laed8 - Man Page

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.

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.

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.

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.

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.