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*> \brief \b DLAED6 used by sstedc. Computes one Newton step in solution of the secular equation.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAED6 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed6.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed6.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed6.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAED6( KNITER, ORGATI, RHO, D, Z, FINIT, TAU, INFO )
*
* .. Scalar Arguments ..
* LOGICAL ORGATI
* INTEGER INFO, KNITER
* DOUBLE PRECISION FINIT, RHO, TAU
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( 3 ), Z( 3 )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAED6 computes the positive or negative root (closest to the origin)
*> of
*> z(1) z(2) z(3)
*> f(x) = rho + --------- + ---------- + ---------
*> d(1)-x d(2)-x d(3)-x
*>
*> It is assumed that
*>
*> if ORGATI = .true. the root is between d(2) and d(3);
*> otherwise it is between d(1) and d(2)
*>
*> This routine will be called by DLAED4 when necessary. In most cases,
*> the root sought is the smallest in magnitude, though it might not be
*> in some extremely rare situations.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] KNITER
*> \verbatim
*> KNITER is INTEGER
*> Refer to DLAED4 for its significance.
*> \endverbatim
*>
*> \param[in] ORGATI
*> \verbatim
*> ORGATI is LOGICAL
*> If ORGATI is true, the needed root is between d(2) and
*> d(3); otherwise it is between d(1) and d(2). See
*> DLAED4 for further details.
*> \endverbatim
*>
*> \param[in] RHO
*> \verbatim
*> RHO is DOUBLE PRECISION
*> Refer to the equation f(x) above.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (3)
*> D satisfies d(1) < d(2) < d(3).
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (3)
*> Each of the elements in z must be positive.
*> \endverbatim
*>
*> \param[in] FINIT
*> \verbatim
*> FINIT is DOUBLE PRECISION
*> The value of f at 0. It is more accurate than the one
*> evaluated inside this routine (if someone wants to do
*> so).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION
*> The root of the equation f(x).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> > 0: if INFO = 1, failure to converge
*> \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
*>
*> 10/02/03: This version has a few statements commented out for thread
*> safety (machine parameters are computed on each entry). SJH.
*>
*> 05/10/06: Modified from a new version of Ren-Cang Li, use
*> Gragg-Thornton-Warner cubic convergent scheme for better stability.
*> \endverbatim
*
*> \par Contributors:
* ==================
*>
*> Ren-Cang Li, Computer Science Division, University of California
*> at Berkeley, USA
*>
* =====================================================================
SUBROUTINE DLAED6( KNITER, ORGATI, RHO, D, Z, FINIT, TAU, INFO )
*
* -- 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 ..
LOGICAL ORGATI
INTEGER INFO, KNITER
DOUBLE PRECISION FINIT, RHO, TAU
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( 3 ), Z( 3 )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER MAXIT
PARAMETER ( MAXIT = 40 )
DOUBLE PRECISION ZERO, ONE, TWO, THREE, FOUR, EIGHT
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0,
$ THREE = 3.0D0, FOUR = 4.0D0, EIGHT = 8.0D0 )
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. Local Arrays ..
DOUBLE PRECISION DSCALE( 3 ), ZSCALE( 3 )
* ..
* .. Local Scalars ..
