path: root/SRC/slasd4.f

*> \brief \b SLASD4 computes the square root of the i-th updated eigenvalue of a positive symmetric rank-one modification to a positive diagonal matrix. Used by sbdsdc.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
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*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE SLASD4( N, I, D, Z, DELTA, RHO, SIGMA, WORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            I, INFO, N
*       REAL               RHO, SIGMA
*       ..
*       .. Array Arguments ..
*       REAL               D( * ), DELTA( * ), WORK( * ), Z( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> This subroutine computes the square root of the I-th updated
*> eigenvalue of a positive symmetric rank-one modification to
*> a positive diagonal matrix whose entries are given as the squares
*> of the corresponding entries in the array d, and that
*>
*>        0 <= D(i) < D(j)  for  i < j
*>
*> and that RHO > 0. This is arranged by the calling routine, and is
*> no loss in generality.  The rank-one modified system is thus
*>
*>        diag( D ) * diag( D ) +  RHO * Z * Z_transpose.
*>
*> where we assume the Euclidean norm of Z is 1.
*>
*> The method consists of approximating the rational functions in the
*> secular equation by simpler interpolating rational functions.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>         The length of all arrays.
*> \endverbatim
*>
*> \param[in] I
*> \verbatim
*>          I is INTEGER
*>         The index of the eigenvalue to be computed.  1 <= I <= N.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is REAL array, dimension ( N )
*>         The original eigenvalues.  It is assumed that they are in
*>         order, 0 <= D(I) < D(J)  for I < J.
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*>          Z is REAL array, dimension ( N )
*>         The components of the updating vector.
*> \endverbatim
*>
*> \param[out] DELTA
*> \verbatim
*>          DELTA is REAL array, dimension ( N )
*>         If N .ne. 1, DELTA contains (D(j) - sigma_I) in its  j-th
*>         component.  If N = 1, then DELTA(1) = 1.  The vector DELTA
*>         contains the information necessary to construct the
*>         (singular) eigenvectors.
*> \endverbatim
*>
*> \param[in] RHO
*> \verbatim
*>          RHO is REAL
*>         The scalar in the symmetric updating formula.
*> \endverbatim
*>
*> \param[out] SIGMA
*> \verbatim
*>          SIGMA is REAL
*>         The computed sigma_I, the I-th updated eigenvalue.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is REAL array, dimension ( N )
*>         If N .ne. 1, WORK contains (D(j) + sigma_I) in its  j-th
*>         component.  If N = 1, then WORK( 1 ) = 1.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>         = 0:  successful exit
*>         > 0:  if INFO = 1, the updating process failed.
*> \endverbatim
*
*> \par Internal Parameters:
*  =========================
*>
*> \verbatim
*>  Logical variable ORGATI (origin-at-i?) is used for distinguishing
*>  whether D(i) or D(i+1) is treated as the origin.
*>
*>            ORGATI = .true.    origin at i
*>            ORGATI = .false.   origin at i+1
*>
*>  Logical variable SWTCH3 (switch-for-3-poles?) is for noting
*>  if we are working with THREE poles!
*>
*>  MAXIT is the maximum number of iterations allowed for each
*>  eigenvalue.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup OTHERauxiliary
*
*> \par Contributors:
*  ==================
*>
*>     Ren-Cang Li, Computer Science Division, University of California
*>     at Berkeley, USA
*>
*  =====================================================================
      SUBROUTINE SLASD4( N, I, D, Z, DELTA, RHO, SIGMA, WORK, INFO )
*
*  -- LAPACK auxiliary routine (version 3.7.0) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     December 2016
*
*     .. Scalar Arguments ..
      INTEGER            I, INFO, N
      REAL   RHO, SIGMA
*     ..
*     .. Array Arguments ..
      REAL   D( * ), DELTA( * ), WORK( * ), Z( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            MAXIT
      PARAMETER          ( MAXIT = 400 )
      REAL               ZERO, ONE, TWO, THREE, FOUR, EIGHT, TEN
      PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0, TWO = 2.0E+0,
     $                   THREE = 3.0E+0, FOUR = 4.0E+0, EIGHT = 8.0E+0,
     $                   TEN = 10.0E+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            ORGATI, SWTCH, SWTCH3, GEOMAVG
      INTEGER            II, IIM1, IIP1, IP1, ITER, J, NITER
      REAL               A, B, C, DELSQ, DELSQ2, SQ2, DPHI, DPSI, DTIIM,
     $                   DTIIP, DTIPSQ, DTISQ, DTNSQ, DTNSQ1, DW, EPS,
     $                   ERRETM, ETA, PHI, PREW, PSI, RHOINV, SGLB,
     $                   SGUB, TAU, TAU2, TEMP, TEMP1, TEMP2, W
*     ..
*     .. Local Arrays ..
      REAL               DD( 3 ), ZZ( 3 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           SLAED6, SLASD5
*     ..
*     .. External Functions ..
      REAL               SLAMCH
      EXTERNAL           SLAMCH
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
*     Since this routine is called in an inner loop, we do no argument
*     checking.
