dlazq4.f

          SUBROUTINE DLAZQ4( I0, N0, Z, PP, N0IN, DMIN, DMIN1, DMIN2, DN,
         $                   DN1, DN2, TAU, TTYPE, G )
    *
    *  -- LAPACK auxiliary routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          INTEGER            I0, N0, N0IN, PP, TTYPE
          DOUBLE PRECISION   DMIN, DMIN1, DMIN2, DN, DN1, DN2, G, TAU
    *     ..
    *     .. Array Arguments ..
          DOUBLE PRECISION   Z( * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  DLAZQ4 computes an approximation TAU to the smallest eigenvalue
    *  using values of d from the previous transform.
    *
    *  I0    (input) INTEGER
    *        First index.
    *
    *  N0    (input) INTEGER
    *        Last index.
    *
    *  Z     (input) DOUBLE PRECISION array, dimension ( 4*N )
    *        Z holds the qd array.
    *
    *  PP    (input) INTEGER
    *        PP=0 for ping, PP=1 for pong.
    *
    *  N0IN  (input) INTEGER
    *        The value of N0 at start of EIGTEST.
    *
    *  DMIN  (input) DOUBLE PRECISION
    *        Minimum value of d.
    *
    *  DMIN1 (input) DOUBLE PRECISION
    *        Minimum value of d, excluding D( N0 ).
    *
    *  DMIN2 (input) DOUBLE PRECISION
    *        Minimum value of d, excluding D( N0 ) and D( N0-1 ).
    *
    *  DN    (input) DOUBLE PRECISION
    *        d(N)
    *
    *  DN1   (input) DOUBLE PRECISION
    *        d(N-1)
    *
    *  DN2   (input) DOUBLE PRECISION
    *        d(N-2)
    *
    *  TAU   (output) DOUBLE PRECISION
    *        This is the shift.
    *
    *  TTYPE (output) INTEGER
    *        Shift type.
    *
    *  G     (input/output) DOUBLE PRECISION
    *        G is passed as an argument in order to save its value between
    *        calls to DLAZQ4
    *
    *  Further Details
    *  ===============
    *  CNST1 = 9/16
    *
    *  This is a thread safe version of DLASQ4, which passes G through the
    *  argument list in place of declaring G in a SAVE statment.
    *
    *  =====================================================================
    *
    *     .. 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
                      B2 = Z( NP-2 )
                      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 DLAZQ4
    *
          END