426 lines
11 KiB
Fortran
426 lines
11 KiB
Fortran
*> \brief \b SLASQ4 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 SLASQ4 + dependencies
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/slasq4.f">
|
|
*> [TGZ]</a>
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/slasq4.f">
|
|
*> [ZIP]</a>
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slasq4.f">
|
|
*> [TXT]</a>
|
|
*> \endhtmlonly
|
|
*
|
|
* Definition:
|
|
* ===========
|
|
*
|
|
* SUBROUTINE SLASQ4( I0, N0, Z, PP, N0IN, DMIN, DMIN1, DMIN2, DN,
|
|
* DN1, DN2, TAU, TTYPE, G )
|
|
*
|
|
* .. Scalar Arguments ..
|
|
* INTEGER I0, N0, N0IN, PP, TTYPE
|
|
* REAL DMIN, DMIN1, DMIN2, DN, DN1, DN2, G, TAU
|
|
* ..
|
|
* .. Array Arguments ..
|
|
* REAL Z( * )
|
|
* ..
|
|
*
|
|
*
|
|
*> \par Purpose:
|
|
* =============
|
|
*>
|
|
*> \verbatim
|
|
*>
|
|
*> SLASQ4 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 REAL 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 REAL
|
|
*> Minimum value of d.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] DMIN1
|
|
*> \verbatim
|
|
*> DMIN1 is REAL
|
|
*> Minimum value of d, excluding D( N0 ).
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] DMIN2
|
|
*> \verbatim
|
|
*> DMIN2 is REAL
|
|
*> Minimum value of d, excluding D( N0 ) and D( N0-1 ).
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] DN
|
|
*> \verbatim
|
|
*> DN is REAL
|
|
*> d(N)
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] DN1
|
|
*> \verbatim
|
|
*> DN1 is REAL
|
|
*> d(N-1)
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] DN2
|
|
*> \verbatim
|
|
*> DN2 is REAL
|
|
*> d(N-2)
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] TAU
|
|
*> \verbatim
|
|
*> TAU is REAL
|
|
*> 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 SLASQ4.
|
|
*> \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 SLASQ4( 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
|
|
REAL DMIN, DMIN1, DMIN2, DN, DN1, DN2, G, TAU
|
|
* ..
|
|
* .. Array Arguments ..
|
|
REAL Z( * )
|
|
* ..
|
|
*
|
|
* =====================================================================
|
|
*
|
|
* .. Parameters ..
|
|
REAL CNST1, CNST2, CNST3
|
|
PARAMETER ( CNST1 = 0.5630E0, CNST2 = 1.010E0,
|
|
$ CNST3 = 1.050E0 )
|
|
REAL QURTR, THIRD, HALF, ZERO, ONE, TWO, HUNDRD
|
|
PARAMETER ( QURTR = 0.250E0, THIRD = 0.3330E0,
|
|
$ HALF = 0.50E0, ZERO = 0.0E0, ONE = 1.0E0,
|
|
$ TWO = 2.0E0, HUNDRD = 100.0E0 )
|
|
* ..
|
|
* .. Local Scalars ..
|
|
INTEGER I4, NN, NP
|
|
REAL 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 SLASQ4
|
|
*
|
|
END
|