slaed4.f

          SUBROUTINE SLAED4( N, I, D, Z, DELTA, RHO, DLAM, INFO )
    *
    *  -- LAPACK routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          INTEGER            I, INFO, N
          REAL               DLAM, RHO
    *     ..
    *     .. Array Arguments ..
          REAL               D( * ), DELTA( * ), Z( * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  This subroutine computes the I-th updated eigenvalue of a symmetric
    *  rank-one modification to a diagonal matrix whose elements are
    *  given in the array d, and that
    *
    *             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 )  +  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.
    *
    *  Arguments
    *  =========
    *
    *  N      (input) INTEGER
    *         The length of all arrays.
    *
    *  I      (input) INTEGER
    *         The index of the eigenvalue to be computed.  1 <= I <= N.
    *
    *  D      (input) REAL array, dimension (N)
    *         The original eigenvalues.  It is assumed that they are in
    *         order, D(I) < D(J)  for I < J.
    *
    *  Z      (input) REAL array, dimension (N)
    *         The components of the updating vector.
    *
    *  DELTA  (output) REAL array, dimension (N)
    *         If N .GT. 2, DELTA contains (D(j) - lambda_I) in its  j-th
    *         component.  If N = 1, then DELTA(1) = 1. If N = 2, see SLAED5
    *         for detail. The vector DELTA contains the information necessary
    *         to construct the eigenvectors by SLAED3 and SLAED9.
    *
    *  RHO    (input) REAL
    *         The scalar in the symmetric updating formula.
    *
    *  DLAM   (output) REAL
    *         The computed lambda_I, the I-th updated eigenvalue.
    *
    *  INFO   (output) INTEGER
    *         = 0:  successful exit
    *         > 0:  if INFO = 1, the updating process failed.
    *
    *  Internal Parameters
    *  ===================
    *
    *  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.
    *
    *  Further Details
    *  ===============
    *
    *  Based on contributions by
    *     Ren-Cang Li, Computer Science Division, University of California
    *     at Berkeley, USA
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          INTEGER            MAXIT
          PARAMETER          ( MAXIT = 30 )
          REAL               ZERO, ONE, TWO, THREE, FOUR, EIGHT, TEN
          PARAMETER          ( ZERO = 0.0E0, ONE = 1.0E0, TWO = 2.0E0,
         $                   THREE = 3.0E0, FOUR = 4.0E0, EIGHT = 8.0E0,
         $                   TEN = 10.0E0 )
    *     ..
    *     .. Local Scalars ..
          LOGICAL            ORGATI, SWTCH, SWTCH3
          INTEGER            II, IIM1, IIP1, IP1, ITER, J, NITER
          REAL               A, B, C, DEL, DLTLB, DLTUB, DPHI, DPSI, DW,
         $                   EPS, ERRETM, ETA, MIDPT, PHI, PREW, PSI,
         $                   RHOINV, TAU, TEMP, TEMP1, W
    *     ..
    *     .. Local Arrays ..
          REAL               ZZ( 3 )
    *     ..
    *     .. External Functions ..
          REAL               SLAMCH
          EXTERNAL           SLAMCH
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           SLAED5, SLAED6
    *     ..
