slaed8.f

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*> \brief \b SLAED8 used by SSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense.
*
*  =========== DOCUMENTATION ===========
*
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*            http://www.netlib.org/lapack/explore-html/
*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE SLAED8( ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ, RHO,
*                          CUTPNT, Z, DLAMBDA, Q2, LDQ2, W, PERM, GIVPTR,
*                          GIVCOL, GIVNUM, INDXP, INDX, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            CUTPNT, GIVPTR, ICOMPQ, INFO, K, LDQ, LDQ2, N,
*      $                   QSIZ
*       REAL               RHO
*       ..
*       .. Array Arguments ..
*       INTEGER            GIVCOL( 2, * ), INDX( * ), INDXP( * ),
*      $                   INDXQ( * ), PERM( * )
*       REAL               D( * ), DLAMBDA( * ), GIVNUM( 2, * ),
*      $                   Q( LDQ, * ), Q2( LDQ2, * ), W( * ), Z( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> 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.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] ICOMPQ
*> \verbatim
*>          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.
*> \endverbatim
*>
*> \param[out] K
*> \verbatim
*>          K is INTEGER
*>         The number of non-deflated eigenvalues, and the order of the
*>         related secular equation.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>         The dimension of the symmetric tridiagonal matrix.  N >= 0.
*> \endverbatim
*>
*> \param[in] QSIZ
*> \verbatim
*>          QSIZ is INTEGER
*>         The dimension of the orthogonal matrix used to reduce
*>         the full matrix to tridiagonal form.  QSIZ >= N if ICOMPQ = 1.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          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.
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*>          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.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*>          LDQ is INTEGER
*>         The leading dimension of the array Q.  LDQ >= max(1,N).
*> \endverbatim
*>
*> \param[in] INDXQ
*> \verbatim
*>          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.
*> \endverbatim
*>
*> \param[in,out] RHO
*> \verbatim
*>          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.
*> \endverbatim
*>
*> \param[in] CUTPNT
*> \verbatim
*>          CUTPNT is INTEGER
*>         The location of the last eigenvalue in the leading
*>         sub-matrix.  min(1,N) <= CUTPNT <= N.
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*>          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.
*> \endverbatim
*>
*> \param[out] DLAMBDA
*> \verbatim
*>          DLAMBDA is REAL array, dimension (N)
*>         A copy of the first K eigenvalues which will be used by
*>         SLAED3 to form the secular equation.
*> \endverbatim
*>
*> \param[out] Q2
*> \verbatim
*>          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.
*> \endverbatim
*>
*> \param[in] LDQ2
*> \verbatim
*>          LDQ2 is INTEGER
*>         The leading dimension of the array Q2.  LDQ2 >= max(1,N).
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is REAL array, dimension (N)
*>         The first k values of the final deflation-altered z-vector and
*>         will be passed to SLAED3.
*> \endverbatim
*>
*> \param[out] PERM
*> \verbatim
*>          PERM is INTEGER array, dimension (N)
*>         The permutations (from deflation and sorting) to be applied
*>         to each eigenblock.
*> \endverbatim
*>
*> \param[out] GIVPTR
*> \verbatim
*>          GIVPTR is INTEGER
*>         The number of Givens rotations which took place in this
*>         subproblem.
*> \endverbatim
*>
*> \param[out] GIVCOL
*> \verbatim
*>          GIVCOL is INTEGER array, dimension (2, N)
*>         Each pair of numbers indicates a pair of columns to take place
*>         in a Givens rotation.
*> \endverbatim
*>
*> \param[out] GIVNUM
*> \verbatim
*>          GIVNUM is REAL array, dimension (2, N)
*>         Each number indicates the S value to be used in the
*>         corresponding Givens rotation.
*> \endverbatim
*>
*> \param[out] INDXP
*> \verbatim
*>          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.
*> \endverbatim
*>
*> \param[out] INDX
*> \verbatim
*>          INDX is INTEGER array, dimension (N)
*>         The permutation used to sort the contents of D into ascending
*>         order.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laed8
*
*> \par Contributors:
*  ==================
*>
*> Jeff Rutter, Computer Science Division, University of California
*> at Berkeley, USA
*
*  =====================================================================
      SUBROUTINE slaed8( ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ, RHO,
     $                   CUTPNT, Z, DLAMBDA, Q2, LDQ2, W, PERM, GIVPTR,
     $                   GIVCOL, GIVNUM, INDXP, INDX, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            CUTPNT, GIVPTR, ICOMPQ, INFO, K, LDQ, LDQ2, N,
     $                   QSIZ
      REAL               RHO
*     ..
