forked from lijiext/lammps
425 lines
11 KiB
Fortran
425 lines
11 KiB
Fortran
*> \brief \b DLASQ4 computes an approximation to the smallest eigenvalue using values of d from the previous transform. Used by sbdsqr.
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*
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* =========== DOCUMENTATION ===========
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*
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* Online html documentation available at
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* http://www.netlib.org/lapack/explore-html/
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*
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*> \htmlonly
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*> Download DLASQ4 + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasq4.f">
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*> [TGZ]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasq4.f">
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*> [ZIP]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasq4.f">
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*> [TXT]</a>
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*> \endhtmlonly
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*
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* Definition:
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* ===========
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*
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* SUBROUTINE DLASQ4( I0, N0, Z, PP, N0IN, DMIN, DMIN1, DMIN2, DN,
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* DN1, DN2, TAU, TTYPE, G )
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*
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* .. Scalar Arguments ..
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* INTEGER I0, N0, N0IN, PP, TTYPE
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* DOUBLE PRECISION DMIN, DMIN1, DMIN2, DN, DN1, DN2, G, TAU
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* ..
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* .. Array Arguments ..
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* DOUBLE PRECISION Z( * )
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* ..
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*
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*
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*> \par Purpose:
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* =============
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*>
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*> \verbatim
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*>
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*> DLASQ4 computes an approximation TAU to the smallest eigenvalue
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*> using values of d from the previous transform.
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*> \endverbatim
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*
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* Arguments:
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* ==========
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*
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*> \param[in] I0
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*> \verbatim
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*> I0 is INTEGER
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*> First index.
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*> \endverbatim
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*>
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*> \param[in] N0
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*> \verbatim
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*> N0 is INTEGER
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*> Last index.
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*> \endverbatim
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*>
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*> \param[in] Z
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*> \verbatim
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*> Z is DOUBLE PRECISION array, dimension ( 4*N0 )
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*> Z holds the qd array.
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*> \endverbatim
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*>
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*> \param[in] PP
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*> \verbatim
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*> PP is INTEGER
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*> PP=0 for ping, PP=1 for pong.
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*> \endverbatim
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*>
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*> \param[in] N0IN
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*> \verbatim
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*> N0IN is INTEGER
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*> The value of N0 at start of EIGTEST.
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*> \endverbatim
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*>
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*> \param[in] DMIN
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*> \verbatim
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*> DMIN is DOUBLE PRECISION
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*> Minimum value of d.
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*> \endverbatim
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*>
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*> \param[in] DMIN1
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*> \verbatim
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*> DMIN1 is DOUBLE PRECISION
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*> Minimum value of d, excluding D( N0 ).
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*> \endverbatim
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*>
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*> \param[in] DMIN2
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*> \verbatim
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*> DMIN2 is DOUBLE PRECISION
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*> Minimum value of d, excluding D( N0 ) and D( N0-1 ).
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*> \endverbatim
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*>
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*> \param[in] DN
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*> \verbatim
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*> DN is DOUBLE PRECISION
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*> d(N)
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*> \endverbatim
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*>
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*> \param[in] DN1
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*> \verbatim
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*> DN1 is DOUBLE PRECISION
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*> d(N-1)
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*> \endverbatim
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*>
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*> \param[in] DN2
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*> \verbatim
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*> DN2 is DOUBLE PRECISION
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*> d(N-2)
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*> \endverbatim
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*>
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*> \param[out] TAU
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*> \verbatim
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*> TAU is DOUBLE PRECISION
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*> This is the shift.
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*> \endverbatim
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*>
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*> \param[out] TTYPE
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*> \verbatim
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*> TTYPE is INTEGER
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*> Shift type.
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*> \endverbatim
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*>
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*> \param[in,out] G
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*> \verbatim
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*> G is DOUBLE PRECISION
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*> G is passed as an argument in order to save its value between
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*> calls to DLASQ4.
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*> \endverbatim
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*
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* Authors:
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* ========
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*
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*> \author Univ. of Tennessee
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*> \author Univ. of California Berkeley
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*> \author Univ. of Colorado Denver
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*> \author NAG Ltd.
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*
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*> \date June 2016
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*
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*> \ingroup auxOTHERcomputational
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*
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*> \par Further Details:
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* =====================
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*>
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*> \verbatim
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*>
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*> CNST1 = 9/16
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*> \endverbatim
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*>
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* =====================================================================
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SUBROUTINE DLASQ4( I0, N0, Z, PP, N0IN, DMIN, DMIN1, DMIN2, DN,
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$ DN1, DN2, TAU, TTYPE, G )
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*
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* -- LAPACK computational routine (version 3.7.1) --
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* -- LAPACK is a software package provided by Univ. of Tennessee, --
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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* June 2016
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*
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* .. Scalar Arguments ..
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INTEGER I0, N0, N0IN, PP, TTYPE
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DOUBLE PRECISION DMIN, DMIN1, DMIN2, DN, DN1, DN2, G, TAU
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* ..
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* .. Array Arguments ..
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DOUBLE PRECISION Z( * )
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* ..