    *     .. 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
    *
             DLAM = D( 1 ) + RHO*Z( 1 )*Z( 1 )
             DELTA( 1 ) = ONE
             RETURN
          END IF
          IF( N.EQ.2 ) THEN
             CALL SLAED5( I, D, Z, DELTA, RHO, DLAM )
             RETURN
          END IF
    *
    *     Compute machine epsilon
    *
          EPS = SLAMCH( 'Epsilon' )
          RHOINV = ONE / RHO
    *
    *     The case I = N
    *
          IF( I.EQ.N ) THEN
    *
    *        Initialize some basic variables
    *
             II = N - 1
             NITER = 1
    *
    *        Calculate initial guess
    *
             MIDPT = RHO / TWO
    *
    *        If ||Z||_2 is not one, then TEMP should be set to
    *        RHO * ||Z||_2^2 / TWO
    *
             DO 10 J = 1, N
                DELTA( J ) = ( D( J )-D( I ) ) - MIDPT
       10    CONTINUE
    *
             PSI = ZERO
             DO 20 J = 1, N - 2
                PSI = PSI + Z( J )*Z( J ) / DELTA( J )
       20    CONTINUE
    *
             C = RHOINV + PSI
             W = C + Z( II )*Z( II ) / DELTA( II ) +
         $       Z( N )*Z( N ) / DELTA( N )
    *
             IF( W.LE.ZERO ) THEN
                TEMP = Z( N-1 )*Z( N-1 ) / ( D( N )-D( N-1 )+RHO ) +
         $             Z( N )*Z( N ) / RHO
                IF( C.LE.TEMP ) THEN
                   TAU = RHO
                ELSE
                   DEL = D( N ) - D( N-1 )
                   A = -C*DEL + Z( N-1 )*Z( N-1 ) + Z( N )*Z( N )
                   B = Z( N )*Z( N )*DEL
                   IF( A.LT.ZERO ) THEN
                      TAU = TWO*B / ( SQRT( A*A+FOUR*B*C )-A )
                   ELSE
                      TAU = ( A+SQRT( A*A+FOUR*B*C ) ) / ( TWO*C )
                   END IF
                END IF
    *
    *           It can be proved that
    *               D(N)+RHO/2 <= LAMBDA(N) < D(N)+TAU <= D(N)+RHO
    *
                DLTLB = MIDPT
                DLTUB = RHO
             ELSE
                DEL = D( N ) - D( N-1 )
                A = -C*DEL + Z( N-1 )*Z( N-1 ) + Z( N )*Z( N )
                B = Z( N )*Z( N )*DEL
                IF( A.LT.ZERO ) THEN
                   TAU = TWO*B / ( SQRT( A*A+FOUR*B*C )-A )
                ELSE
                   TAU = ( A+SQRT( A*A+FOUR*B*C ) ) / ( TWO*C )
                END IF
    *
    *           It can be proved that
    *               D(N) < D(N)+TAU < LAMBDA(N) < D(N)+RHO/2
    *
                DLTLB = ZERO
                DLTUB = MIDPT
             END IF
    *
             DO 30 J = 1, N
                DELTA( J ) = ( D( J )-D( I ) ) - 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 )
                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 )
             PHI = Z( N )*TEMP
             DPHI = TEMP*TEMP
             ERRETM = EIGHT*( -PHI-PSI ) + ERRETM - PHI + RHOINV +
         $            ABS( TAU )*( DPSI+DPHI )
    *
             W = RHOINV + PHI + PSI
    *
    *        Test for convergence
    *
             IF( ABS( W ).LE.EPS*ERRETM ) THEN
                DLAM = D( I ) + TAU
                GO TO 250
             END IF
    *
             IF( W.LE.ZERO ) THEN
                DLTLB = MAX( DLTLB, TAU )
             ELSE
                DLTUB = MIN( DLTUB, TAU )
             END IF
    *
    *        Calculate the new step
    *
             NITER = NITER + 1
             C = W - DELTA( N-1 )*DPSI - DELTA( N )*DPHI
             A = ( DELTA( N-1 )+DELTA( N ) )*W -
         $       DELTA( N-1 )*DELTA( N )*( DPSI+DPHI )
             B = DELTA( N-1 )*DELTA( N )*W
             IF( C.LT.ZERO )
         $      C = ABS( C )
             IF( C.EQ.ZERO ) THEN
    *          ETA = B/A
    *           ETA = RHO - TAU
                ETA = DLTUB - TAU
             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 = TAU + ETA
             IF( TEMP.GT.DLTUB .OR. TEMP.LT.DLTLB ) THEN
                IF( W.LT.ZERO ) THEN
                   ETA = ( DLTUB-TAU ) / TWO
                ELSE