*
*     Quick return for N=1 and 2.
*
      INFO = 0
      IF( N.EQ.1 ) THEN
*
*        Presumably, I=1 upon entry
*
         SIGMA = SQRT( D( 1 )*D( 1 )+RHO*Z( 1 )*Z( 1 ) )
         DELTA( 1 ) = ONE
         WORK( 1 ) = ONE
         RETURN
      END IF
      IF( N.EQ.2 ) THEN
         CALL SLASD5( I, D, Z, DELTA, RHO, SIGMA, WORK )
         RETURN
      END IF
*
*     Compute machine epsilon
*
      EPS = SLAMCH( 'Epsilon' )
      RHOINV = ONE / RHO
      TAU2= ZERO
*
*     The case I = N
*
      IF( I.EQ.N ) THEN
*
*        Initialize some basic variables
*
         II = N - 1
         NITER = 1
*
*        Calculate initial guess
*
         TEMP = RHO / TWO
*
*        If ||Z||_2 is not one, then TEMP should be set to
*        RHO * ||Z||_2^2 / TWO
*
         TEMP1 = TEMP / ( D( N )+SQRT( D( N )*D( N )+TEMP ) )
         DO 10 J = 1, N
            WORK( J ) = D( J ) + D( N ) + TEMP1
            DELTA( J ) = ( D( J )-D( N ) ) - TEMP1
   10    CONTINUE
*
         PSI = ZERO
         DO 20 J = 1, N - 2
            PSI = PSI + Z( J )*Z( J ) / ( DELTA( J )*WORK( J ) )
   20    CONTINUE
*
         C = RHOINV + PSI
         W = C + Z( II )*Z( II ) / ( DELTA( II )*WORK( II ) ) +
     $       Z( N )*Z( N ) / ( DELTA( N )*WORK( N ) )
*
         IF( W.LE.ZERO ) THEN
            TEMP1 = SQRT( D( N )*D( N )+RHO )
            TEMP = Z( N-1 )*Z( N-1 ) / ( ( D( N-1 )+TEMP1 )*
     $             ( D( N )-D( N-1 )+RHO / ( D( N )+TEMP1 ) ) ) +
     $             Z( N )*Z( N ) / RHO
*
*           The following TAU2 is to approximate
*           SIGMA_n^2 - D( N )*D( N )
*
            IF( C.LE.TEMP ) THEN
               TAU = RHO
            ELSE
               DELSQ = ( D( N )-D( N-1 ) )*( D( N )+D( N-1 ) )
               A = -C*DELSQ + Z( N-1 )*Z( N-1 ) + Z( N )*Z( N )
               B = Z( N )*Z( N )*DELSQ
               IF( A.LT.ZERO ) THEN
                  TAU2 = TWO*B / ( SQRT( A*A+FOUR*B*C )-A )
               ELSE
                  TAU2 = ( A+SQRT( A*A+FOUR*B*C ) ) / ( TWO*C )
               END IF
               TAU = TAU2 / ( D( N )+SQRT( D( N )*D( N )+TAU2 ) )
            END IF
*
*           It can be proved that
*               D(N)^2+RHO/2 <= SIGMA_n^2 < D(N)^2+TAU2 <= D(N)^2+RHO
*
         ELSE
            DELSQ = ( D( N )-D( N-1 ) )*( D( N )+D( N-1 ) )
            A = -C*DELSQ + Z( N-1 )*Z( N-1 ) + Z( N )*Z( N )
            B = Z( N )*Z( N )*DELSQ
*
*           The following TAU2 is to approximate
*           SIGMA_n^2 - D( N )*D( N )
*
            IF( A.LT.ZERO ) THEN
               TAU2 = TWO*B / ( SQRT( A*A+FOUR*B*C )-A )
            ELSE
               TAU2 = ( A+SQRT( A*A+FOUR*B*C ) ) / ( TWO*C )
            END IF
            TAU = TAU2 / ( D( N )+SQRT( D( N )*D( N )+TAU2 ) )

*
*           It can be proved that
*           D(N)^2 < D(N)^2+TAU2 < SIGMA(N)^2 < D(N)^2+RHO/2
*
         END IF
*
*        The following TAU is to approximate SIGMA_n - D( N )
*
*         TAU = TAU2 / ( D( N )+SQRT( D( N )*D( N )+TAU2 ) )
*
         SIGMA = D( N ) + TAU
         DO 30 J = 1, N
            DELTA( J ) = ( D( J )-D( N ) ) - TAU
            WORK( J ) = D( J ) + D( N ) + TAU
   30    CONTINUE
*
*        Evaluate PSI and the derivative DPSI
*
         DPSI = ZERO
         PSI = ZERO
         ERRETM = ZERO
         DO 40 J = 1, II
            TEMP = Z( J ) / ( DELTA( J )*WORK( J ) )
            PSI = PSI + Z( J )*TEMP
            DPSI = DPSI + TEMP*TEMP
