*> \brief \b DLAED6 used by DSTEDC. Computes one Newton step in solution of the secular equation.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
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*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAED6( KNITER, ORGATI, RHO, D, Z, FINIT, TAU, INFO )
*
*       .. Scalar Arguments ..
*       LOGICAL            ORGATI
*       INTEGER            INFO, KNITER
*       DOUBLE PRECISION   FINIT, RHO, TAU
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   D( 3 ), Z( 3 )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLAED6 computes the positive or negative root (closest to the origin)
*> of
*>                  z(1)        z(2)        z(3)
*> f(x) =   rho + --------- + ---------- + ---------
*>                 d(1)-x      d(2)-x      d(3)-x
*>
*> It is assumed that
*>
*>       if ORGATI = .true. the root is between d(2) and d(3);
*>       otherwise it is between d(1) and d(2)
*>
*> This routine will be called by DLAED4 when necessary. In most cases,
*> the root sought is the smallest in magnitude, though it might not be
*> in some extremely rare situations.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] KNITER
*> \verbatim
*>          KNITER is INTEGER
*>               Refer to DLAED4 for its significance.
*> \endverbatim
*>
*> \param[in] ORGATI
*> \verbatim
*>          ORGATI is LOGICAL
*>               If ORGATI is true, the needed root is between d(2) and
*>               d(3); otherwise it is between d(1) and d(2).  See
*>               DLAED4 for further details.
*> \endverbatim
*>
*> \param[in] RHO
*> \verbatim
*>          RHO is DOUBLE PRECISION
*>               Refer to the equation f(x) above.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (3)
*>               D satisfies d(1) < d(2) < d(3).
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (3)
*>               Each of the elements in z must be positive.
*> \endverbatim
*>
*> \param[in] FINIT
*> \verbatim
*>          FINIT is DOUBLE PRECISION
*>               The value of f at 0. It is more accurate than the one
*>               evaluated inside this routine (if someone wants to do
*>               so).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION
*>               The root of the equation f(x).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>               = 0: successful exit
*>               > 0: if INFO = 1, failure to converge
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laed6
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  10/02/03: This version has a few statements commented out for thread
*>  safety (machine parameters are computed on each entry). SJH.
*>
*>  05/10/06: Modified from a new version of Ren-Cang Li, use
*>     Gragg-Thornton-Warner cubic convergent scheme for better stability.
*> \endverbatim
*
*> \par Contributors:
*  ==================
*>
*>     Ren-Cang Li, Computer Science Division, University of California
*>     at Berkeley, USA
*>
*  =====================================================================
      SUBROUTINE DLAED6( KNITER, ORGATI, RHO, D, Z, FINIT, TAU,
     $                   INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      LOGICAL            ORGATI
      INTEGER            INFO, KNITER
      DOUBLE PRECISION   FINIT, RHO, TAU
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   D( 3 ), Z( 3 )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            MAXIT
      PARAMETER          ( MAXIT = 40 )
      DOUBLE PRECISION   ZERO, ONE, TWO, THREE, FOUR, EIGHT
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0,
     $                   THREE = 3.0D0, FOUR = 4.0D0, EIGHT = 8.0D0 )
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DLAMCH
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   DSCALE( 3 ), ZSCALE( 3 )
*     ..
*     .. Local Scalars ..
