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    1:       SUBROUTINE DLAED1( N, D, Q, LDQ, INDXQ, RHO, CUTPNT, WORK, IWORK,
    2:      $                   INFO )
    3: *
    4: *  -- LAPACK routine (version 3.2) --
    5: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
    6: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
    7: *     November 2006
    8: *
    9: *     .. Scalar Arguments ..
   10:       INTEGER            CUTPNT, INFO, LDQ, N
   11:       DOUBLE PRECISION   RHO
   12: *     ..
   13: *     .. Array Arguments ..
   14:       INTEGER            INDXQ( * ), IWORK( * )
   15:       DOUBLE PRECISION   D( * ), Q( LDQ, * ), WORK( * )
   16: *     ..
   17: *
   18: *  Purpose
   19: *  =======
   20: *
   21: *  DLAED1 computes the updated eigensystem of a diagonal
   22: *  matrix after modification by a rank-one symmetric matrix.  This
   23: *  routine is used only for the eigenproblem which requires all
   24: *  eigenvalues and eigenvectors of a tridiagonal matrix.  DLAED7 handles
   25: *  the case in which eigenvalues only or eigenvalues and eigenvectors
   26: *  of a full symmetric matrix (which was reduced to tridiagonal form)
   27: *  are desired.
   28: *
   29: *    T = Q(in) ( D(in) + RHO * Z*Z' ) Q'(in) = Q(out) * D(out) * Q'(out)
   30: *
   31: *     where Z = Q'u, u is a vector of length N with ones in the
   32: *     CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.
   33: *
   34: *     The eigenvectors of the original matrix are stored in Q, and the
   35: *     eigenvalues are in D.  The algorithm consists of three stages:
   36: *
   37: *        The first stage consists of deflating the size of the problem
   38: *        when there are multiple eigenvalues or if there is a zero in
   39: *        the Z vector.  For each such occurence the dimension of the
   40: *        secular equation problem is reduced by one.  This stage is
   41: *        performed by the routine DLAED2.
   42: *
   43: *        The second stage consists of calculating the updated
   44: *        eigenvalues. This is done by finding the roots of the secular
   45: *        equation via the routine DLAED4 (as called by DLAED3).
   46: *        This routine also calculates the eigenvectors of the current
   47: *        problem.
   48: *
   49: *        The final stage consists of computing the updated eigenvectors
   50: *        directly using the updated eigenvalues.  The eigenvectors for
   51: *        the current problem are multiplied with the eigenvectors from
   52: *        the overall problem.
   53: *
   54: *  Arguments
   55: *  =========
   56: *
   57: *  N      (input) INTEGER
   58: *         The dimension of the symmetric tridiagonal matrix.  N >= 0.
   59: *
   60: *  D      (input/output) DOUBLE PRECISION array, dimension (N)
   61: *         On entry, the eigenvalues of the rank-1-perturbed matrix.
   62: *         On exit, the eigenvalues of the repaired matrix.
   63: *
   64: *  Q      (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
   65: *         On entry, the eigenvectors of the rank-1-perturbed matrix.
   66: *         On exit, the eigenvectors of the repaired tridiagonal matrix.
   67: *
   68: *  LDQ    (input) INTEGER
   69: *         The leading dimension of the array Q.  LDQ >= max(1,N).
   70: *
   71: *  INDXQ  (input/output) INTEGER array, dimension (N)
   72: *         On entry, the permutation which separately sorts the two
   73: *         subproblems in D into ascending order.
   74: *         On exit, the permutation which will reintegrate the
   75: *         subproblems back into sorted order,
   76: *         i.e. D( INDXQ( I = 1, N ) ) will be in ascending order.
   77: *
   78: *  RHO    (input) DOUBLE PRECISION
   79: *         The subdiagonal entry used to create the rank-1 modification.
   80: *
   81: *  CUTPNT (input) INTEGER
   82: *         The location of the last eigenvalue in the leading sub-matrix.
   83: *         min(1,N) <= CUTPNT <= N/2.
   84: *
   85: *  WORK   (workspace) DOUBLE PRECISION array, dimension (4*N + N**2)
   86: *
   87: *  IWORK  (workspace) INTEGER array, dimension (4*N)
   88: *
   89: *  INFO   (output) INTEGER
   90: *          = 0:  successful exit.
   91: *          < 0:  if INFO = -i, the i-th argument had an illegal value.