            ERRETM = ERRETM + PSI
   40    CONTINUE
         ERRETM = ABS( ERRETM )
*
*        Evaluate PHI and the derivative DPHI
*
         TEMP = Z( N ) / ( DELTA( N )*WORK( N ) )
         PHI = Z( N )*TEMP
         DPHI = TEMP*TEMP
         ERRETM = EIGHT*( -PHI-PSI ) + ERRETM - PHI + RHOINV
*    $          + ABS( TAU2 )*( DPSI+DPHI )
*
         W = RHOINV + PHI + PSI
*
*        Test for convergence
*
         IF( ABS( W ).LE.EPS*ERRETM ) THEN
            GO TO 240
         END IF
*
*        Calculate the new step
*
         NITER = NITER + 1
         DTNSQ1 = WORK( N-1 )*DELTA( N-1 )
         DTNSQ = WORK( N )*DELTA( N )
         C = W - DTNSQ1*DPSI - DTNSQ*DPHI
         A = ( DTNSQ+DTNSQ1 )*W - DTNSQ*DTNSQ1*( DPSI+DPHI )
         B = DTNSQ*DTNSQ1*W
         IF( C.LT.ZERO )
     $      C = ABS( C )
         IF( C.EQ.ZERO ) THEN
            ETA = RHO - SIGMA*SIGMA
         ELSE IF( A.GE.ZERO ) THEN
            ETA = ( A+SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
         ELSE
            ETA = TWO*B / ( A-SQRT( ABS( A*A-FOUR*B*C ) ) )
         END IF
*
*        Note, eta should be positive if w is negative, and
*        eta should be negative otherwise. However,
*        if for some reason caused by roundoff, eta*w > 0,
*        we simply use one Newton step instead. This way
*        will guarantee eta*w < 0.
*
         IF( W*ETA.GT.ZERO )
     $      ETA = -W / ( DPSI+DPHI )
         TEMP = ETA - DTNSQ
         IF( TEMP.GT.RHO )
     $      ETA = RHO + DTNSQ
*
         ETA = ETA / ( SIGMA+SQRT( ETA+SIGMA*SIGMA ) )
         TAU = TAU + ETA
         SIGMA = SIGMA + ETA
*
         DO 50 J = 1, N
            DELTA( J ) = DELTA( J ) - ETA
            WORK( J ) = WORK( J ) + ETA
   50    CONTINUE
*
*        Evaluate PSI and the derivative DPSI
*
         DPSI = ZERO
         PSI = ZERO
         ERRETM = ZERO
         DO 60 J = 1, II
            TEMP = Z( J ) / ( WORK( J )*DELTA( J ) )
            PSI = PSI + Z( J )*TEMP
            DPSI = DPSI + TEMP*TEMP
            ERRETM = ERRETM + PSI
   60    CONTINUE
         ERRETM = ABS( ERRETM )
*
*        Evaluate PHI and the derivative DPHI
*
         TAU2 = WORK( N )*DELTA( N )
         TEMP = Z( N ) / TAU2
         PHI = Z( N )*TEMP
         DPHI = TEMP*TEMP
         ERRETM = EIGHT*( -PHI-PSI ) + ERRETM - PHI + RHOINV
*    $          + ABS( TAU2 )*( DPSI+DPHI )
*
         W = RHOINV + PHI + PSI
*
*        Main loop to update the values of the array   DELTA
*
         ITER = NITER + 1
*
         DO 90 NITER = ITER, MAXIT
*
*           Test for convergence
*
            IF( ABS( W ).LE.EPS*ERRETM ) THEN
               GO TO 240
            END IF
*
*           Calculate the new step
*
            DTNSQ1 = WORK( N-1 )*DELTA( N-1 )
            DTNSQ = WORK( N )*DELTA( N )
            C = W - DTNSQ1*DPSI - DTNSQ*DPHI
            A = ( DTNSQ+DTNSQ1 )*W - DTNSQ1*DTNSQ*( DPSI+DPHI )
            B = DTNSQ1*DTNSQ*W
            IF( A.GE.ZERO ) THEN
               ETA = ( A+SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
            ELSE
               ETA = TWO*B / ( A-SQRT( ABS( A*A-FOUR*B*C ) ) )
            END IF
*
*           Note, eta should be positive if w is negative, and
*           eta should be negative otherwise. However,
*           if for some reason caused by roundoff, eta*w > 0,
*           we simply use one Newton step instead. This way
*           will guarantee eta*w < 0.
