dlaed8.f

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      SUBROUTINE DLAED8( ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ, RHO,
     $                   CUTPNT, Z, DLAMDA, Q2, LDQ2, W, PERM, GIVPTR,
     $                   GIVCOL, GIVNUM, INDXP, INDX, INFO )
*
*  -- LAPACK routine (instrumented to count operations, version 3.0) --
*     Univ. of Tennessee, Oak Ridge National Lab, Argonne National Lab,
*     Courant Institute, NAG Ltd., and Rice University
*     September 30, 1994
*
*     .. Scalar Arguments ..
      INTEGER            CUTPNT, GIVPTR, ICOMPQ, INFO, K, LDQ, LDQ2, N,
     $                   QSIZ
      DOUBLE PRECISION   RHO
*     ..
*     .. Array Arguments ..
      INTEGER            GIVCOL( 2, * ), INDX( * ), INDXP( * ),
     $                   INDXQ( * ), PERM( * )
      DOUBLE PRECISION   D( * ), DLAMDA( * ), GIVNUM( 2, * ),
     $                   Q( LDQ, * ), Q2( LDQ2, * ), W( * ), Z( * )
*     ..
*     Common block to return operation count and iteration count
*     ITCNT is unchanged, OPS is only incremented
*     .. Common blocks ..
      COMMON             / LATIME / OPS, ITCNT
*     ..
*     .. Scalars in Common ..
      DOUBLE PRECISION   ITCNT, OPS
*     ..
*
*  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.
*
*  Arguments
*  =========
*
*  ICOMPQ  (input) 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      (output) INTEGER
*         The number of non-deflated eigenvalues, and the order of the
*         related secular equation.
*
*  N      (input) INTEGER
*         The dimension of the symmetric tridiagonal matrix.  N >= 0.
*
*  QSIZ   (input) INTEGER
*         The dimension of the orthogonal matrix used to reduce
*         the full matrix to tridiagonal form.  QSIZ >= N if ICOMPQ = 1.
*
*  D      (input/output) 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      (input/output) 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    (input) INTEGER
*         The leading dimension of the array Q.  LDQ >= max(1,N).
*
*  INDXQ  (input) 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    (input/output) 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 (input) INTEGER
*         The location of the last eigenvalue in the leading
*         sub-matrix.  min(1,N) <= CUTPNT <= N.
*
*  Z      (input) 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.
*
*  DLAMDA (output) DOUBLE PRECISION array, dimension (N)
*         A copy of the first K eigenvalues which will be used by
*         DLAED3 to form the secular equation.
*
*  Q2     (output) 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   (input) INTEGER
*         The leading dimension of the array Q2.  LDQ2 >= max(1,N).
*
*  W      (output) DOUBLE PRECISION array, dimension (N)
*         The first k values of the final deflation-altered z-vector and
*         will be passed to DLAED3.
*
*  PERM   (output) INTEGER array, dimension (N)
*         The permutations (from deflation and sorting) to be applied
*         to each eigenblock.
*
*  GIVPTR (output) INTEGER
*         The number of Givens rotations which took place in this
*         subproblem.
*
*  GIVCOL (output) INTEGER array, dimension (2, N)
*         Each pair of numbers indicates a pair of columns to take place
*         in a Givens rotation.
*
*  GIVNUM (output) DOUBLE PRECISION array, dimension (2, N)
*         Each number indicates the S value to be used in the
*         corresponding Givens rotation.
*
*  INDXP  (workspace) 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   (workspace) INTEGER array, dimension (N)
*         The permutation used to sort the contents of D into ascending
*         order.
*
*  INFO   (output) INTEGER
*          = 0:  successful exit.
*          < 0:  if INFO = -i, the i-th argument had an illegal value.