*     .. Array Arguments ..
      INTEGER            GIVCOL( 2, * ), INDX( * ), INDXP( * ),
     $                   INDXQ( * ), PERM( * )
      REAL               D( * ), DLAMBDA( * ), GIVNUM( 2, * ),
     $                   Q( LDQ, * ), Q2( LDQ2, * ), W( * ), Z( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               MONE, ZERO, ONE, TWO, EIGHT
      PARAMETER          ( MONE = -1.0e0, zero = 0.0e0, one = 1.0e0,
     $                   two = 2.0e0, eight = 8.0e0 )
*     ..
*     .. Local Scalars ..
*
      INTEGER            I, IMAX, J, JLAM, JMAX, JP, K2, N1, N1P1, N2
      REAL               C, EPS, S, T, TAU, TOL
*     ..
*     .. External Functions ..
      INTEGER            ISAMAX
      REAL               SLAMCH, SLAPY2
      EXTERNAL           isamax, slamch, slapy2
*     ..
*     .. External Subroutines ..
      EXTERNAL           scopy, slacpy, slamrg, srot, sscal,
     $                   xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, max, min, sqrt
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      info = 0
*
      IF( icompq.LT.0 .OR. icompq.GT.1 ) THEN
         info = -1
      ELSE IF( n.LT.0 ) THEN
         info = -3
      ELSE IF( icompq.EQ.1 .AND. qsiz.LT.n ) THEN
         info = -4
      ELSE IF( ldq.LT.max( 1, n ) ) THEN
         info = -7
      ELSE IF( cutpnt.LT.min( 1, n ) .OR. cutpnt.GT.n ) THEN
         info = -10
      ELSE IF( ldq2.LT.max( 1, n ) ) THEN
         info = -14
      END IF
      IF( info.NE.0 ) THEN
         CALL xerbla( 'SLAED8', -info )
         RETURN
      END IF
*
*     Need to initialize GIVPTR to O here in case of quick exit
*     to prevent an unspecified code behavior (usually sigfault)
*     when IWORK array on entry to *stedc is not zeroed
*     (or at least some IWORK entries which used in *laed7 for GIVPTR).
*
      givptr = 0
*
*     Quick return if possible
*
      IF( n.EQ.0 )
     $   RETURN
*
      n1 = cutpnt
      n2 = n - n1
      n1p1 = n1 + 1
*
      IF( rho.LT.zero ) THEN
         CALL sscal( n2, mone, z( n1p1 ), 1 )
      END IF
*
*     Normalize z so that norm(z) = 1
*
      t = one / sqrt( two )
      DO 10 j = 1, n
         indx( j ) = j
   10 CONTINUE
      CALL sscal( n, t, z, 1 )
      rho = abs( two*rho )
*
*     Sort the eigenvalues into increasing order
*
      DO 20 i = cutpnt + 1, n
         indxq( i ) = indxq( i ) + cutpnt
   20 CONTINUE
      DO 30 i = 1, n
         dlambda( i ) = d( indxq( i ) )
         w( i ) = z( indxq( i ) )
   30 CONTINUE
      i = 1
      j = cutpnt + 1
      CALL slamrg( n1, n2, dlambda, 1, 1, indx )
      DO 40 i = 1, n
         d( i ) = dlambda( indx( i ) )
         z( i ) = w( indx( i ) )
   40 CONTINUE
*
*     Calculate the allowable deflation tolerance
*
      imax = isamax( n, z, 1 )
      jmax = isamax( n, d, 1 )
      eps = slamch( 'Epsilon' )
      tol = eight*eps*abs( d( jmax ) )
*
*     If the rank-1 modifier is small enough, no more needs to be done
*     except to reorganize Q so that its columns correspond with the
*     elements in D.
*
      IF( rho*abs( z( imax ) ).LE.tol ) THEN
         k = 0
         IF( icompq.EQ.0 ) THEN
            DO 50 j = 1, n
               perm( j ) = indxq( indx( j ) )
   50       CONTINUE
         ELSE
            DO 60 j = 1, n
               perm( j ) = indxq( indx( j ) )
               CALL scopy( qsiz, q( 1, perm( j ) ), 1, q2( 1, j ),
     $                     1 )
   60       CONTINUE
            CALL slacpy( 'A', qsiz, n, q2( 1, 1 ), ldq2, q( 1, 1 ),
     $                   ldq )
         END IF
         RETURN
      END IF
*
*     If there are multiple eigenvalues then the problem deflates.  Here
*     the number of equal eigenvalues are found.  As each equal
*     eigenvalue is found, an elementary reflector is computed to rotate
*     the corresponding eigensubspace so that the corresponding
*     components of Z are zero in this new basis.
