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laed8(3) LAPACK 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.

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

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

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.

Author

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