.TH "laed8" 3 "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
laed8 \- laed8: D&C step: deflation
.SH SYNOPSIS
.br
.PP
.SS "Functions"

.in +1c
.ti -1c
.RI "subroutine \fBclaed8\fP (k, n, qsiz, q, ldq, d, rho, cutpnt, z, dlambda, q2, ldq2, w, indxp, indx, indxq, perm, givptr, givcol, givnum, info)"
.br
.RI "\fBCLAED8\fP used by CSTEDC\&. Merges eigenvalues and deflates secular equation\&. Used when the original matrix is dense\&. "
.ti -1c
.RI "subroutine \fBdlaed8\fP (icompq, k, n, qsiz, d, q, ldq, indxq, rho, cutpnt, z, dlambda, q2, ldq2, w, perm, givptr, givcol, givnum, indxp, indx, info)"
.br
.RI "\fBDLAED8\fP used by DSTEDC\&. Merges eigenvalues and deflates secular equation\&. Used when the original matrix is dense\&. "
.ti -1c
.RI "subroutine \fBslaed8\fP (icompq, k, n, qsiz, d, q, ldq, indxq, rho, cutpnt, z, dlambda, q2, ldq2, w, perm, givptr, givcol, givnum, indxp, indx, info)"
.br
.RI "\fBSLAED8\fP used by SSTEDC\&. Merges eigenvalues and deflates secular equation\&. Used when the original matrix is dense\&. "
.ti -1c
.RI "subroutine \fBzlaed8\fP (k, n, qsiz, q, ldq, d, rho, cutpnt, z, dlambda, q2, ldq2, w, indxp, indx, indxq, perm, givptr, givcol, givnum, info)"
.br
.RI "\fBZLAED8\fP used by ZSTEDC\&. Merges eigenvalues and deflates secular equation\&. Used when the original matrix is dense\&. "
.in -1c
.SH "Detailed Description"
.PP 

.SH "Function Documentation"
.PP 
.SS "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)"

.PP
\fBCLAED8\fP used by CSTEDC\&. Merges eigenvalues and deflates secular equation\&. Used when the original matrix is dense\&.  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
!>
!> 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\&.
!> 
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIK\fP 
.PP
.nf
!>          K is INTEGER
!>         Contains the number of non-deflated eigenvalues\&.
!>         This is the order of the related secular equation\&.
!> 
.fi
.PP
.br
\fIN\fP 
.PP
.nf
!>          N is INTEGER
!>         The dimension of the symmetric tridiagonal matrix\&.  N >= 0\&.
!> 
.fi
.PP
.br
\fIQSIZ\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIQ\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fILDQ\fP 
.PP
.nf
!>          LDQ is INTEGER
!>         The leading dimension of the array Q\&.  LDQ >= max( 1, N )\&.
!> 
.fi
.PP
.br
\fID\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIRHO\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fICUTPNT\fP 
.PP
.nf
!>          CUTPNT is INTEGER
!>         Contains the location of the last eigenvalue in the leading
!>         sub-matrix\&.  MIN(1,N) <= CUTPNT <= N\&.
!> 
.fi
.PP
.br
\fIZ\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIDLAMBDA\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIQ2\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fILDQ2\fP 
.PP
.nf
!>          LDQ2 is INTEGER
!>         The leading dimension of the array Q2\&.  LDQ2 >= max( 1, N )\&.
!> 
.fi
.PP
.br
\fIW\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIINDXP\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIINDX\fP 
.PP
.nf
!>          INDX is INTEGER array, dimension (N)
!>         This will contain the permutation used to sort the contents of
!>         D into ascending order\&.
!> 
.fi
.PP
.br
\fIINDXQ\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIPERM\fP 
.PP
.nf
!>          PERM is INTEGER array, dimension (N)
!>         Contains the permutations (from deflation and sorting) to be
!>         applied to each eigenblock\&.
!> 
.fi
.PP
.br
\fIGIVPTR\fP 
.PP
.nf
!>          GIVPTR is INTEGER
!>         Contains the number of Givens rotations which took place in
!>         this subproblem\&.
!> 
.fi
.PP
.br
\fIGIVCOL\fP 
.PP
.nf
!>          GIVCOL is INTEGER array, dimension (2, N)
!>         Each pair of numbers indicates a pair of columns to take place
!>         in a Givens rotation\&.
!> 
.fi
.PP
.br
\fIGIVNUM\fP 
.PP
.nf
!>          GIVNUM is REAL array, dimension (2, N)
!>         Each number indicates the S value to be used in the
!>         corresponding Givens rotation\&.
!> 
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
!>          INFO is INTEGER
!>          = 0:  successful exit\&.
!>          < 0:  if INFO = -i, the i-th argument had an illegal value\&.
!> 
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 