      LOGICAL            SCALE
      INTEGER            I, ITER, NITER
      DOUBLE PRECISION   A, B, BASE, C, DDF, DF, EPS, ERRETM, ETA, F,
     $                   FC, SCLFAC, SCLINV, SMALL1, SMALL2, SMINV1,
     $                   SMINV2, TEMP, TEMP1, TEMP2, TEMP3, TEMP4,
     $                   LBD, UBD
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, INT, LOG, MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
      INFO = 0
*
      IF( ORGATI ) THEN
         LBD = D(2)
         UBD = D(3)
      ELSE
         LBD = D(1)
         UBD = D(2)
      END IF
      IF( FINIT .LT. ZERO )THEN
         LBD = ZERO
      ELSE
         UBD = ZERO
      END IF
*
      NITER = 1
      TAU = ZERO
      IF( KNITER.EQ.2 ) THEN
         IF( ORGATI ) THEN
            TEMP = ( D( 3 )-D( 2 ) ) / TWO
            C = RHO + Z( 1 ) / ( ( D( 1 )-D( 2 ) )-TEMP )
            A = C*( D( 2 )+D( 3 ) ) + Z( 2 ) + Z( 3 )
            B = C*D( 2 )*D( 3 ) + Z( 2 )*D( 3 ) + Z( 3 )*D( 2 )
         ELSE
            TEMP = ( D( 1 )-D( 2 ) ) / TWO
            C = RHO + Z( 3 ) / ( ( D( 3 )-D( 2 ) )-TEMP )
            A = C*( D( 1 )+D( 2 ) ) + Z( 1 ) + Z( 2 )
            B = C*D( 1 )*D( 2 ) + Z( 1 )*D( 2 ) + Z( 2 )*D( 1 )
         END IF
         TEMP = MAX( ABS( A ), ABS( B ), ABS( C ) )
         A = A / TEMP
         B = B / TEMP
         C = C / TEMP
         IF( C.EQ.ZERO ) THEN
            TAU = B / A
         ELSE IF( A.LE.ZERO ) THEN
            TAU = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
         ELSE
            TAU = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) )
         END IF
         IF( TAU .LT. LBD .OR. TAU .GT. UBD )
     $      TAU = ( LBD+UBD )/TWO
         IF( D(1).EQ.TAU .OR. D(2).EQ.TAU .OR. D(3).EQ.TAU ) THEN
            TAU = ZERO
         ELSE
            TEMP = FINIT + TAU*Z(1)/( D(1)*( D( 1 )-TAU ) ) +
     $                     TAU*Z(2)/( D(2)*( D( 2 )-TAU ) ) +
     $                     TAU*Z(3)/( D(3)*( D( 3 )-TAU ) )
            IF( TEMP .LE. ZERO )THEN
               LBD = TAU
            ELSE
               UBD = TAU
            END IF
            IF( ABS( FINIT ).LE.ABS( TEMP ) )
     $         TAU = ZERO
         END IF
      END IF
*
*     get machine parameters for possible scaling to avoid overflow
*
*     modified by Sven: parameters SMALL1, SMINV1, SMALL2,
*     SMINV2, EPS are not SAVEd anymore between one call to the
*     others but recomputed at each call
*
      EPS = DLAMCH( 'Epsilon' )
      BASE = DLAMCH( 'Base' )
      SMALL1 = BASE**( INT( LOG( DLAMCH( 'SafMin' ) ) / LOG( BASE ) /
     $         THREE ) )
      SMINV1 = ONE / SMALL1
      SMALL2 = SMALL1*SMALL1
      SMINV2 = SMINV1*SMINV1
*
*     Determine if scaling of inputs necessary to avoid overflow
*     when computing 1/TEMP**3
*
      IF( ORGATI ) THEN
         TEMP = MIN( ABS( D( 2 )-TAU ), ABS( D( 3 )-TAU ) )
      ELSE
         TEMP = MIN( ABS( D( 1 )-TAU ), ABS( D( 2 )-TAU ) )
      END IF
      SCALE = .FALSE.
      IF( TEMP.LE.SMALL1 ) THEN
         SCALE = .TRUE.