                   ETA = ( DLTLB-TAU ) / TWO
                END IF
             END IF
             DO 50 J = 1, N
                DELTA( J ) = DELTA( J ) - ETA
       50    CONTINUE
    *
             TAU = TAU + ETA
    *
    *        Evaluate PSI and the derivative DPSI
    *
             DPSI = ZERO
             PSI = ZERO
             ERRETM = ZERO
             DO 60 J = 1, II
                TEMP = Z( 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
    *
             TEMP = Z( N ) / DELTA( N )
             PHI = Z( N )*TEMP
             DPHI = TEMP*TEMP
             ERRETM = EIGHT*( -PHI-PSI ) + ERRETM - PHI + RHOINV +
         $            ABS( TAU )*( 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
                   DLAM = D( I ) + TAU
                   GO TO 250
                END IF
    *
                IF( W.LE.ZERO ) THEN
                   DLTLB = MAX( DLTLB, TAU )
                ELSE
                   DLTUB = MIN( DLTUB, TAU )
                END IF
    *
    *           Calculate the new step
    *
                C = W - DELTA( N-1 )*DPSI - DELTA( N )*DPHI
                A = ( DELTA( N-1 )+DELTA( N ) )*W -
         $          DELTA( N-1 )*DELTA( N )*( DPSI+DPHI )
                B = DELTA( N-1 )*DELTA( N )*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 = TAU + ETA
                IF( TEMP.GT.DLTUB .OR. TEMP.LT.DLTLB ) THEN
                   IF( W.LT.ZERO ) THEN
                      ETA = ( DLTUB-TAU ) / TWO
                   ELSE
                      ETA = ( DLTLB-TAU ) / TWO
                   END IF
                END IF
                DO 70 J = 1, N
                   DELTA( J ) = DELTA( J ) - ETA
       70       CONTINUE
    *
                TAU = TAU + ETA
    *
    *           Evaluate PSI and the derivative DPSI
    *
                DPSI = ZERO
                PSI = ZERO
                ERRETM = ZERO
                DO 80 J = 1, II
                   TEMP = Z( 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
    *
                TEMP = Z( N ) / DELTA( N )
                PHI = Z( N )*TEMP
                DPHI = TEMP*TEMP
                ERRETM = EIGHT*( -PHI-PSI ) + ERRETM - PHI + RHOINV +
         $               ABS( TAU )*( DPSI+DPHI )
    *
                W = RHOINV + PHI + PSI
       90    CONTINUE
    *
    *        Return with INFO = 1, NITER = MAXIT and not converged
    *
             INFO = 1
             DLAM = D( I ) + TAU
             GO TO 250
    *
    *        End for the case I = N
    *
          ELSE
    *
    *        The case for I < N
    *
             NITER = 1
             IP1 = I + 1
    *
    *        Calculate initial guess
    *
             DEL = D( IP1 ) - D( I )
             MIDPT = DEL / TWO
             DO 100 J = 1, N
                DELTA( J ) = ( D( J )-D( I ) ) - MIDPT
      100    CONTINUE
    *
             PSI = ZERO
             DO 110 J = 1, I - 1
                PSI = PSI + Z( J )*Z( J ) / DELTA( J )
      110    CONTINUE
    *
             PHI = ZERO
             DO 120 J = N, I + 2, -1
                PHI = PHI + Z( J )*Z( J ) / DELTA( J )
      120    CONTINUE
             C = RHOINV + PSI + PHI
             W = C + Z( I )*Z( I ) / DELTA( I ) +
         $       Z( IP1 )*Z( IP1 ) / DELTA( IP1 )
    *
             IF( W.GT.ZERO ) THEN
    *
    *           d(i)< the ith eigenvalue < (d(i)+d(i+1))/2
    *
    *           We choose d(i) as origin.
    *
                ORGATI = .TRUE.
                A = C*DEL + Z( I )*Z( I ) + Z( IP1 )*Z( IP1 )
                B = Z( I )*Z( I )*DEL
                IF( A.GT.ZERO ) THEN
                   TAU = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) )
                ELSE
                   TAU = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
                END IF
                DLTLB = ZERO
                DLTUB = MIDPT
             ELSE
    *
    *           (d(i)+d(i+1))/2 <= the ith eigenvalue < d(i+1)
    *
    *           We choose d(i+1) as origin.
