slaed1.f

          SUBROUTINE SLAED1( N, D, Q, LDQ, INDXQ, RHO, CUTPNT, WORK, IWORK,
         $                   INFO )
    *
    *  -- LAPACK routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          INTEGER            CUTPNT, INFO, LDQ, N
          REAL               RHO
    *     ..
    *     .. Array Arguments ..
          INTEGER            INDXQ( * ), IWORK( * )
          REAL               D( * ), Q( LDQ, * ), WORK( * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  SLAED1 computes the updated eigensystem of a diagonal
    *  matrix after modification by a rank-one symmetric matrix.  This
    *  routine is used only for the eigenproblem which requires all
    *  eigenvalues and eigenvectors of a tridiagonal matrix.  SLAED7 handles
    *  the case in which eigenvalues only or eigenvalues and eigenvectors
    *  of a full symmetric matrix (which was reduced to tridiagonal form)
    *  are desired.
    *
    *    T = Q(in) ( D(in) + RHO * Z*Z' ) Q'(in) = Q(out) * D(out) * Q'(out)
    *
    *     where Z = Q'u, u is a vector of length N with ones in the
    *     CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.
    *
    *     The eigenvectors of the original matrix are stored in Q, and the
    *     eigenvalues are in D.  The algorithm consists of three stages:
    *
    *        The first stage consists of deflating the size of the problem
    *        when there are multiple eigenvalues or if there is a zero in
    *        the Z vector.  For each such occurence the dimension of the
    *        secular equation problem is reduced by one.  This stage is
    *        performed by the routine SLAED2.
    *
    *        The second stage consists of calculating the updated
    *        eigenvalues. This is done by finding the roots of the secular
    *        equation via the routine SLAED4 (as called by SLAED3).
    *        This routine also calculates the eigenvectors of the current
    *        problem.
    *
    *        The final stage consists of computing the updated eigenvectors
    *        directly using the updated eigenvalues.  The eigenvectors for
    *        the current problem are multiplied with the eigenvectors from
    *        the overall problem.
    *
    *  Arguments
    *  =========
    *
    *  N      (input) INTEGER
    *         The dimension of the symmetric tridiagonal matrix.  N >= 0.
    *
    *  D      (input/output) REAL array, dimension (N)
    *         On entry, the eigenvalues of the rank-1-perturbed matrix.
    *         On exit, the eigenvalues of the repaired matrix.
    *
    *  Q      (input/output) REAL array, dimension (LDQ,N)
    *         On entry, the eigenvectors of the rank-1-perturbed matrix.
    *         On exit, the eigenvectors of the repaired tridiagonal matrix.
    *
    *  LDQ    (input) INTEGER
    *         The leading dimension of the array Q.  LDQ >= max(1,N).
    *
    *  INDXQ  (input/output) INTEGER array, dimension (N)
    *         On entry, the permutation which separately sorts the two
    *         subproblems in D into ascending order.
    *         On exit, the permutation which will reintegrate the
    *         subproblems back into sorted order,
    *         i.e. D( INDXQ( I = 1, N ) ) will be in ascending order.
    *
    *  RHO    (input) REAL
    *         The subdiagonal entry used to create the rank-1 modification.
    *
    *  CUTPNT (input) INTEGER
    *         The location of the last eigenvalue in the leading sub-matrix.
    *         min(1,N) <= CUTPNT <= N/2.
    *
    *  WORK   (workspace) REAL array, dimension (4*N + N**2)
    *
    *  IWORK  (workspace) INTEGER array, dimension (4*N)
    *
    *  INFO   (output) INTEGER
    *          = 0:  successful exit.
    *          < 0:  if INFO = -i, the i-th argument had an illegal value.
    *          > 0:  if INFO = 1, an eigenvalue did not converge
    *
    *  Further Details
    *  ===============
    *
    *  Based on contributions by
    *     Jeff Rutter, Computer Science Division, University of California
    *     at Berkeley, USA
    *  Modified by Francoise Tisseur, University of Tennessee.
    *
    *  =====================================================================
    *
    *     .. Local Scalars ..
          INTEGER            COLTYP, CPP1, I, IDLMDA, INDX, INDXC, INDXP,
         $                   IQ2, IS, IW, IZ, K, N1, N2
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           SCOPY, SLAED2, SLAED3, SLAMRG, XERBLA
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          MAX, MIN
    *     ..
    *     .. Executable Statements ..
    *
    *     Test the input parameters.
    *
          INFO = 0
    *
          IF( N.LT.0 ) THEN
             INFO = -1
          ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
             INFO = -4
          ELSE IF( MIN( 1, N / 2 ).GT.CUTPNT .OR. ( N / 2 ).LT.CUTPNT ) THEN
             INFO = -7
          END IF
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'SLAED1', -INFO )
             RETURN
          END IF
    *
    *     Quick return if possible
    *
          IF( N.EQ.0 )
         $   RETURN
    *
    *     The following values are integer pointers which indicate
    *     the portion of the workspace
    *     used by a particular array in SLAED2 and SLAED3.
    *
          IZ = 1
          IDLMDA = IZ + N
          IW = IDLMDA + N
          IQ2 = IW + N
    *
          INDX = 1
          INDXC = INDX + N
          COLTYP = INDXC + N
          INDXP = COLTYP + N
    *
    *
    *     Form the z-vector which consists of the last row of Q_1 and the
    *     first row of Q_2.
    *
          CALL SCOPY( CUTPNT, Q( CUTPNT, 1 ), LDQ, WORK( IZ ), 1 )
          CPP1 = CUTPNT + 1
          CALL SCOPY( N-CUTPNT, Q( CPP1, CPP1 ), LDQ, WORK( IZ+CUTPNT ), 1 )
    *
    *     Deflate eigenvalues.
    *
          CALL SLAED2( K, N, CUTPNT, D, Q, LDQ, INDXQ, RHO, WORK( IZ ),
         $             WORK( IDLMDA ), WORK( IW ), WORK( IQ2 ),
         $             IWORK( INDX ), IWORK( INDXC ), IWORK( INDXP ),
         $             IWORK( COLTYP ), INFO )
    *
          IF( INFO.NE.0 )
         $   GO TO 20
    *
    *     Solve Secular Equation.
    *
          IF( K.NE.0 ) THEN
             IS = ( IWORK( COLTYP )+IWORK( COLTYP+1 ) )*CUTPNT +
         $        ( IWORK( COLTYP+1 )+IWORK( COLTYP+2 ) )*( N-CUTPNT ) + IQ2
             CALL SLAED3( K, N, CUTPNT, D, Q, LDQ, RHO, WORK( IDLMDA ),
         $                WORK( IQ2 ), IWORK( INDXC ), IWORK( COLTYP ),
         $                WORK( IW ), WORK( IS ), INFO )
             IF( INFO.NE.0 )
         $      GO TO 20
    *
    *     Prepare the INDXQ sorting permutation.
    *
             N1 = K
             N2 = N - K
             CALL SLAMRG( N1, N2, D, 1, -1, INDXQ )
          ELSE
             DO 10 I = 1, N
                INDXQ( I ) = I
       10    CONTINUE
          END IF
    *
       20 CONTINUE
          RETURN
    *
    *     End of SLAED1
    *
          END