*> \brief \b DLASQ4 computes an approximation to the smallest eigenvalue using values of d from the previous transform. Used by sbdsqr.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASQ4 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasq4.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasq4.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasq4.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLASQ4( I0, N0, Z, PP, N0IN, DMIN, DMIN1, DMIN2, DN,
*                          DN1, DN2, TAU, TTYPE, G )
*
*       .. Scalar Arguments ..
*       INTEGER            I0, N0, N0IN, PP, TTYPE
*       DOUBLE PRECISION   DMIN, DMIN1, DMIN2, DN, DN1, DN2, G, TAU
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   Z( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLASQ4 computes an approximation TAU to the smallest eigenvalue
*> using values of d from the previous transform.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] I0
*> \verbatim
*>          I0 is INTEGER
*>        First index.
*> \endverbatim
*>
*> \param[in] N0
*> \verbatim
*>          N0 is INTEGER
*>        Last index.
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension ( 4*N0 )
*>        Z holds the qd array.
*> \endverbatim
*>
*> \param[in] PP
*> \verbatim
*>          PP is INTEGER
*>        PP=0 for ping, PP=1 for pong.
*> \endverbatim
*>
*> \param[in] N0IN
*> \verbatim
*>          N0IN is INTEGER
*>        The value of N0 at start of EIGTEST.
*> \endverbatim
*>
*> \param[in] DMIN
*> \verbatim
*>          DMIN is DOUBLE PRECISION
*>        Minimum value of d.
*> \endverbatim
*>
*> \param[in] DMIN1
*> \verbatim
*>          DMIN1 is DOUBLE PRECISION
*>        Minimum value of d, excluding D( N0 ).
*> \endverbatim
*>
*> \param[in] DMIN2
*> \verbatim
*>          DMIN2 is DOUBLE PRECISION
*>        Minimum value of d, excluding D( N0 ) and D( N0-1 ).
*> \endverbatim
*>
*> \param[in] DN
*> \verbatim
*>          DN is DOUBLE PRECISION
*>        d(N)
*> \endverbatim
*>
*> \param[in] DN1
*> \verbatim
*>          DN1 is DOUBLE PRECISION
*>        d(N-1)
*> \endverbatim
*>
*> \param[in] DN2
*> \verbatim
*>          DN2 is DOUBLE PRECISION
*>        d(N-2)
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION
*>        This is the shift.
*> \endverbatim
*>
*> \param[out] TTYPE
*> \verbatim
*>          TTYPE is INTEGER
*>        Shift type.
*> \endverbatim
*>
*> \param[in,out] G
*> \verbatim
*>          G is DOUBLE PRECISION
*>        G is passed as an argument in order to save its value between
*>        calls to DLASQ4.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lasq4
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  CNST1 = 9/16
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DLASQ4( I0, N0, Z, PP, N0IN, DMIN, DMIN1, DMIN2, DN,
     $                   DN1, DN2, TAU, TTYPE, G )
*
*  -- 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 ..
      INTEGER            I0, N0, N0IN, PP, TTYPE
      DOUBLE PRECISION   DMIN, DMIN1, DMIN2, DN, DN1, DN2, G, TAU
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   Z( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   CNST1, CNST2, CNST3
      PARAMETER          ( CNST1 = 0.5630D0, CNST2 = 1.010D0,
     $                   CNST3 = 1.050D0 )
      DOUBLE PRECISION   QURTR, THIRD, HALF, ZERO, ONE, TWO, HUNDRD
      PARAMETER          ( QURTR = 0.250D0, THIRD = 0.3330D0,
     $                   HALF = 0.50D0, ZERO = 0.0D0, ONE = 1.0D0,
     $                   TWO = 2.0D0, HUNDRD = 100.0D0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I4, NN, NP
      DOUBLE PRECISION   A2, B1, B2, GAM, GAP1, GAP2, S
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
*     A negative DMIN forces the shift to take that absolute value
*     TTYPE records the type of shift.
*
      IF( DMIN.LE.ZERO ) THEN
         TAU = -DMIN
         TTYPE = -1
         RETURN
      END IF
*
      NN = 4*N0 + PP
      IF( N0IN.EQ.N0 ) THEN
*
*        No eigenvalues deflated.
