laed8(3)

NAME
laed8 - laed8: D&C step: deflation

SYNOPSIS

  Functions
subroutine claed8 (k, n, qsiz, q, ldq, d, rho, cutpnt, z, dlambda, q2, ldq2, w, indxp, indx, indxq, perm, givptr, givcol, givnum, info)
CLAED8 used by CSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense.
subroutine dlaed8 (icompq, k, n, qsiz, d, q, ldq, indxq, rho, cutpnt, z, dlambda, q2, ldq2, w, perm, givptr, givcol, givnum, indxp, indx, info)
DLAED8 used by DSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense.
subroutine slaed8 (icompq, k, n, qsiz, d, q, ldq, indxq, rho, cutpnt, z, dlambda, q2, ldq2, w, perm, givptr, givcol, givnum, indxp, indx, info)
SLAED8 used by SSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense.
subroutine zlaed8 (k, n, qsiz, q, ldq, d, rho, cutpnt, z, dlambda, q2, ldq2, w, indxp, indx, indxq, perm, givptr, givcol, givnum, info)
ZLAED8 used by ZSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense.

Detailed Description

Function Documentation

  subroutine claed8 (integer k, integer n, integer qsiz, complex, dimension( ldq, * ) q, integer ldq, real, dimension( * ) d, real rho, integer cutpnt, real, dimension( * ) z, real, dimension( * ) dlambda, complex, dimension( ldq2, * ) q2, integer ldq2, real, dimension( * ) w, integer, dimension( * ) indxp, integer, dimension( * ) indx, integer, dimension( * ) indxq, integer, dimension( * ) perm, integer givptr, integer, dimension( 2, * ) givcol, real, dimension( 2, * ) givnum, integer info)

CLAED8 used by CSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense.

Purpose:

     CLAED8 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.

Parameters
    K

              K is INTEGER
             Contains the number of non-deflated eigenvalues.
             This is 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 unitary matrix used to reduce
             the dense or band matrix to tridiagonal form.
             QSIZ >= N if ICOMPQ = 1.

    Q

              Q is COMPLEX array, dimension (LDQ,N)
             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 ).

    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.

    RHO

              RHO is REAL
             Contains the off diagonal element associated with the rank-1
             cut which originally split the two submatrices which are now
             being recombined. RHO is modified during the computation to
             the value required by SLAED3.

    CUTPNT

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

    Z

              Z is REAL array, dimension (N)
             On input this vector contains the updating vector (the last
             row of the first sub-eigenvector matrix and the first row of
             the second sub-eigenvector matrix).  The contents of Z are
             destroyed during the updating process.

    DLAMBDA

              DLAMBDA is REAL array, dimension (N)
             Contains a copy of the first K eigenvalues which will be used
             by SLAED3 to form the secular equation.

    Q2

              Q2 is COMPLEX array, dimension (LDQ2,N)
             If ICOMPQ = 0, Q2 is not referenced.  Otherwise,
             Contains 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)
             This will hold the first k values of the final
             deflation-altered z-vector and will be passed to SLAED3.

    INDXP

              INDXP is INTEGER array, dimension (N)
             This will contain the permutation used to place deflated
             values of D at the end of the array. On output 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)
             This will contain the permutation used to sort the contents of
             D into ascending order.

    INDXQ

              INDXQ is INTEGER array, dimension (N)
             This contains 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.

    PERM

              PERM is INTEGER array, dimension (N)
             Contains the permutations (from deflation and sorting) to be
             applied to each eigenblock.

    GIVPTR

              GIVPTR is INTEGER
             Contains 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.

    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.

Definition at line 225 of file claed8.f.

  subroutine dlaed8 (integer icompq, integer k, integer n, integer qsiz, double precision, dimension( * ) d, double precision, dimension( ldq, * ) q, integer ldq, integer, dimension( * ) indxq, double precision rho, integer cutpnt, double precision, dimension( * ) z, double precision, dimension( * ) dlambda, double precision, dimension( ldq2, * ) q2, integer ldq2, double precision, dimension( * ) w, integer, dimension( * ) perm, integer givptr, integer, dimension( 2, * ) givcol, double precision, dimension( 2, * ) givnum, integer, dimension( * ) indxp, integer, dimension( * ) indx, integer info)

DLAED8 used by DSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense.

Purpose:

     DLAED8 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.

Parameters
    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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 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.

    CUTPNT

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

    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 are 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.