   92: *          > 0:  if INFO = 1, an eigenvalue did not converge
   93: *
   94: *  Further Details
   95: *  ===============
   96: *
   97: *  Based on contributions by
   98: *     Jeff Rutter, Computer Science Division, University of California
   99: *     at Berkeley, USA
  100: *  Modified by Francoise Tisseur, University of Tennessee.
  101: *
  102: *  =====================================================================
  103: *
  104: *     .. Local Scalars ..
  105:       INTEGER            COLTYP, I, IDLMDA, INDX, INDXC, INDXP, IQ2, IS,
  106:      $                   IW, IZ, K, N1, N2, ZPP1
  107: *     ..
  108: *     .. External Subroutines ..
  109:       EXTERNAL           DCOPY, DLAED2, DLAED3, DLAMRG, XERBLA
  110: *     ..
  111: *     .. Intrinsic Functions ..
  112:       INTRINSIC          MAX, MIN
  113: *     ..
  114: *     .. Executable Statements ..
  115: *
  116: *     Test the input parameters.
  117: *
  118:       INFO = 0
  119: *
  120:       IF( N.LT.0 ) THEN
  121:          INFO = -1
  122:       ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
  123:          INFO = -4
  124:       ELSE IF( MIN( 1, N / 2 ).GT.CUTPNT .OR. ( N / 2 ).LT.CUTPNT ) THEN
  125:          INFO = -7
  126:       END IF
  127:       IF( INFO.NE.0 ) THEN
  128:          CALL XERBLA( 'DLAED1', -INFO )
  129:          RETURN
  130:       END IF
  131: *
  132: *     Quick return if possible
  133: *
  134:       IF( N.EQ.0 )
  135:      $   RETURN
  136: *
  137: *     The following values are integer pointers which indicate
  138: *     the portion of the workspace
  139: *     used by a particular array in DLAED2 and DLAED3.
  140: *
  141:       IZ = 1
  142:       IDLMDA = IZ + N
  143:       IW = IDLMDA + N
  144:       IQ2 = IW + N
  145: *
  146:       INDX = 1
  147:       INDXC = INDX + N
  148:       COLTYP = INDXC + N
  149:       INDXP = COLTYP + N
  150: *
  151: *
  152: *     Form the z-vector which consists of the last row of Q_1 and the
  153: *     first row of Q_2.
  154: *
  155:       CALL DCOPY( CUTPNT, Q( CUTPNT, 1 ), LDQ, WORK( IZ ), 1 )
  156:       ZPP1 = CUTPNT + 1
  157:       CALL DCOPY( N-CUTPNT, Q( ZPP1, ZPP1 ), LDQ, WORK( IZ+CUTPNT ), 1 )
  158: *
  159: *     Deflate eigenvalues.
  160: *
  161:       CALL DLAED2( K, N, CUTPNT, D, Q, LDQ, INDXQ, RHO, WORK( IZ ),
  162:      $             WORK( IDLMDA ), WORK( IW ), WORK( IQ2 ),
  163:      $             IWORK( INDX ), IWORK( INDXC ), IWORK( INDXP ),
  164:      $             IWORK( COLTYP ), INFO )
  165: *
  166:       IF( INFO.NE.0 )
  167:      $   GO TO 20
  168: *
  169: *     Solve Secular Equation.
  170: *
  171:       IF( K.NE.0 ) THEN
  172:          IS = ( IWORK( COLTYP )+IWORK( COLTYP+1 ) )*CUTPNT +
  173:      $        ( IWORK( COLTYP+1 )+IWORK( COLTYP+2 ) )*( N-CUTPNT ) + IQ2
  174:          CALL DLAED3( K, N, CUTPNT, D, Q, LDQ, RHO, WORK( IDLMDA ),
  175:      $                WORK( IQ2 ), IWORK( INDXC ), IWORK( COLTYP ),
  176:      $                WORK( IW ), WORK( IS ), INFO )
  177:          IF( INFO.NE.0 )
  178:      $      GO TO 20
  179: *
  180: *     Prepare the INDXQ sorting permutation.
  181: *
  182:          N1 = K
  183:          N2 = N - K
  184:          CALL DLAMRG( N1, N2, D, 1, -1, INDXQ )
  185:       ELSE
  186:          DO 10 I = 1, N
  187:             INDXQ( I ) = I
  188:    10    CONTINUE
  189:       END IF
  190: *
  191:    20 CONTINUE
  192:       RETURN
  193: *
  194: *     End of DLAED1
  195: *
  196:       END