*
         IF( DMIN.EQ.DN .OR. DMIN.EQ.DN1 ) THEN
*
            B1 = SQRT( Z( NN-3 ) )*SQRT( Z( NN-5 ) )
            B2 = SQRT( Z( NN-7 ) )*SQRT( Z( NN-9 ) )
            A2 = Z( NN-7 ) + Z( NN-5 )
*
*           Cases 2 and 3.
*
            IF( DMIN.EQ.DN .AND. DMIN1.EQ.DN1 ) THEN
               GAP2 = DMIN2 - A2 - DMIN2*QURTR
               IF( GAP2.GT.ZERO .AND. GAP2.GT.B2 ) THEN
                  GAP1 = A2 - DN - ( B2 / GAP2 )*B2
               ELSE
                  GAP1 = A2 - DN - ( B1+B2 )
               END IF
               IF( GAP1.GT.ZERO .AND. GAP1.GT.B1 ) THEN
                  S = MAX( DN-( B1 / GAP1 )*B1, HALF*DMIN )
                  TTYPE = -2
               ELSE
                  S = ZERO
                  IF( DN.GT.B1 )
     $               S = DN - B1
                  IF( A2.GT.( B1+B2 ) )
     $               S = MIN( S, A2-( B1+B2 ) )
                  S = MAX( S, THIRD*DMIN )
                  TTYPE = -3
               END IF
            ELSE
*
*              Case 4.
*
               TTYPE = -4
               S = QURTR*DMIN
               IF( DMIN.EQ.DN ) THEN
                  GAM = DN
                  A2 = ZERO
                  IF( Z( NN-5 ) .GT. Z( NN-7 ) )
     $               RETURN
                  B2 = Z( NN-5 ) / Z( NN-7 )
                  NP = NN - 9
               ELSE
                  NP = NN - 2*PP
                  GAM = DN1
                  IF( Z( NP-4 ) .GT. Z( NP-2 ) )
     $               RETURN
                  A2 = Z( NP-4 ) / Z( NP-2 )
                  IF( Z( NN-9 ) .GT. Z( NN-11 ) )
     $               RETURN
                  B2 = Z( NN-9 ) / Z( NN-11 )
                  NP = NN - 13
               END IF
*
*              Approximate contribution to norm squared from I < NN-1.
*
               A2 = A2 + B2
               DO 10 I4 = NP, 4*I0 - 1 + PP, -4
                  IF( B2.EQ.ZERO )
     $               GO TO 20
                  B1 = B2
                  IF( Z( I4 ) .GT. Z( I4-2 ) )
     $               RETURN
                  B2 = B2*( Z( I4 ) / Z( I4-2 ) )
                  A2 = A2 + B2
                  IF( HUNDRD*MAX( B2, B1 ).LT.A2 .OR. CNST1.LT.A2 )
     $               GO TO 20
   10          CONTINUE
   20          CONTINUE
               A2 = CNST3*A2
*
*              Rayleigh quotient residual bound.
*
               IF( A2.LT.CNST1 )
     $            S = GAM*( ONE-SQRT( A2 ) ) / ( ONE+A2 )
            END IF
         ELSE IF( DMIN.EQ.DN2 ) THEN
*
*           Case 5.
*
            TTYPE = -5
            S = QURTR*DMIN
*
*           Compute contribution to norm squared from I > NN-2.
*
            NP = NN - 2*PP
            B1 = Z( NP-2 )
            B2 = Z( NP-6 )
            GAM = DN2
            IF( Z( NP-8 ).GT.B2 .OR. Z( NP-4 ).GT.B1 )
     $         RETURN
            A2 = ( Z( NP-8 ) / B2 )*( ONE+Z( NP-4 ) / B1 )
*
*           Approximate contribution to norm squared from I < NN-2.
*
            IF( N0-I0.GT.2 ) THEN
               B2 = Z( NN-13 ) / Z( NN-15 )
               A2 = A2 + B2
               DO 30 I4 = NN - 17, 4*I0 - 1 + PP, -4
                  IF( B2.EQ.ZERO )
     $               GO TO 40
                  B1 = B2
                  IF( Z( I4 ) .GT. Z( I4-2 ) )
     $               RETURN
                  B2 = B2*( Z( I4 ) / Z( I4-2 ) )
                  A2 = A2 + B2
                  IF( HUNDRD*MAX( B2, B1 ).LT.A2 .OR. CNST1.LT.A2 )
     $               GO TO 40
   30          CONTINUE
   40          CONTINUE
               A2 = CNST3*A2
            END IF
*
            IF( A2.LT.CNST1 )
     $         S = GAM*( ONE-SQRT( A2 ) ) / ( ONE+A2 )
         ELSE
*
*           Case 6, no information to guide us.