*
            IF( W*ETA.GT.ZERO )
     $         ETA = -W / ( DPSI+DPHI )
            TEMP = ETA - DTNSQ
            IF( TEMP.LE.ZERO )
     $         ETA = ETA / TWO
*
            ETA = ETA / ( SIGMA+SQRT( ETA+SIGMA*SIGMA ) )
            TAU = TAU + ETA
            SIGMA = SIGMA + ETA
*
            DO 70 J = 1, N
               DELTA( J ) = DELTA( J ) - ETA
               WORK( J ) = WORK( J ) + ETA
   70       CONTINUE
*
*           Evaluate PSI and the derivative DPSI
*
            DPSI = ZERO
            PSI = ZERO
            ERRETM = ZERO
            DO 80 J = 1, II
               TEMP = Z( J ) / ( WORK( J )*DELTA( J ) )
               PSI = PSI + Z( J )*TEMP
               DPSI = DPSI + TEMP*TEMP
               ERRETM = ERRETM + PSI
   80       CONTINUE
            ERRETM = ABS( ERRETM )
*
*           Evaluate PHI and the derivative DPHI
*
            TAU2 = WORK( N )*DELTA( N )
            TEMP = Z( N ) / TAU2
            PHI = Z( N )*TEMP
            DPHI = TEMP*TEMP
            ERRETM = EIGHT*( -PHI-PSI ) + ERRETM - PHI + RHOINV
*    $             + ABS( TAU2 )*( DPSI+DPHI )
*
            W = RHOINV + PHI + PSI
   90    CONTINUE
*
*        Return with INFO = 1, NITER = MAXIT and not converged
*
         INFO = 1
         GO TO 240
*
*        End for the case I = N
*
      ELSE
*
*        The case for I < N
*
         NITER = 1
         IP1 = I + 1
*
*        Calculate initial guess
*
         DELSQ = ( D( IP1 )-D( I ) )*( D( IP1 )+D( I ) )
         DELSQ2 = DELSQ / TWO
         SQ2=SQRT( ( D( I )*D( I )+D( IP1 )*D( IP1 ) ) / TWO )
         TEMP = DELSQ2 / ( D( I )+SQ2 )
         DO 100 J = 1, N
            WORK( J ) = D( J ) + D( I ) + TEMP
            DELTA( J ) = ( D( J )-D( I ) ) - TEMP
  100    CONTINUE
*
         PSI = ZERO
         DO 110 J = 1, I - 1
            PSI = PSI + Z( J )*Z( J ) / ( WORK( J )*DELTA( J ) )
  110    CONTINUE
*
         PHI = ZERO
         DO 120 J = N, I + 2, -1
            PHI = PHI + Z( J )*Z( J ) / ( WORK( J )*DELTA( J ) )
  120    CONTINUE
         C = RHOINV + PSI + PHI
         W = C + Z( I )*Z( I ) / ( WORK( I )*DELTA( I ) ) +
     $       Z( IP1 )*Z( IP1 ) / ( WORK( IP1 )*DELTA( IP1 ) )
*
         GEOMAVG = .FALSE.
         IF( W.GT.ZERO ) THEN
*
*           d(i)^2 < the ith sigma^2 < (d(i)^2+d(i+1)^2)/2
*
*           We choose d(i) as origin.
*
            ORGATI = .TRUE.
            II = I
            SGLB = ZERO
            SGUB = DELSQ2  / ( D( I )+SQ2 )
            A = C*DELSQ + Z( I )*Z( I ) + Z( IP1 )*Z( IP1 )
            B = Z( I )*Z( I )*DELSQ
            IF( A.GT.ZERO ) THEN
               TAU2 = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) )
            ELSE
               TAU2 = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
            END IF
*
*           TAU2 now is an estimation of SIGMA^2 - D( I )^2. The
*           following, however, is the corresponding estimation of
*           SIGMA - D( I ).
*
            TAU = TAU2 / ( D( I )+SQRT( D( I )*D( I )+TAU2 ) )
            TEMP = SQRT(EPS)
            IF( (D(I).LE.TEMP*D(IP1)).AND.(ABS(Z(I)).LE.TEMP)
     $                               .AND.(D(I).GT.ZERO) ) THEN
               TAU = MIN( TEN*D(I), SGUB )
               GEOMAVG = .TRUE.
            END IF
         ELSE
*
*           (d(i)^2+d(i+1)^2)/2 <= the ith sigma^2 < d(i+1)^2/2
*
*           We choose d(i+1) as origin.
*
            ORGATI = .FALSE.
            II = IP1
            SGLB = -DELSQ2  / ( D( II )+SQ2 )
            SGUB = ZERO
            A = C*DELSQ - Z( I )*Z( I ) - Z( IP1 )*Z( IP1 )
            B = Z( IP1 )*Z( IP1 )*DELSQ
            IF( A.LT.ZERO ) THEN
               TAU2 = TWO*B / ( A-SQRT( ABS( A*A+FOUR*B*C ) ) )
            ELSE
               TAU2 = -( A+SQRT( ABS( A*A+FOUR*B*C ) ) ) / ( TWO*C )
            END IF
*
*           TAU2 now is an estimation of SIGMA^2 - D( IP1 )^2. The
*           following, however, is the corresponding estimation of
*           SIGMA - D( IP1 ).