.PP
Univ\&. of California Berkeley 

.PP
Univ\&. of Colorado Denver 

.PP
NAG Ltd\&. 
.RE
.PP

.PP
Definition at line \fB225\fP of file \fBclaed8\&.f\fP\&.
.SS "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)"

.PP
\fBDLAED8\fP used by DSTEDC\&. Merges eigenvalues and deflates secular equation\&. Used when the original matrix is dense\&.  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
!>
!> 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\&.
!> 
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIICOMPQ\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIK\fP 
.PP
.nf
!>          K is INTEGER
!>         The number of non-deflated eigenvalues, and the order of the
!>         related secular equation\&.
!> 
.fi
.PP
.br
\fIN\fP 
.PP
.nf
!>          N is INTEGER
!>         The dimension of the symmetric tridiagonal matrix\&.  N >= 0\&.
!> 
.fi
.PP
.br
\fIQSIZ\fP 
.PP
.nf
!>          QSIZ is INTEGER
!>         The dimension of the orthogonal matrix used to reduce
!>         the full matrix to tridiagonal form\&.  QSIZ >= N if ICOMPQ = 1\&.
!> 
.fi
.PP
.br
\fID\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIQ\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fILDQ\fP 
.PP
.nf
!>          LDQ is INTEGER
!>         The leading dimension of the array Q\&.  LDQ >= max(1,N)\&.
!> 
.fi
.PP
.br
\fIINDXQ\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIRHO\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fICUTPNT\fP 
.PP
.nf
!>          CUTPNT is INTEGER
!>         The location of the last eigenvalue in the leading
!>         sub-matrix\&.  min(1,N) <= CUTPNT <= N\&.
!> 
.fi
.PP
.br
\fIZ\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIDLAMBDA\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIQ2\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fILDQ2\fP 
.PP
.nf
!>          LDQ2 is INTEGER
!>         The leading dimension of the array Q2\&.  LDQ2 >= max(1,N)\&.
!> 
.fi
.PP
.br
\fIW\fP 
.PP
.nf
!>          W is DOUBLE PRECISION array, dimension (N)
!>         The first k values of the final deflation-altered z-vector and
!>         will be passed to DLAED3\&.
!> 
.fi
.PP
.br
\fIPERM\fP 
.PP
.nf
!>          PERM is INTEGER array, dimension (N)
!>         The permutations (from deflation and sorting) to be applied
!>         to each eigenblock\&.
!> 
.fi
.PP
.br
\fIGIVPTR\fP 
.PP
.nf
!>          GIVPTR is INTEGER
!>         The number of Givens rotations which took place in this
!>         subproblem\&.
!> 
.fi
.PP
.br
\fIGIVCOL\fP 
.PP
.nf
!>          GIVCOL is INTEGER array, dimension (2, N)
!>         Each pair of numbers indicates a pair of columns to take place
!>         in a Givens rotation\&.
!> 
.fi
.PP
.br
\fIGIVNUM\fP 
.PP
.nf
!>          GIVNUM is DOUBLE PRECISION array, dimension (2, N)
!>         Each number indicates the S value to be used in the
!>         corresponding Givens rotation\&.
!> 
.fi
.PP
.br
\fIINDXP\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIINDX\fP 
.PP
.nf
!>          INDX is INTEGER array, dimension (N)
!>         The permutation used to sort the contents of D into ascending
!>         order\&.
!> 
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
!>          INFO is INTEGER
!>          = 0:  successful exit\&.
!>          < 0:  if INFO = -i, the i-th argument had an illegal value\&.
!> 
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 