*
            IF( TTYPE.EQ.-6 ) THEN
               G = G + THIRD*( ONE-G )
            ELSE IF( TTYPE.EQ.-18 ) THEN
               G = QURTR*THIRD
            ELSE
               G = QURTR
            END IF
            S = G*DMIN
            TTYPE = -6
         END IF
*
      ELSE IF( N0IN.EQ.( N0+1 ) ) THEN
*
*        One eigenvalue just deflated. Use DMIN1, DN1 for DMIN and DN.
*
         IF( DMIN1.EQ.DN1 .AND. DMIN2.EQ.DN2 ) THEN
*
*           Cases 7 and 8.
*
            TTYPE = -7
            S = THIRD*DMIN1
            IF( Z( NN-5 ).GT.Z( NN-7 ) )
     $         RETURN
            B1 = Z( NN-5 ) / Z( NN-7 )
            B2 = B1
            IF( B2.EQ.ZERO )
     $         GO TO 60
            DO 50 I4 = 4*N0 - 9 + PP, 4*I0 - 1 + PP, -4
               A2 = B1
               IF( Z( I4 ).GT.Z( I4-2 ) )
     $            RETURN
               B1 = B1*( Z( I4 ) / Z( I4-2 ) )
               B2 = B2 + B1
               IF( HUNDRD*MAX( B1, A2 ).LT.B2 )
     $            GO TO 60
   50       CONTINUE
   60       CONTINUE
            B2 = SQRT( CNST3*B2 )
            A2 = DMIN1 / ( ONE+B2**2 )
            GAP2 = HALF*DMIN2 - A2
            IF( GAP2.GT.ZERO .AND. GAP2.GT.B2*A2 ) THEN
               S = MAX( S, A2*( ONE-CNST2*A2*( B2 / GAP2 )*B2 ) )
            ELSE
               S = MAX( S, A2*( ONE-CNST2*B2 ) )
               TTYPE = -8
            END IF
         ELSE
*
*           Case 9.
*
            S = QURTR*DMIN1
            IF( DMIN1.EQ.DN1 )
     $         S = HALF*DMIN1
            TTYPE = -9
         END IF
*
      ELSE IF( N0IN.EQ.( N0+2 ) ) THEN
*
*        Two eigenvalues deflated. Use DMIN2, DN2 for DMIN and DN.
*
*        Cases 10 and 11.
*
         IF( DMIN2.EQ.DN2 .AND. TWO*Z( NN-5 ).LT.Z( NN-7 ) ) THEN
            TTYPE = -10
            S = THIRD*DMIN2
            IF( Z( NN-5 ).GT.Z( NN-7 ) )
     $         RETURN
            B1 = Z( NN-5 ) / Z( NN-7 )
            B2 = B1
            IF( B2.EQ.ZERO )
     $         GO TO 80
            DO 70 I4 = 4*N0 - 9 + PP, 4*I0 - 1 + PP, -4
               IF( Z( I4 ).GT.Z( I4-2 ) )
     $            RETURN
               B1 = B1*( Z( I4 ) / Z( I4-2 ) )
               B2 = B2 + B1
               IF( HUNDRD*B1.LT.B2 )
     $            GO TO 80
   70       CONTINUE
   80       CONTINUE
            B2 = SQRT( CNST3*B2 )
            A2 = DMIN2 / ( ONE+B2**2 )
            GAP2 = Z( NN-7 ) + Z( NN-9 ) -
     $             SQRT( Z( NN-11 ) )*SQRT( Z( NN-9 ) ) - A2
            IF( GAP2.GT.ZERO .AND. GAP2.GT.B2*A2 ) THEN
               S = MAX( S, A2*( ONE-CNST2*A2*( B2 / GAP2 )*B2 ) )
            ELSE
               S = MAX( S, A2*( ONE-CNST2*B2 ) )
            END IF
         ELSE
            S = QURTR*DMIN2
            TTYPE = -11
         END IF
      ELSE IF( N0IN.GT.( N0+2 ) ) THEN
*
*        Case 12, more than two eigenvalues deflated. No information.
*
         S = ZERO
         TTYPE = -12
      END IF
*
      TAU = S
      RETURN
*
*     End of DLASQ4
*
      END