    *
                ORGATI = .FALSE.
                A = C*DEL - Z( I )*Z( I ) - Z( IP1 )*Z( IP1 )
                B = Z( IP1 )*Z( IP1 )*DEL
                IF( A.LT.ZERO ) THEN
                   TAU = TWO*B / ( A-SQRT( ABS( A*A+FOUR*B*C ) ) )
                ELSE
                   TAU = -( A+SQRT( ABS( A*A+FOUR*B*C ) ) ) / ( TWO*C )
                END IF
                DLTLB = -MIDPT
                DLTUB = ZERO
             END IF
    *
             IF( ORGATI ) THEN
                DO 130 J = 1, N
                   DELTA( J ) = ( D( J )-D( I ) ) - TAU
      130       CONTINUE
             ELSE
                DO 140 J = 1, N
                   DELTA( J ) = ( D( J )-D( IP1 ) ) - TAU
      140       CONTINUE
             END IF
             IF( ORGATI ) THEN
                II = I
             ELSE
                II = I + 1
             END IF
             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 ) / 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 ) / 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 ) / 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( TAU )*DW
    *
    *        Test for convergence
    *
             IF( ABS( W ).LE.EPS*ERRETM ) THEN
                IF( ORGATI ) THEN
                   DLAM = D( I ) + TAU
                ELSE
                   DLAM = D( IP1 ) + TAU
                END IF
                GO TO 250
             END IF
    *
             IF( W.LE.ZERO ) THEN
                DLTLB = MAX( DLTLB, TAU )
             ELSE
                DLTUB = MIN( DLTUB, TAU )
             END IF
    *
    *        Calculate the new step
    *
             NITER = NITER + 1
             IF( .NOT.SWTCH3 ) THEN
                IF( ORGATI ) THEN
                   C = W - DELTA( IP1 )*DW - ( D( I )-D( IP1 ) )*
         $             ( Z( I ) / DELTA( I ) )**2
                ELSE
                   C = W - DELTA( I )*DW - ( D( IP1 )-D( I ) )*
         $             ( Z( IP1 ) / DELTA( IP1 ) )**2
                END IF
                A = ( DELTA( I )+DELTA( IP1 ) )*W -
         $          DELTA( I )*DELTA( IP1 )*DW
                B = DELTA( I )*DELTA( IP1 )*W
                IF( C.EQ.ZERO ) THEN
                   IF( A.EQ.ZERO ) THEN
                      IF( ORGATI ) THEN
                         A = Z( I )*Z( I ) + DELTA( IP1 )*DELTA( IP1 )*
         $                   ( DPSI+DPHI )
                      ELSE
                         A = Z( IP1 )*Z( IP1 ) + DELTA( I )*DELTA( I )*
         $                   ( 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
    *
                TEMP = RHOINV + PSI + PHI
                IF( ORGATI ) THEN
                   TEMP1 = Z( IIM1 ) / DELTA( IIM1 )
                   TEMP1 = TEMP1*TEMP1
                   C = TEMP - DELTA( IIP1 )*( DPSI+DPHI ) -
         $             ( D( IIM1 )-D( IIP1 ) )*TEMP1
                   ZZ( 1 ) = Z( IIM1 )*Z( IIM1 )
                   ZZ( 3 ) = DELTA( IIP1 )*DELTA( IIP1 )*
         $                   ( ( DPSI-TEMP1 )+DPHI )
                ELSE
                   TEMP1 = Z( IIP1 ) / DELTA( IIP1 )
                   TEMP1 = TEMP1*TEMP1
                   C = TEMP - DELTA( IIM1 )*( DPSI+DPHI ) -
         $             ( D( IIP1 )-D( IIM1 ) )*TEMP1
                   ZZ( 1 ) = DELTA( IIM1 )*DELTA( IIM1 )*
         $                   ( DPSI+( DPHI-TEMP1 ) )
                   ZZ( 3 ) = Z( IIP1 )*Z( IIP1 )
                END IF
                ZZ( 2 ) = Z( II )*Z( II )
                CALL SLAED6( NITER, ORGATI, C, DELTA( IIM1 ), ZZ, W, ETA,
         $                   INFO )
                IF( INFO.NE.0 )
