+++ /dev/null
-*> \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*N )
-*> 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 REAL
-*> 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.
-*
-*> \date September 2012
-*
-*> \ingroup auxOTHERcomputational
-*
-*> \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 (version 3.4.2) --
-* -- LAPACK is a software package provided by Univ. of Tennessee, --
-* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
-* September 2012
-*
-* .. 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
- 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 DLASQ4
-*
- END
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