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*
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* =====================================================================
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*
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* .. Parameters ..
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DOUBLE PRECISION CNST1, CNST2, CNST3
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PARAMETER ( CNST1 = 0.5630D0, CNST2 = 1.010D0,
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$ CNST3 = 1.050D0 )
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DOUBLE PRECISION QURTR, THIRD, HALF, ZERO, ONE, TWO, HUNDRD
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PARAMETER ( QURTR = 0.250D0, THIRD = 0.3330D0,
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$ HALF = 0.50D0, ZERO = 0.0D0, ONE = 1.0D0,
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$ TWO = 2.0D0, HUNDRD = 100.0D0 )
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* ..
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* .. Local Scalars ..
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INTEGER I4, NN, NP
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DOUBLE PRECISION A2, B1, B2, GAM, GAP1, GAP2, S
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC MAX, MIN, SQRT
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* ..
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* .. Executable Statements ..
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*
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* A negative DMIN forces the shift to take that absolute value
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* TTYPE records the type of shift.
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*
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IF( DMIN.LE.ZERO ) THEN
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TAU = -DMIN
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TTYPE = -1
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RETURN
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END IF
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*
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NN = 4*N0 + PP
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IF( N0IN.EQ.N0 ) THEN
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*
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* No eigenvalues deflated.
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*
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IF( DMIN.EQ.DN .OR. DMIN.EQ.DN1 ) THEN
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*
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B1 = SQRT( Z( NN-3 ) )*SQRT( Z( NN-5 ) )
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B2 = SQRT( Z( NN-7 ) )*SQRT( Z( NN-9 ) )
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A2 = Z( NN-7 ) + Z( NN-5 )
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*
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* Cases 2 and 3.
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*
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IF( DMIN.EQ.DN .AND. DMIN1.EQ.DN1 ) THEN
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GAP2 = DMIN2 - A2 - DMIN2*QURTR
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IF( GAP2.GT.ZERO .AND. GAP2.GT.B2 ) THEN
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GAP1 = A2 - DN - ( B2 / GAP2 )*B2
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ELSE
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GAP1 = A2 - DN - ( B1+B2 )
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END IF
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IF( GAP1.GT.ZERO .AND. GAP1.GT.B1 ) THEN
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S = MAX( DN-( B1 / GAP1 )*B1, HALF*DMIN )
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TTYPE = -2
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ELSE
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S = ZERO
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IF( DN.GT.B1 )
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$ S = DN - B1
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IF( A2.GT.( B1+B2 ) )
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$ S = MIN( S, A2-( B1+B2 ) )
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S = MAX( S, THIRD*DMIN )
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TTYPE = -3
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END IF
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ELSE
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*
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* Case 4.
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*
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TTYPE = -4
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S = QURTR*DMIN
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IF( DMIN.EQ.DN ) THEN
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GAM = DN
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A2 = ZERO
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IF( Z( NN-5 ) .GT. Z( NN-7 ) )
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$ RETURN
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B2 = Z( NN-5 ) / Z( NN-7 )
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NP = NN - 9
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ELSE
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NP = NN - 2*PP
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GAM = DN1
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IF( Z( NP-4 ) .GT. Z( NP-2 ) )
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$ RETURN
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A2 = Z( NP-4 ) / Z( NP-2 )
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IF( Z( NN-9 ) .GT. Z( NN-11 ) )
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$ RETURN
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B2 = Z( NN-9 ) / Z( NN-11 )
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NP = NN - 13
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END IF
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*
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* Approximate contribution to norm squared from I < NN-1.
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*
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A2 = A2 + B2
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DO 10 I4 = NP, 4*I0 - 1 + PP, -4
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IF( B2.EQ.ZERO )
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$ GO TO 20
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B1 = B2
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IF( Z( I4 ) .GT. Z( I4-2 ) )
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$ RETURN
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B2 = B2*( Z( I4 ) / Z( I4-2 ) )
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A2 = A2 + B2
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IF( HUNDRD*MAX( B2, B1 ).LT.A2 .OR. CNST1.LT.A2 )
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$ GO TO 20
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10 CONTINUE
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20 CONTINUE
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A2 = CNST3*A2
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*
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* Rayleigh quotient residual bound.
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*
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IF( A2.LT.CNST1 )
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$ S = GAM*( ONE-SQRT( A2 ) ) / ( ONE+A2 )
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END IF
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ELSE IF( DMIN.EQ.DN2 ) THEN
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*
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* Case 5.
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*
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TTYPE = -5
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S = QURTR*DMIN
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*
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* Compute contribution to norm squared from I > NN-2.
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*
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NP = NN - 2*PP
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B1 = Z( NP-2 )
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B2 = Z( NP-6 )
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GAM = DN2
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IF( Z( NP-8 ).GT.B2 .OR. Z( NP-4 ).GT.B1 )
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$ RETURN
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A2 = ( Z( NP-8 ) / B2 )*( ONE+Z( NP-4 ) / B1 )
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*
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* Approximate contribution to norm squared from I < NN-2.
