laed2(3)

NAME
laed2 - laed2: D&C step: deflation

SYNOPSIS

  Functions

subroutine dlaed2 (k, n, n1, d, q, ldq, indxq, rho, z, dlambda, w, q2, indx, indxc, indxp, coltyp, info)
DLAED2 used by DSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is tridiagonal.
subroutine slaed2 (k, n, n1, d, q, ldq, indxq, rho, z, dlambda, w, q2, indx, indxc, indxp, coltyp, info)
SLAED2 used by SSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is tridiagonal.

Detailed Description

Function Documentation

  subroutine dlaed2 (integer k, integer n, integer n1, double precision, dimension( * ) d, double precision, dimension( ldq, * ) q, integer ldq, integer, dimension( * ) indxq, double precision rho, double precision, dimension( * ) z, double precision, dimension( * ) dlambda, double precision, dimension( * ) w, double precision, dimension( * ) q2, integer, dimension( * ) indx, integer, dimension( * ) indxc, integer, dimension( * ) indxp, integer, dimension( * ) coltyp, integer info)

DLAED2 used by DSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is tridiagonal.

Purpose:

    !>
    !> 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.
    !>

Parameters
    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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

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

  subroutine slaed2 (integer k, integer n, integer n1, real, dimension( * ) d, real, dimension( ldq, * ) q, integer ldq, integer, dimension( * ) indxq, real rho, real, dimension( * ) z, real, dimension( * ) dlambda, real, dimension( * ) w, real, dimension( * ) q2, integer, dimension( * ) indx, integer, dimension( * ) indxc, integer, dimension( * ) indxp, integer, dimension( * ) coltyp, integer info)

SLAED2 used by SSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is tridiagonal.

Purpose:

    !>
    !> 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.
    !>

Parameters
    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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

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

Author

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