.PP
Univ\&. of California Berkeley 

.PP
Univ\&. of Colorado Denver 

.PP
NAG Ltd\&. 
.RE
.PP
\fBContributors:\fP
.RS 4
Jeff Rutter, Computer Science Division, University of California at Berkeley, USA 
.RE
.PP

.PP
Definition at line \fB240\fP of file \fBdlaed8\&.f\fP\&.
.SS "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)"

.PP
\fBSLAED8\fP used by SSTEDC\&. Merges eigenvalues and deflates secular equation\&. Used when the original matrix is dense\&.  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
!>
!> 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\&.
!> 
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIICOMPQ\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIK\fP 
.PP
.nf
!>          K is INTEGER
!>         The number of non-deflated eigenvalues, and the order of the
!>         related secular equation\&.
!> 
.fi
.PP
.br
\fIN\fP 
.PP
.nf
!>          N is INTEGER
!>         The dimension of the symmetric tridiagonal matrix\&.  N >= 0\&.
!> 
.fi
.PP
.br
\fIQSIZ\fP 
.PP
.nf
!>          QSIZ is INTEGER
!>         The dimension of the orthogonal matrix used to reduce
!>         the full matrix to tridiagonal form\&.  QSIZ >= N if ICOMPQ = 1\&.
!> 
.fi
.PP
.br
\fID\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIQ\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fILDQ\fP 
.PP
.nf
!>          LDQ is INTEGER
!>         The leading dimension of the array Q\&.  LDQ >= max(1,N)\&.
!> 
.fi
.PP
.br
\fIINDXQ\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIRHO\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fICUTPNT\fP 
.PP
.nf
!>          CUTPNT is INTEGER
!>         The location of the last eigenvalue in the leading
!>         sub-matrix\&.  min(1,N) <= CUTPNT <= N\&.
!> 
.fi
.PP
.br
\fIZ\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIDLAMBDA\fP 
.PP
.nf
!>          DLAMBDA is REAL array, dimension (N)
!>         A copy of the first K eigenvalues which will be used by
!>         SLAED3 to form the secular equation\&.
!> 
.fi
.PP
.br
\fIQ2\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fILDQ2\fP 
.PP
.nf
!>          LDQ2 is INTEGER
!>         The leading dimension of the array Q2\&.  LDQ2 >= max(1,N)\&.
!> 
.fi
.PP
.br
\fIW\fP 
.PP
.nf
!>          W is REAL array, dimension (N)
!>         The first k values of the final deflation-altered z-vector and
!>         will be passed to SLAED3\&.
!> 
.fi
.PP
.br
\fIPERM\fP 
.PP
.nf
!>          PERM is INTEGER array, dimension (N)
!>         The permutations (from deflation and sorting) to be applied
!>         to each eigenblock\&.
!> 
.fi
.PP
.br
\fIGIVPTR\fP 
.PP
.nf
!>          GIVPTR is INTEGER
!>         The number of Givens rotations which took place in this
!>         subproblem\&.
!> 
.fi
.PP
.br
\fIGIVCOL\fP 
.PP
.nf
!>          GIVCOL is INTEGER array, dimension (2, N)
!>         Each pair of numbers indicates a pair of columns to take place
!>         in a Givens rotation\&.
!> 
.fi
.PP
.br
\fIGIVNUM\fP 
.PP
.nf
!>          GIVNUM is REAL array, dimension (2, N)
!>         Each number indicates the S value to be used in the
!>         corresponding Givens rotation\&.
!> 
.fi
.PP
.br
\fIINDXP\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIINDX\fP 
.PP
.nf
!>          INDX is INTEGER array, dimension (N)
!>         The permutation used to sort the contents of D into ascending
!>         order\&.
!> 
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
!>          INFO is INTEGER
!>          = 0:  successful exit\&.
!>          < 0:  if INFO = -i, the i-th argument had an illegal value\&.
!> 
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 