         IF( TEMP.LE.SMALL2 ) THEN
*
*        Scale up by power of radix nearest 1/SAFMIN**(2/3)
*
            SCLFAC = SMINV2
            SCLINV = SMALL2
         ELSE
*
*        Scale up by power of radix nearest 1/SAFMIN**(1/3)
*
            SCLFAC = SMINV1
            SCLINV = SMALL1
         END IF
*
*        Scaling up safe because D, Z, TAU scaled elsewhere to be O(1)
*
         DO 10 I = 1, 3
            DSCALE( I ) = D( I )*SCLFAC
            ZSCALE( I ) = Z( I )*SCLFAC
   10    CONTINUE
         TAU = TAU*SCLFAC
         LBD = LBD*SCLFAC
         UBD = UBD*SCLFAC
      ELSE
*
*        Copy D and Z to DSCALE and ZSCALE
*
         DO 20 I = 1, 3
            DSCALE( I ) = D( I )
            ZSCALE( I ) = Z( I )
   20    CONTINUE
      END IF
*
      FC = ZERO
      DF = ZERO
      DDF = ZERO
      DO 30 I = 1, 3
         TEMP = ONE / ( DSCALE( I )-TAU )
         TEMP1 = ZSCALE( I )*TEMP
         TEMP2 = TEMP1*TEMP
         TEMP3 = TEMP2*TEMP
         FC = FC + TEMP1 / DSCALE( I )
         DF = DF + TEMP2
         DDF = DDF + TEMP3
   30 CONTINUE
      F = FINIT + TAU*FC
*
      IF( ABS( F ).LE.ZERO )
     $   GO TO 60
      IF( F .LE. ZERO )THEN
         LBD = TAU
      ELSE
         UBD = TAU
      END IF
*
*        Iteration begins -- Use Gragg-Thornton-Warner cubic convergent
*                            scheme
*
*     It is not hard to see that
*
*           1) Iterations will go up monotonically
*              if FINIT < 0;
*
*           2) Iterations will go down monotonically
*              if FINIT > 0.
*
      ITER = NITER + 1
*
      DO 50 NITER = ITER, MAXIT
*
         IF( ORGATI ) THEN
            TEMP1 = DSCALE( 2 ) - TAU
            TEMP2 = DSCALE( 3 ) - TAU
         ELSE
            TEMP1 = DSCALE( 1 ) - TAU
            TEMP2 = DSCALE( 2 ) - TAU
         END IF
         A = ( TEMP1+TEMP2 )*F - TEMP1*TEMP2*DF
         B = TEMP1*TEMP2*F
         C = F - ( TEMP1+TEMP2 )*DF + TEMP1*TEMP2*DDF
         TEMP = MAX( ABS( A ), ABS( B ), ABS( C ) )
         A = A / TEMP
         B = B / TEMP
         C = C / TEMP
         IF( C.EQ.ZERO ) THEN
            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
         IF( F*ETA.GE.ZERO ) THEN
            ETA = -F / DF
         END IF
*
         TAU = TAU + ETA
         IF( TAU .LT. LBD .OR. TAU .GT. UBD )
     $      TAU = ( LBD + UBD )/TWO
*
         FC = ZERO
         ERRETM = ZERO
         DF = ZERO
         DDF = ZERO
         DO 40 I = 1, 3
            IF ( ( DSCALE( I )-TAU ).NE.ZERO ) THEN
               TEMP = ONE / ( DSCALE( I )-TAU )
               TEMP1 = ZSCALE( I )*TEMP
               TEMP2 = TEMP1*TEMP
               TEMP3 = TEMP2*TEMP
               TEMP4 = TEMP1 / DSCALE( I )
               FC = FC + TEMP4
               ERRETM = ERRETM + ABS( TEMP4 )
               DF = DF + TEMP2
               DDF = DDF + TEMP3
            ELSE
               GO TO 60
            END IF
   40    CONTINUE
         F = FINIT + TAU*FC
         ERRETM = EIGHT*( ABS( FINIT )+ABS( TAU )*ERRETM ) +
     $            ABS( TAU )*DF
         IF( ( ABS( F ).LE.FOUR*EPS*ERRETM ) .OR.
     $      ( (UBD-LBD).LE.FOUR*EPS*ABS(TAU) )  )
     $      GO TO 60
         IF( F .LE. ZERO )THEN
            LBD = TAU
         ELSE
            UBD = TAU
         END IF
   50 CONTINUE
      INFO = 1
   60 CONTINUE
*
*     Undo scaling
*
      IF( SCALE )
     $   TAU = TAU*SCLINV
      RETURN
*
*     End of DLAED6
*
      END