*
            TAU = TAU2 / ( D( IP1 )+SQRT( ABS( D( IP1 )*D( IP1 )+
     $            TAU2 ) ) )
         END IF
*
         SIGMA = D( II ) + TAU
         DO 130 J = 1, N
            WORK( J ) = D( J ) + D( II ) + TAU
            DELTA( J ) = ( D( J )-D( II ) ) - TAU
  130    CONTINUE
         IIM1 = II - 1
         IIP1 = II + 1
*
*        Evaluate PSI and the derivative DPSI
*
         DPSI = ZERO
         PSI = ZERO
         ERRETM = ZERO
         DO 150 J = 1, IIM1
            TEMP = Z( J ) / ( WORK( J )*DELTA( J ) )
            PSI = PSI + Z( J )*TEMP
            DPSI = DPSI + TEMP*TEMP
            ERRETM = ERRETM + PSI
  150    CONTINUE
         ERRETM = ABS( ERRETM )
*
*        Evaluate PHI and the derivative DPHI
*
         DPHI = ZERO
         PHI = ZERO
         DO 160 J = N, IIP1, -1
            TEMP = Z( J ) / ( WORK( J )*DELTA( J ) )
            PHI = PHI + Z( J )*TEMP
            DPHI = DPHI + TEMP*TEMP
            ERRETM = ERRETM + PHI
  160    CONTINUE
*
         W = RHOINV + PHI + PSI
*
*        W is the value of the secular function with
*        its ii-th element removed.
*
         SWTCH3 = .FALSE.
         IF( ORGATI ) THEN
            IF( W.LT.ZERO )
     $         SWTCH3 = .TRUE.
         ELSE
            IF( W.GT.ZERO )
     $         SWTCH3 = .TRUE.
         END IF
         IF( II.EQ.1 .OR. II.EQ.N )
     $      SWTCH3 = .FALSE.
*
         TEMP = Z( II ) / ( WORK( II )*DELTA( II ) )
         DW = DPSI + DPHI + TEMP*TEMP
         TEMP = Z( II )*TEMP
         W = W + TEMP
         ERRETM = EIGHT*( PHI-PSI ) + ERRETM + TWO*RHOINV
     $          + THREE*ABS( TEMP )
*    $          + ABS( TAU2 )*DW
*
*        Test for convergence
*
         IF( ABS( W ).LE.EPS*ERRETM ) THEN
            GO TO 240
         END IF
*
         IF( W.LE.ZERO ) THEN
            SGLB = MAX( SGLB, TAU )
         ELSE
            SGUB = MIN( SGUB, TAU )
         END IF
*
*        Calculate the new step
*
         NITER = NITER + 1
         IF( .NOT.SWTCH3 ) THEN
            DTIPSQ = WORK( IP1 )*DELTA( IP1 )
            DTISQ = WORK( I )*DELTA( I )
            IF( ORGATI ) THEN
               C = W - DTIPSQ*DW + DELSQ*( Z( I ) / DTISQ )**2
            ELSE
               C = W - DTISQ*DW - DELSQ*( Z( IP1 ) / DTIPSQ )**2
            END IF
            A = ( DTIPSQ+DTISQ )*W - DTIPSQ*DTISQ*DW
            B = DTIPSQ*DTISQ*W
            IF( C.EQ.ZERO ) THEN
               IF( A.EQ.ZERO ) THEN
                  IF( ORGATI ) THEN
                     A = Z( I )*Z( I ) + DTIPSQ*DTIPSQ*( DPSI+DPHI )
                  ELSE
                     A = Z( IP1 )*Z( IP1 ) + DTISQ*DTISQ*( DPSI+DPHI )
                  END IF
               END IF
               ETA = B / A
            ELSE IF( A.LE.ZERO ) THEN
               ETA = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
            ELSE
               ETA = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) )
            END IF
         ELSE
*
*           Interpolation using THREE most relevant poles
*
            DTIIM = WORK( IIM1 )*DELTA( IIM1 )
            DTIIP = WORK( IIP1 )*DELTA( IIP1 )
            TEMP = RHOINV + PSI + PHI
            IF( ORGATI ) THEN
               TEMP1 = Z( IIM1 ) / DTIIM
               TEMP1 = TEMP1*TEMP1
               C = ( TEMP - DTIIP*( DPSI+DPHI ) ) -
     $             ( D( IIM1 )-D( IIP1 ) )*( D( IIM1 )+D( IIP1 ) )*TEMP1
               ZZ( 1 ) = Z( IIM1 )*Z( IIM1 )
               IF( DPSI.LT.TEMP1 ) THEN
                  ZZ( 3 ) = DTIIP*DTIIP*DPHI
               ELSE
                  ZZ( 3 ) = DTIIP*DTIIP*( ( DPSI-TEMP1 )+DPHI )
               END IF
            ELSE
               TEMP1 = Z( IIP1 ) / DTIIP
               TEMP1 = TEMP1*TEMP1
               C = ( TEMP - DTIIM*( DPSI+DPHI ) ) -
     $             ( D( IIP1 )-D( IIM1 ) )*( D( IIM1 )+D( IIP1 ) )*TEMP1
               IF( DPHI.LT.TEMP1 ) THEN
                  ZZ( 1 ) = DTIIM*DTIIM*DPSI
               ELSE
                  ZZ( 1 ) = DTIIM*DTIIM*( DPSI+( DPHI-TEMP1 ) )
               END IF
               ZZ( 3 ) = Z( IIP1 )*Z( IIP1 )
            END IF
            ZZ( 2 ) = Z( II )*Z( II )
            DD( 1 ) = DTIIM
            DD( 2 ) = DELTA( II )*WORK( II )
            DD( 3 ) = DTIIP
            CALL SLAED6( NITER, ORGATI, C, DD, ZZ, W, ETA, INFO )
*
            IF( INFO.NE.0 ) THEN
*
*              If INFO is not 0, i.e., SLAED6 failed, switch back
*              to 2 pole interpolation.