         $         GO TO 250
             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
             TEMP = TAU + ETA
             IF( TEMP.GT.DLTUB .OR. TEMP.LT.DLTLB ) THEN
                IF( W.LT.ZERO ) THEN
                   ETA = ( DLTUB-TAU ) / TWO
                ELSE
                   ETA = ( DLTLB-TAU ) / TWO
                END IF
             END IF
    *
             PREW = W
    *
             DO 180 J = 1, N
                DELTA( J ) = DELTA( J ) - ETA
      180    CONTINUE
    *
    *        Evaluate PSI and the derivative DPSI
    *
             DPSI = ZERO
             PSI = ZERO
             ERRETM = ZERO
             DO 190 J = 1, IIM1
                TEMP = Z( J ) / DELTA( J )
                PSI = PSI + Z( J )*TEMP
                DPSI = DPSI + TEMP*TEMP
                ERRETM = ERRETM + PSI
      190    CONTINUE
             ERRETM = ABS( ERRETM )
    *
    *        Evaluate PHI and the derivative DPHI
    *
             DPHI = ZERO
             PHI = ZERO
             DO 200 J = N, IIP1, -1
                TEMP = Z( J ) / DELTA( J )
                PHI = PHI + Z( J )*TEMP
                DPHI = DPHI + TEMP*TEMP
                ERRETM = ERRETM + PHI
      200    CONTINUE
    *
             TEMP = Z( II ) / DELTA( II )
             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( TAU+ETA )*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
    *
             TAU = TAU + ETA
    *
    *        Main loop to update the values of the array   DELTA
    *
             ITER = NITER + 1
    *
             DO 240 NITER = ITER, MAXIT
    *
    *           Test for convergence
    *
                IF( ABS( W ).LE.EPS*ERRETM ) THEN
                   IF( ORGATI ) THEN
                      DLAM = D( I ) + TAU
                   ELSE
                      DLAM = D( IP1 ) + TAU
                   END IF
                   GO TO 250
                END IF
    *
                IF( W.LE.ZERO ) THEN
                   DLTLB = MAX( DLTLB, TAU )
                ELSE
                   DLTUB = MIN( DLTUB, TAU )
                END IF
    *
    *           Calculate the new step
    *
                IF( .NOT.SWTCH3 ) THEN
                   IF( .NOT.SWTCH ) THEN
                      IF( ORGATI ) THEN
                         C = W - DELTA( IP1 )*DW -
         $                   ( D( I )-D( IP1 ) )*( Z( I ) / DELTA( I ) )**2
                      ELSE
                         C = W - DELTA( I )*DW - ( D( IP1 )-D( I ) )*
         $                   ( Z( IP1 ) / DELTA( IP1 ) )**2
                      END IF
                   ELSE
                      TEMP = Z( II ) / DELTA( II )
                      IF( ORGATI ) THEN
                         DPSI = DPSI + TEMP*TEMP
                      ELSE
                         DPHI = DPHI + TEMP*TEMP
                      END IF
                      C = W - DELTA( I )*DPSI - DELTA( IP1 )*DPHI
                   END IF
                   A = ( DELTA( I )+DELTA( IP1 ) )*W -
         $             DELTA( I )*DELTA( IP1 )*DW
                   B = DELTA( I )*DELTA( IP1 )*W
                   IF( C.EQ.ZERO ) THEN
                      IF( A.EQ.ZERO ) THEN
                         IF( .NOT.SWTCH ) THEN
                            IF( ORGATI ) THEN
                               A = Z( I )*Z( I ) + DELTA( IP1 )*
         $                         DELTA( IP1 )*( DPSI+DPHI )
                            ELSE
                               A = Z( IP1 )*Z( IP1 ) +
         $                         DELTA( I )*DELTA( I )*( DPSI+DPHI )
                            END IF