    Q2

              Q2 is DOUBLE PRECISION array, dimension (LDQ2,N)
             If ICOMPQ = 0, Q2 is not referenced.  Otherwise,
             a copy of the first K eigenvectors which will be used by
             DLAED7 in a matrix multiply (DGEMM) to update the new
             eigenvectors.

    LDQ2

              LDQ2 is INTEGER
             The leading dimension of the array Q2.  LDQ2 >= max(1,N).

    W

              W is DOUBLE PRECISION array, dimension (N)
             The first k values of the final deflation-altered z-vector and
             will be passed to DLAED3.

    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 DOUBLE PRECISION 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.

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

Definition at line 240 of file dlaed8.f.

  subroutine slaed8 (integer icompq, integer k, integer n, integer qsiz, real, dimension( * ) d, real, dimension( ldq, * ) q, integer ldq, integer, dimension( * ) indxq, real rho, integer cutpnt, real, dimension( * ) z, real, dimension( * ) dlambda, real, dimension( ldq2, * ) q2, integer ldq2, real, dimension( * ) w, integer, dimension( * ) perm, integer givptr, integer, dimension( 2, * ) givcol, real, dimension( 2, * ) givnum, integer, dimension( * ) indxp, integer, dimension( * ) indx, integer info)

SLAED8 used by SSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense.

Purpose:

     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.

Parameters
    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.

    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.

    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.

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

Definition at line 240 of file slaed8.f.

  subroutine zlaed8 (integer k, integer n, integer qsiz, complex*16, dimension( ldq, * ) q, integer ldq, double precision, dimension( * ) d, double precision rho, integer cutpnt, double precision, dimension( * ) z, double precision, dimension( * ) dlambda, complex*16, dimension( ldq2, * ) q2, integer ldq2, double precision, dimension( * ) w, integer, dimension( * ) indxp, integer, dimension( * ) indx, integer, dimension( * ) indxq, integer, dimension( * ) perm, integer givptr, integer, dimension( 2, * ) givcol, double precision, dimension( 2, * ) givnum, integer info)

ZLAED8 used by ZSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense.

Purpose:

     ZLAED8 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.

Parameters
    K

              K is INTEGER
             Contains the number of non-deflated eigenvalues.
             This is 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 unitary matrix used to reduce
             the dense or band matrix to tridiagonal form.
             QSIZ >= N if ICOMPQ = 1.

    Q

              Q is COMPLEX*16 array, dimension (LDQ,N)
             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 ).

    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.

    RHO

              RHO is DOUBLE PRECISION
             Contains the off diagonal element associated with the rank-1
             cut which originally split the two submatrices which are now
             being recombined. RHO is modified during the computation to
             the value required by DLAED3.

    CUTPNT

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

    Z

              Z is DOUBLE PRECISION array, dimension (N)
             On input this vector contains the updating vector (the last
             row of the first sub-eigenvector matrix and the first row of
             the second sub-eigenvector matrix).  The contents of Z are
             destroyed during the updating process.

    DLAMBDA

              DLAMBDA is DOUBLE PRECISION array, dimension (N)
             Contains a copy of the first K eigenvalues which will be used
             by DLAED3 to form the secular equation.

    Q2

              Q2 is COMPLEX*16 array, dimension (LDQ2,N)
             If ICOMPQ = 0, Q2 is not referenced.  Otherwise,
             Contains a copy of the first K eigenvectors which will be used
             by DLAED7 in a matrix multiply (DGEMM) to update the new
             eigenvectors.

    LDQ2

              LDQ2 is INTEGER
             The leading dimension of the array Q2.  LDQ2 >= max( 1, N ).

    W

              W is DOUBLE PRECISION array, dimension (N)
             This will hold the first k values of the final
             deflation-altered z-vector and will be passed to DLAED3.

    INDXP

              INDXP is INTEGER array, dimension (N)
             This will contain the permutation used to place deflated
             values of D at the end of the array. On output 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)
             This will contain the permutation used to sort the contents of
             D into ascending order.

    INDXQ

              INDXQ is INTEGER array, dimension (N)
             This contains 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.

    PERM

              PERM is INTEGER array, dimension (N)
             Contains the permutations (from deflation and sorting) to be
             applied to each eigenblock.

    GIVPTR

              GIVPTR is INTEGER
             Contains 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 DOUBLE PRECISION array, dimension (2, N)
             Each number indicates the S value to be used in the
             corresponding Givens rotation.

    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.

Definition at line 225 of file zlaed8.f.

Author

Generated automatically by Doxygen for LAPACK from the source code.