*
               SWTCH3 = .FALSE.
               INFO = 0
               DTIPSQ = WORK( IP1 )*DELTA( IP1 )
               DTISQ = WORK( I )*DELTA( I )
               IF( ORGATI ) THEN
                  C = W - DTIPSQ*DW + DELSQ*( Z( I ) / DTISQ )**2
               ELSE
                  C = W - DTISQ*DW - DELSQ*( Z( IP1 ) / DTIPSQ )**2
               END IF
               A = ( DTIPSQ+DTISQ )*W - DTIPSQ*DTISQ*DW
               B = DTIPSQ*DTISQ*W
               IF( C.EQ.ZERO ) THEN
                  IF( A.EQ.ZERO ) THEN
                     IF( ORGATI ) THEN
                        A = Z( I )*Z( I ) + DTIPSQ*DTIPSQ*( DPSI+DPHI )
                     ELSE
                        A = Z( IP1 )*Z( IP1 ) + DTISQ*DTISQ*( DPSI+DPHI)
                     END IF
                  END IF
                  ETA = B / A
               ELSE IF( A.LE.ZERO ) THEN
                  ETA = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
               ELSE
                  ETA = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) )
               END IF
            END IF
         END IF
*
*        Note, eta should be positive if w is negative, and
*        eta should be negative otherwise. However,
*        if for some reason caused by roundoff, eta*w > 0,
*        we simply use one Newton step instead. This way
*        will guarantee eta*w < 0.
*
         IF( W*ETA.GE.ZERO )
     $      ETA = -W / DW
*
         ETA = ETA / ( SIGMA+SQRT( SIGMA*SIGMA+ETA ) )
         TEMP = TAU + ETA
         IF( TEMP.GT.SGUB .OR. TEMP.LT.SGLB ) THEN
            IF( W.LT.ZERO ) THEN
               ETA = ( SGUB-TAU ) / TWO
            ELSE
               ETA = ( SGLB-TAU ) / TWO
            END IF
            IF( GEOMAVG ) THEN
               IF( W .LT. ZERO ) THEN
                  IF( TAU .GT. ZERO ) THEN
                     ETA = SQRT(SGUB*TAU)-TAU
                  END IF
               ELSE
                  IF( SGLB .GT. ZERO ) THEN
                     ETA = SQRT(SGLB*TAU)-TAU
                  END IF
               END IF
            END IF
         END IF
*
         PREW = W
*
         TAU = TAU + ETA
         SIGMA = SIGMA + ETA
*
         DO 170 J = 1, N
            WORK( J ) = WORK( J ) + ETA
            DELTA( J ) = DELTA( J ) - ETA
  170    CONTINUE
*
*        Evaluate PSI and the derivative DPSI
*
         DPSI = ZERO
         PSI = ZERO
         ERRETM = ZERO
         DO 180 J = 1, IIM1
            TEMP = Z( J ) / ( WORK( J )*DELTA( J ) )
            PSI = PSI + Z( J )*TEMP
            DPSI = DPSI + TEMP*TEMP
            ERRETM = ERRETM + PSI
  180    CONTINUE
         ERRETM = ABS( ERRETM )
*
*        Evaluate PHI and the derivative DPHI
*
         DPHI = ZERO
         PHI = ZERO
         DO 190 J = N, IIP1, -1
            TEMP = Z( J ) / ( WORK( J )*DELTA( J ) )
            PHI = PHI + Z( J )*TEMP
            DPHI = DPHI + TEMP*TEMP
            ERRETM = ERRETM + PHI
  190    CONTINUE
*
         TAU2 = WORK( II )*DELTA( II )
         TEMP = Z( II ) / TAU2
         DW = DPSI + DPHI + TEMP*TEMP
         TEMP = Z( II )*TEMP
         W = RHOINV + PHI + PSI + TEMP
         ERRETM = EIGHT*( PHI-PSI ) + ERRETM + TWO*RHOINV
     $          + THREE*ABS( TEMP )
*    $          + ABS( TAU2 )*DW
*
         SWTCH = .FALSE.
         IF( ORGATI ) THEN
            IF( -W.GT.ABS( PREW ) / TEN )
     $         SWTCH = .TRUE.
         ELSE
            IF( W.GT.ABS( PREW ) / TEN )
     $         SWTCH = .TRUE.