                         ELSE
                            A = DELTA( I )*DELTA( I )*DPSI +
         $                      DELTA( IP1 )*DELTA( IP1 )*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
    *
                   TEMP = RHOINV + PSI + PHI
                   IF( SWTCH ) THEN
                      C = TEMP - DELTA( IIM1 )*DPSI - DELTA( IIP1 )*DPHI
                      ZZ( 1 ) = DELTA( IIM1 )*DELTA( IIM1 )*DPSI
                      ZZ( 3 ) = DELTA( IIP1 )*DELTA( IIP1 )*DPHI
                   ELSE
                      IF( ORGATI ) THEN
                         TEMP1 = Z( IIM1 ) / DELTA( IIM1 )
                         TEMP1 = TEMP1*TEMP1
                         C = TEMP - DELTA( IIP1 )*( DPSI+DPHI ) -
         $                   ( D( IIM1 )-D( IIP1 ) )*TEMP1
                         ZZ( 1 ) = Z( IIM1 )*Z( IIM1 )
                         ZZ( 3 ) = DELTA( IIP1 )*DELTA( IIP1 )*
         $                         ( ( DPSI-TEMP1 )+DPHI )
                      ELSE
                         TEMP1 = Z( IIP1 ) / DELTA( IIP1 )
                         TEMP1 = TEMP1*TEMP1
                         C = TEMP - DELTA( IIM1 )*( DPSI+DPHI ) -
         $                   ( D( IIP1 )-D( IIM1 ) )*TEMP1
                         ZZ( 1 ) = DELTA( IIM1 )*DELTA( IIM1 )*
         $                         ( DPSI+( DPHI-TEMP1 ) )
                         ZZ( 3 ) = Z( IIP1 )*Z( IIP1 )
                      END IF
                   END IF
                   CALL SLAED6( NITER, ORGATI, C, DELTA( IIM1 ), ZZ, W, ETA,
         $                      INFO )
                   IF( INFO.NE.0 )
         $            GO TO 250
                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
                TEMP = TAU + ETA
                IF( TEMP.GT.DLTUB .OR. TEMP.LT.DLTLB ) THEN
                   IF( W.LT.ZERO ) THEN
                      ETA = ( DLTUB-TAU ) / TWO
                   ELSE
                      ETA = ( DLTLB-TAU ) / TWO
                   END IF
                END IF
    *
                DO 210 J = 1, N
                   DELTA( J ) = DELTA( J ) - ETA
      210       CONTINUE
    *
                TAU = TAU + ETA
                PREW = W
    *
    *           Evaluate PSI and the derivative DPSI
    *
                DPSI = ZERO
                PSI = ZERO
                ERRETM = ZERO
                DO 220 J = 1, IIM1
                   TEMP = Z( J ) / DELTA( J )
                   PSI = PSI + Z( J )*TEMP
                   DPSI = DPSI + TEMP*TEMP
                   ERRETM = ERRETM + PSI
      220       CONTINUE
                ERRETM = ABS( ERRETM )
    *
    *           Evaluate PHI and the derivative DPHI
    *
                DPHI = ZERO
                PHI = ZERO
                DO 230 J = N, IIP1, -1
                   TEMP = Z( J ) / DELTA( J )
                   PHI = PHI + Z( J )*TEMP
                   DPHI = DPHI + TEMP*TEMP
                   ERRETM = ERRETM + PHI
      230       CONTINUE
    *
                TEMP = Z( II ) / DELTA( II )
                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( TAU )*DW
                IF( W*PREW.GT.ZERO .AND. ABS( W ).GT.ABS( PREW ) / TEN )
         $         SWTCH = .NOT.SWTCH
    *
      240    CONTINUE
    *
    *        Return with INFO = 1, NITER = MAXIT and not converged
    *
             INFO = 1
             IF( ORGATI ) THEN
                DLAM = D( I ) + TAU
             ELSE
                DLAM = D( IP1 ) + TAU
             END IF
    *
          END IF
    *
      250 CONTINUE
    *
          RETURN
    *
    *     End of SLAED4
    *
          END