.PP
Univ\&. of California Berkeley 

.PP
Univ\&. of Colorado Denver 

.PP
NAG Ltd\&. 
.RE
.PP
\fBContributors:\fP
.RS 4
Jeff Rutter, Computer Science Division, University of California at Berkeley, USA 
.RE
.PP

.PP
Definition at line \fB240\fP of file \fBslaed8\&.f\fP\&.
.SS "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)"

.PP
\fBZLAED8\fP used by ZSTEDC\&. Merges eigenvalues and deflates secular equation\&. Used when the original matrix is dense\&.  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
!>
!> 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\&.
!> 
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIK\fP 
.PP
.nf
!>          K is INTEGER
!>         Contains the number of non-deflated eigenvalues\&.
!>         This is the order of the related secular equation\&.
!> 
.fi
.PP
.br
\fIN\fP 
.PP
.nf
!>          N is INTEGER
!>         The dimension of the symmetric tridiagonal matrix\&.  N >= 0\&.
!> 
.fi
.PP
.br
\fIQSIZ\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIQ\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fILDQ\fP 
.PP
.nf
!>          LDQ is INTEGER
!>         The leading dimension of the array Q\&.  LDQ >= max( 1, N )\&.
!> 
.fi
.PP
.br
\fID\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIRHO\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fICUTPNT\fP 
.PP
.nf
!>          CUTPNT is INTEGER
!>         Contains the location of the last eigenvalue in the leading
!>         sub-matrix\&.  MIN(1,N) <= CUTPNT <= N\&.
!> 
.fi
.PP
.br
\fIZ\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIDLAMBDA\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIQ2\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fILDQ2\fP 
.PP
.nf
!>          LDQ2 is INTEGER
!>         The leading dimension of the array Q2\&.  LDQ2 >= max( 1, N )\&.
!> 
.fi
.PP
.br
\fIW\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIINDXP\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIINDX\fP 
.PP
.nf
!>          INDX is INTEGER array, dimension (N)
!>         This will contain the permutation used to sort the contents of
!>         D into ascending order\&.
!> 
.fi
.PP
.br
\fIINDXQ\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIPERM\fP 
.PP
.nf
!>          PERM is INTEGER array, dimension (N)
!>         Contains the permutations (from deflation and sorting) to be
!>         applied to each eigenblock\&.
!> 
.fi
.PP
.br
\fIGIVPTR\fP 
.PP
.nf
!>          GIVPTR is INTEGER
!>         Contains the number of Givens rotations which took place in
!>         this subproblem\&.
!> 
.fi
.PP
.br
\fIGIVCOL\fP 
.PP
.nf
!>          GIVCOL is INTEGER array, dimension (2, N)
!>         Each pair of numbers indicates a pair of columns to take place
!>         in a Givens rotation\&.
!> 
.fi
.PP
.br
\fIGIVNUM\fP 
.PP
.nf
!>          GIVNUM is DOUBLE PRECISION array, dimension (2, N)
!>         Each number indicates the S value to be used in the
!>         corresponding Givens rotation\&.
!> 
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
!>          INFO is INTEGER
!>          = 0:  successful exit\&.
!>          < 0:  if INFO = -i, the i-th argument had an illegal value\&.
!> 
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 

.PP
Univ\&. of California Berkeley 

.PP
Univ\&. of Colorado Denver 

.PP
NAG Ltd\&. 
.RE
.PP

.PP
Definition at line \fB225\fP of file \fBzlaed8\&.f\fP\&.
.SH "Author"
.PP 
Generated automatically by Doxygen for LAPACK from the source code\&.