         END IF
*
*        Main loop to update the values of the array   DELTA and WORK
*
         ITER = NITER + 1
*
         DO 230 NITER = ITER, MAXIT
*
*           Test for convergence
*
            IF( ABS( W ).LE.EPS*ERRETM ) THEN
*     $          .OR. (SGUB-SGLB).LE.EIGHT*ABS(SGUB+SGLB) ) THEN
               GO TO 240
            END IF
*
            IF( W.LE.ZERO ) THEN
               SGLB = MAX( SGLB, TAU )
            ELSE
               SGUB = MIN( SGUB, TAU )
            END IF
*
*           Calculate the new step
*
            IF( .NOT.SWTCH3 ) THEN
               DTIPSQ = WORK( IP1 )*DELTA( IP1 )
               DTISQ = WORK( I )*DELTA( I )
               IF( .NOT.SWTCH ) THEN
                  IF( ORGATI ) THEN
                     C = W - DTIPSQ*DW + DELSQ*( Z( I ) / DTISQ )**2
                  ELSE
                     C = W - DTISQ*DW - DELSQ*( Z( IP1 ) / DTIPSQ )**2
                  END IF
               ELSE
                  TEMP = Z( II ) / ( WORK( II )*DELTA( II ) )
                  IF( ORGATI ) THEN
                     DPSI = DPSI + TEMP*TEMP
                  ELSE
                     DPHI = DPHI + TEMP*TEMP
                  END IF
                  C = W - DTISQ*DPSI - DTIPSQ*DPHI
               END IF
               A = ( DTIPSQ+DTISQ )*W - DTIPSQ*DTISQ*DW
               B = DTIPSQ*DTISQ*W
               IF( C.EQ.ZERO ) THEN
                  IF( A.EQ.ZERO ) THEN
                     IF( .NOT.SWTCH ) THEN
                        IF( ORGATI ) THEN
                           A = Z( I )*Z( I ) + DTIPSQ*DTIPSQ*
     $                         ( DPSI+DPHI )
                        ELSE
                           A = Z( IP1 )*Z( IP1 ) +
     $                         DTISQ*DTISQ*( DPSI+DPHI )
                        END IF
                     ELSE
                        A = DTISQ*DTISQ*DPSI + DTIPSQ*DTIPSQ*DPHI
                     END IF
                  END IF
                  ETA = B / A
               ELSE IF( A.LE.ZERO ) THEN
                  ETA = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
               ELSE
                  ETA = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) )
               END IF
            ELSE
*
*              Interpolation using THREE most relevant poles
*
               DTIIM = WORK( IIM1 )*DELTA( IIM1 )
               DTIIP = WORK( IIP1 )*DELTA( IIP1 )
               TEMP = RHOINV + PSI + PHI
               IF( SWTCH ) THEN
                  C = TEMP - DTIIM*DPSI - DTIIP*DPHI
                  ZZ( 1 ) = DTIIM*DTIIM*DPSI
                  ZZ( 3 ) = DTIIP*DTIIP*DPHI
               ELSE
                  IF( ORGATI ) THEN
                     TEMP1 = Z( IIM1 ) / DTIIM
                     TEMP1 = TEMP1*TEMP1
                     TEMP2 = ( D( IIM1 )-D( IIP1 ) )*
     $                       ( D( IIM1 )+D( IIP1 ) )*TEMP1
                     C = TEMP - DTIIP*( DPSI+DPHI ) - TEMP2
                     ZZ( 1 ) = Z( IIM1 )*Z( IIM1 )
                     IF( DPSI.LT.TEMP1 ) THEN
                        ZZ( 3 ) = DTIIP*DTIIP*DPHI
                     ELSE
                        ZZ( 3 ) = DTIIP*DTIIP*( ( DPSI-TEMP1 )+DPHI )
                     END IF
                  ELSE
                     TEMP1 = Z( IIP1 ) / DTIIP
                     TEMP1 = TEMP1*TEMP1
                     TEMP2 = ( D( IIP1 )-D( IIM1 ) )*
     $                       ( D( IIM1 )+D( IIP1 ) )*TEMP1
                     C = TEMP - DTIIM*( DPSI+DPHI ) - TEMP2
                     IF( DPHI.LT.TEMP1 ) THEN
                        ZZ( 1 ) = DTIIM*DTIIM*DPSI
                     ELSE
                        ZZ( 1 ) = DTIIM*DTIIM*( DPSI+( DPHI-TEMP1 ) )
                     END IF
                     ZZ( 3 ) = Z( IIP1 )*Z( IIP1 )
                  END IF
               END IF
               DD( 1 ) = DTIIM
               DD( 2 ) = DELTA( II )*WORK( II )
               DD( 3 ) = DTIIP
               CALL SLAED6( NITER, ORGATI, C, DD, ZZ, W, ETA, INFO )
*
               IF( INFO.NE.0 ) THEN
*
*                 If INFO is not 0, i.e., SLAED6 failed, switch
*                 back to two pole interpolation
*
                  SWTCH3 = .FALSE.