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*
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IF( N0-I0.GT.2 ) THEN
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B2 = Z( NN-13 ) / Z( NN-15 )
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A2 = A2 + B2
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DO 30 I4 = NN - 17, 4*I0 - 1 + PP, -4
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IF( B2.EQ.ZERO )
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$ GO TO 40
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B1 = B2
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IF( Z( I4 ) .GT. Z( I4-2 ) )
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$ RETURN
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B2 = B2*( Z( I4 ) / Z( I4-2 ) )
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A2 = A2 + B2
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IF( HUNDRD*MAX( B2, B1 ).LT.A2 .OR. CNST1.LT.A2 )
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$ GO TO 40
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30 CONTINUE
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40 CONTINUE
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A2 = CNST3*A2
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END IF
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*
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IF( A2.LT.CNST1 )
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$ S = GAM*( ONE-SQRT( A2 ) ) / ( ONE+A2 )
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ELSE
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*
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* Case 6, no information to guide us.
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*
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IF( TTYPE.EQ.-6 ) THEN
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G = G + THIRD*( ONE-G )
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ELSE IF( TTYPE.EQ.-18 ) THEN
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G = QURTR*THIRD
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ELSE
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G = QURTR
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END IF
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S = G*DMIN
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TTYPE = -6
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END IF
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*
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ELSE IF( N0IN.EQ.( N0+1 ) ) THEN
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*
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* One eigenvalue just deflated. Use DMIN1, DN1 for DMIN and DN.
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*
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IF( DMIN1.EQ.DN1 .AND. DMIN2.EQ.DN2 ) THEN
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*
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* Cases 7 and 8.
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*
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TTYPE = -7
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S = THIRD*DMIN1
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IF( Z( NN-5 ).GT.Z( NN-7 ) )
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$ RETURN
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B1 = Z( NN-5 ) / Z( NN-7 )
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B2 = B1
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IF( B2.EQ.ZERO )
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$ GO TO 60
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DO 50 I4 = 4*N0 - 9 + PP, 4*I0 - 1 + PP, -4
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A2 = B1
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IF( Z( I4 ).GT.Z( I4-2 ) )
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$ RETURN
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B1 = B1*( Z( I4 ) / Z( I4-2 ) )
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B2 = B2 + B1
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IF( HUNDRD*MAX( B1, A2 ).LT.B2 )
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$ GO TO 60
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50 CONTINUE
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60 CONTINUE
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B2 = SQRT( CNST3*B2 )
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A2 = DMIN1 / ( ONE+B2**2 )
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GAP2 = HALF*DMIN2 - A2
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IF( GAP2.GT.ZERO .AND. GAP2.GT.B2*A2 ) THEN
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S = MAX( S, A2*( ONE-CNST2*A2*( B2 / GAP2 )*B2 ) )
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ELSE
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S = MAX( S, A2*( ONE-CNST2*B2 ) )
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TTYPE = -8
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END IF
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ELSE
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*
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* Case 9.
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*
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S = QURTR*DMIN1
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IF( DMIN1.EQ.DN1 )
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$ S = HALF*DMIN1
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TTYPE = -9
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END IF
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*
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ELSE IF( N0IN.EQ.( N0+2 ) ) THEN
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*
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* Two eigenvalues deflated. Use DMIN2, DN2 for DMIN and DN.
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*
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* Cases 10 and 11.
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*
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IF( DMIN2.EQ.DN2 .AND. TWO*Z( NN-5 ).LT.Z( NN-7 ) ) THEN
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TTYPE = -10
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S = THIRD*DMIN2
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IF( Z( NN-5 ).GT.Z( NN-7 ) )
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$ RETURN
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B1 = Z( NN-5 ) / Z( NN-7 )
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B2 = B1
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IF( B2.EQ.ZERO )
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$ GO TO 80
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DO 70 I4 = 4*N0 - 9 + PP, 4*I0 - 1 + PP, -4
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IF( Z( I4 ).GT.Z( I4-2 ) )
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$ RETURN
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B1 = B1*( Z( I4 ) / Z( I4-2 ) )
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B2 = B2 + B1
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IF( HUNDRD*B1.LT.B2 )
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$ GO TO 80
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70 CONTINUE
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80 CONTINUE
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B2 = SQRT( CNST3*B2 )
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A2 = DMIN2 / ( ONE+B2**2 )
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GAP2 = Z( NN-7 ) + Z( NN-9 ) -
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$ SQRT( Z( NN-11 ) )*SQRT( Z( NN-9 ) ) - A2
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IF( GAP2.GT.ZERO .AND. GAP2.GT.B2*A2 ) THEN
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S = MAX( S, A2*( ONE-CNST2*A2*( B2 / GAP2 )*B2 ) )
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ELSE
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S = MAX( S, A2*( ONE-CNST2*B2 ) )
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END IF
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ELSE
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S = QURTR*DMIN2
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TTYPE = -11
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END IF
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ELSE IF( N0IN.GT.( N0+2 ) ) THEN
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*
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* Case 12, more than two eigenvalues deflated. No information.
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*
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S = ZERO
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TTYPE = -12
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END IF
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*
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TAU = S
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RETURN
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*
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* End of DLASQ4
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*
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END
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