSIZ*N)

• IWORK – [out] IWORK is INTEGER array, dimension (4*N)

• INFO – [out]

  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