                  INFO = 0
                  DTIPSQ = WORK( IP1 )*DELTA( IP1 )
                  DTISQ = WORK( I )*DELTA( I )
                  IF( .NOT.SWTCH ) THEN
                     IF( ORGATI ) THEN
                        C = W - DTIPSQ*DW + DELSQ*( Z( I )/DTISQ )**2
                     ELSE
                        C = W - DTISQ*DW - DELSQ*( Z( IP1 )/DTIPSQ )**2
                     END IF
                  ELSE
                     TEMP = Z( II ) / ( WORK( II )*DELTA( II ) )
                     IF( ORGATI ) THEN
                        DPSI = DPSI + TEMP*TEMP
                     ELSE
                        DPHI = DPHI + TEMP*TEMP
                     END IF
                     C = W - DTISQ*DPSI - DTIPSQ*DPHI
                  END IF
                  A = ( DTIPSQ+DTISQ )*W - DTIPSQ*DTISQ*DW
                  B = DTIPSQ*DTISQ*W
                  IF( C.EQ.ZERO ) THEN
                     IF( A.EQ.ZERO ) THEN
                        IF( .NOT.SWTCH ) THEN
                           IF( ORGATI ) THEN
                              A = Z( I )*Z( I ) + DTIPSQ*DTIPSQ*
     $                            ( DPSI+DPHI )
                           ELSE
                              A = Z( IP1 )*Z( IP1 ) +
     $                            DTISQ*DTISQ*( DPSI+DPHI )
                           END IF
                        ELSE
                           A = DTISQ*DTISQ*DPSI + DTIPSQ*DTIPSQ*DPHI
                        END IF
                     END IF
                     ETA = B / A
                  ELSE IF( A.LE.ZERO ) THEN
                     ETA = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
                  ELSE
                     ETA = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) )
                  END IF
               END IF
            END IF
*
*           Note, eta should be positive if w is negative, and
*           eta should be negative otherwise. However,
*           if for some reason caused by roundoff, eta*w > 0,
*           we simply use one Newton step instead. This way
*           will guarantee eta*w < 0.
*
            IF( W*ETA.GE.ZERO )
     $         ETA = -W / DW
*
            ETA = ETA / ( SIGMA+SQRT( SIGMA*SIGMA+ETA ) )
            TEMP=TAU+ETA
            IF( TEMP.GT.SGUB .OR. TEMP.LT.SGLB ) THEN
               IF( W.LT.ZERO ) THEN
                  ETA = ( SGUB-TAU ) / TWO
               ELSE
                  ETA = ( SGLB-TAU ) / TWO
               END IF
               IF( GEOMAVG ) THEN
                  IF( W .LT. ZERO ) THEN
                     IF( TAU .GT. ZERO ) THEN
                        ETA = SQRT(SGUB*TAU)-TAU
                     END IF
                  ELSE
                     IF( SGLB .GT. ZERO ) THEN
                        ETA = SQRT(SGLB*TAU)-TAU
                     END IF
                  END IF
               END IF
            END IF
*
            PREW = W
*
            TAU = TAU + ETA
            SIGMA = SIGMA + ETA
*
            DO 200 J = 1, N
               WORK( J ) = WORK( J ) + ETA
               DELTA( J ) = DELTA( J ) - ETA
  200       CONTINUE
*
*           Evaluate PSI and the derivative DPSI
*
            DPSI = ZERO
            PSI = ZERO
            ERRETM = ZERO
            DO 210 J = 1, IIM1
               TEMP = Z( J ) / ( WORK( J )*DELTA( J ) )
               PSI = PSI + Z( J )*TEMP
               DPSI = DPSI + TEMP*TEMP
               ERRETM = ERRETM + PSI
  210       CONTINUE
            ERRETM = ABS( ERRETM )
*
*           Evaluate PHI and the derivative DPHI
*
            DPHI = ZERO
            PHI = ZERO
            DO 220 J = N, IIP1, -1
               TEMP = Z( J ) / ( WORK( J )*DELTA( J ) )
               PHI = PHI + Z( J )*TEMP
               DPHI = DPHI + TEMP*TEMP
               ERRETM = ERRETM + PHI
  220       CONTINUE
*
            TAU2 = WORK( II )*DELTA( II )
            TEMP = Z( II ) / TAU2
            DW = DPSI + DPHI + TEMP*TEMP
            TEMP = Z( II )*TEMP
            W = RHOINV + PHI + PSI + TEMP
            ERRETM = EIGHT*( PHI-PSI ) + ERRETM + TWO*RHOINV
     $             + THREE*ABS( TEMP )
*    $             + ABS( TAU2 )*DW
*
            IF( W*PREW.GT.ZERO .AND. ABS( W ).GT.ABS( PREW ) / TEN )
     $         SWTCH = .NOT.SWTCH
*
  230    CONTINUE
*
*        Return with INFO = 1, NITER = MAXIT and not converged
*
         INFO = 1
*
      END IF
*
  240 CONTINUE
      RETURN
*
*     End of SLASD4
*
      END