*
*  Further Details
*  ===============
*
*  Based on contributions by
*     Jeff Rutter, Computer Science Division, University of California
*     at Berkeley, USA
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   MONE, ZERO, ONE, TWO, EIGHT
      PARAMETER          ( MONE = -1.0D0, ZERO = 0.0D0, ONE = 1.0D0,
     $                   TWO = 2.0D0, EIGHT = 8.0D0 )
*     ..
*     .. Local Scalars ..
*
      INTEGER            I, IMAX, J, JLAM, JMAX, JP, K2, N1, N1P1, N2
      DOUBLE PRECISION   C, EPS, S, T, TAU, TOL
*     ..
*     .. External Functions ..
      INTEGER            IDAMAX
      DOUBLE PRECISION   DLAMCH, DLAPY2
      EXTERNAL           IDAMAX, DLAMCH, DLAPY2
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DLACPY, DLAMRG, DROT, DSCAL, 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( 'DLAED8', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
      N1 = CUTPNT
      N2 = N - N1
      N1P1 = N1 + 1
*
      IF( RHO.LT.ZERO ) THEN
         OPS = OPS + N2
         CALL DSCAL( N2, MONE, Z( N1P1 ), 1 )
      END IF
*
*     Normalize z so that norm(z) = 1
*
      OPS = OPS + N + 6
      T = ONE / SQRT( TWO )
      DO 10 J = 1, N
         INDX( J ) = J
   10 CONTINUE
      CALL DSCAL( 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
         DLAMDA( I ) = D( INDXQ( I ) )
         W( I ) = Z( INDXQ( I ) )
   30 CONTINUE
      I = 1
      J = CUTPNT + 1
      CALL DLAMRG( N1, N2, DLAMDA, 1, 1, INDX )
      DO 40 I = 1, N
         D( I ) = DLAMDA( INDX( I ) )
         Z( I ) = W( INDX( I ) )
   40 CONTINUE
*
*     Calculate the allowable deflation tolerence
*
      IMAX = IDAMAX( N, Z, 1 )
      JMAX = IDAMAX( N, D, 1 )
      EPS = DLAMCH( '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 DCOPY( QSIZ, Q( 1, PERM( J ) ), 1, Q2( 1, J ), 1 )
   60       CONTINUE
            CALL DLACPY( '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
      GIVPTR = 0
      K2 = N + 1
      DO 70 J = 1, N
         OPS = OPS + 1
         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
      OPS = OPS + 1
      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.
*
         OPS = OPS + 10
         TAU = DLAPY2( 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
               OPS = OPS + 6*QSIZ
               CALL DROT( QSIZ, Q( 1, INDXQ( INDX( JLAM ) ) ), 1,
     $                    Q( 1, INDXQ( INDX( J ) ) ), 1, C, S )
            END IF
            OPS = OPS + 10
            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 )
            DLAMDA( 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 )
      DLAMDA( K ) = D( JLAM )
      INDXP( K ) = JLAM
*
  110 CONTINUE
*
*     Sort the eigenvalues and corresponding eigenvectors into DLAMDA
*     and Q2 respectively.  The eigenvalues/vectors which were not
*     deflated go into the first K slots of DLAMDA 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 )
            DLAMDA( J ) = D( JP )
            PERM( J ) = INDXQ( INDX( JP ) )
  120    CONTINUE
      ELSE
         DO 130 J = 1, N
            JP = INDXP( J )
            DLAMDA( J ) = D( JP )
            PERM( J ) = INDXQ( INDX( JP ) )
            CALL DCOPY( 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 DCOPY( N-K, DLAMDA( K+1 ), 1, D( K+1 ), 1 )
         ELSE
            CALL DCOPY( N-K, DLAMDA( K+1 ), 1, D( K+1 ), 1 )
            CALL DLACPY( 'A', QSIZ, N-K, Q2( 1, K+1 ), LDQ2,
     $                   Q( 1, K+1 ), LDQ )
         END IF
      END IF
*
      RETURN
*
*     End of DLAED8
*
      END