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dlasq4.f
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*> \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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