*
      k = 0
      k2 = n + 1
      DO 70 j = 1, n
         IF( rho*abs( z( j ) ).LE.tol ) THEN
*
*           Deflate due to small z component.
*
            k2 = k2 - 1
            indxp( k2 ) = j
            IF( j.EQ.n )
     $         GO TO 110
         ELSE
            jlam = j
            GO TO 80
         END IF
   70 CONTINUE
   80 CONTINUE
      j = j + 1
      IF( j.GT.n )
     $   GO TO 100
      IF( rho*abs( z( j ) ).LE.tol ) THEN
*
*        Deflate due to small z component.
*
         k2 = k2 - 1
         indxp( k2 ) = j
      ELSE
*
*        Check if eigenvalues are close enough to allow deflation.
*
         s = z( jlam )
         c = z( j )
*
*        Find sqrt(a**2+b**2) without overflow or
*        destructive underflow.
*
         tau = slapy2( c, s )
         t = d( j ) - d( jlam )
         c = c / tau
         s = -s / tau
         IF( abs( t*c*s ).LE.tol ) THEN
*
*           Deflation is possible.
*
            z( j ) = tau
            z( jlam ) = zero
*
*           Record the appropriate Givens rotation
*
            givptr = givptr + 1
            givcol( 1, givptr ) = indxq( indx( jlam ) )
            givcol( 2, givptr ) = indxq( indx( j ) )
            givnum( 1, givptr ) = c
            givnum( 2, givptr ) = s
            IF( icompq.EQ.1 ) THEN
               CALL srot( qsiz, q( 1, indxq( indx( jlam ) ) ), 1,
     $                    q( 1, indxq( indx( j ) ) ), 1, c, s )
            END IF
            t = d( jlam )*c*c + d( j )*s*s
            d( j ) = d( jlam )*s*s + d( j )*c*c
            d( jlam ) = t
            k2 = k2 - 1
            i = 1
   90       CONTINUE
            IF( k2+i.LE.n ) THEN
               IF( d( jlam ).LT.d( indxp( k2+i ) ) ) THEN
                  indxp( k2+i-1 ) = indxp( k2+i )
                  indxp( k2+i ) = jlam
                  i = i + 1
                  GO TO 90
               ELSE
                  indxp( k2+i-1 ) = jlam
               END IF
            ELSE
               indxp( k2+i-1 ) = jlam
            END IF
            jlam = j
         ELSE
            k = k + 1
            w( k ) = z( jlam )
            dlambda( k ) = d( jlam )
            indxp( k ) = jlam
            jlam = j
         END IF
      END IF
      GO TO 80
  100 CONTINUE
*
*     Record the last eigenvalue.
*
      k = k + 1
      w( k ) = z( jlam )
      dlambda( k ) = d( jlam )
      indxp( k ) = jlam
*
  110 CONTINUE
*
*     Sort the eigenvalues and corresponding eigenvectors into DLAMBDA
*     and Q2 respectively.  The eigenvalues/vectors which were not
*     deflated go into the first K slots of DLAMBDA and Q2 respectively,
*     while those which were deflated go into the last N - K slots.
*
      IF( icompq.EQ.0 ) THEN
         DO 120 j = 1, n
            jp = indxp( j )
            dlambda( j ) = d( jp )
            perm( j ) = indxq( indx( jp ) )
  120    CONTINUE
      ELSE
         DO 130 j = 1, n
            jp = indxp( j )
            dlambda( j ) = d( jp )
            perm( j ) = indxq( indx( jp ) )
            CALL scopy( qsiz, q( 1, perm( j ) ), 1, q2( 1, j ), 1 )
  130    CONTINUE
      END IF
*
*     The deflated eigenvalues and their corresponding vectors go back
*     into the last N - K slots of D and Q respectively.
*
      IF( k.LT.n ) THEN
         IF( icompq.EQ.0 ) THEN
            CALL scopy( n-k, dlambda( k+1 ), 1, d( k+1 ), 1 )
         ELSE
            CALL scopy( n-k, dlambda( k+1 ), 1, d( k+1 ), 1 )
            CALL slacpy( 'A', qsiz, n-k, q2( 1, k+1 ), ldq2,
     $                   q( 1, k+1 ), ldq )
         END IF
      END IF
*
      RETURN
*
*     End of SLAED8
*
      SUBROUTINE slaed8( ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ, RHO, …
      END