LOGICAL SCALE
INTEGER I, ITER, NITER
DOUBLE PRECISION A, B, BASE, C, DDF, DF, EPS, ERRETM, ETA, F,
$ FC, SCLFAC, SCLINV, SMALL1, SMALL2, SMINV1,
$ SMINV2, TEMP, TEMP1, TEMP2, TEMP3, TEMP4,
$ LBD, UBD
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, INT, LOG, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
INFO = 0
*
IF( ORGATI ) THEN
LBD = D(2)
UBD = D(3)
ELSE
LBD = D(1)
UBD = D(2)
END IF
IF( FINIT .LT. ZERO )THEN
LBD = ZERO
ELSE
UBD = ZERO
END IF
*
NITER = 1
TAU = ZERO
IF( KNITER.EQ.2 ) THEN
IF( ORGATI ) THEN
TEMP = ( D( 3 )-D( 2 ) ) / TWO
C = RHO + Z( 1 ) / ( ( D( 1 )-D( 2 ) )-TEMP )
A = C*( D( 2 )+D( 3 ) ) + Z( 2 ) + Z( 3 )
B = C*D( 2 )*D( 3 ) + Z( 2 )*D( 3 ) + Z( 3 )*D( 2 )
ELSE
TEMP = ( D( 1 )-D( 2 ) ) / TWO
C = RHO + Z( 3 ) / ( ( D( 3 )-D( 2 ) )-TEMP )
A = C*( D( 1 )+D( 2 ) ) + Z( 1 ) + Z( 2 )
B = C*D( 1 )*D( 2 ) + Z( 1 )*D( 2 ) + Z( 2 )*D( 1 )
END IF
TEMP = MAX( ABS( A ), ABS( B ), ABS( C ) )
A = A / TEMP
B = B / TEMP
C = C / TEMP
IF( C.EQ.ZERO ) THEN
TAU = B / A
ELSE IF( A.LE.ZERO ) THEN
TAU = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
ELSE
TAU = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) )
END IF
IF( TAU .LT. LBD .OR. TAU .GT. UBD )
$ TAU = ( LBD+UBD )/TWO
IF( D(1).EQ.TAU .OR. D(2).EQ.TAU .OR. D(3).EQ.TAU ) THEN
TAU = ZERO
ELSE
TEMP = FINIT + TAU*Z(1)/( D(1)*( D( 1 )-TAU ) ) +
$ TAU*Z(2)/( D(2)*( D( 2 )-TAU ) ) +
$ TAU*Z(3)/( D(3)*( D( 3 )-TAU ) )
IF( TEMP .LE. ZERO )THEN
LBD = TAU
ELSE
UBD = TAU
END IF
IF( ABS( FINIT ).LE.ABS( TEMP ) )
$ TAU = ZERO
END IF
END IF
*
* get machine parameters for possible scaling to avoid overflow
*
* modified by Sven: parameters SMALL1, SMINV1, SMALL2,
* SMINV2, EPS are not SAVEd anymore between one call to the
* others but recomputed at each call
*
EPS = DLAMCH( 'Epsilon' )
BASE = DLAMCH( 'Base' )
SMALL1 = BASE**( INT( LOG( DLAMCH( 'SafMin' ) ) / LOG( BASE ) /
$ THREE ) )
SMINV1 = ONE / SMALL1
SMALL2 = SMALL1*SMALL1
SMINV2 = SMINV1*SMINV1
*
* Determine if scaling of inputs necessary to avoid overflow
* when computing 1/TEMP**3
*
IF( ORGATI ) THEN
TEMP = MIN( ABS( D( 2 )-TAU ), ABS( D( 3 )-TAU ) )
ELSE
TEMP = MIN( ABS( D( 1 )-TAU ), ABS( D( 2 )-TAU ) )
END IF
SCALE = .FALSE.
IF( TEMP.LE.SMALL1 ) THEN
SCALE = .TRUE.
IF( TEMP.LE.SMALL2 ) THEN
*
* Scale up by power of radix nearest 1/SAFMIN**(2/3)
*
SCLFAC = SMINV2
SCLINV = SMALL2
ELSE
*
* Scale up by power of radix nearest 1/SAFMIN**(1/3)
*
SCLFAC = SMINV1
SCLINV = SMALL1
END IF
*
* Scaling up safe because D, Z, TAU scaled elsewhere to be O(1)
*
DO 10 I = 1, 3
DSCALE( I ) = D( I )*SCLFAC
ZSCALE( I ) = Z( I )*SCLFAC
10 CONTINUE
TAU = TAU*SCLFAC
LBD = LBD*SCLFAC
UBD = UBD*SCLFAC
ELSE
*
* Copy D and Z to DSCALE and ZSCALE
*
DO 20 I = 1, 3
DSCALE( I ) = D( I )
ZSCALE( I ) = Z( I )
20 CONTINUE
END IF
*
FC = ZERO
DF = ZERO
DDF = ZERO
DO 30 I = 1, 3
TEMP = ONE / ( DSCALE( I )-TAU )
TEMP1 = ZSCALE( I )*TEMP
TEMP2 = TEMP1*TEMP
TEMP3 = TEMP2*TEMP
FC = FC + TEMP1 / DSCALE( I )
DF = DF + TEMP2
DDF = DDF + TEMP3
30 CONTINUE
F = FINIT + TAU*FC
*
IF( ABS( F ).LE.ZERO )
$ GO TO 60
IF( F .LE. ZERO )THEN
LBD = TAU
ELSE
UBD = TAU
END IF
*
* Iteration begins -- Use Gragg-Thornton-Warner cubic convergent
* scheme
*
* It is not hard to see that
*
* 1) Iterations will go up monotonically
* if FINIT < 0;
*
* 2) Iterations will go down monotonically
* if FINIT > 0.
*
ITER = NITER + 1
*
DO 50 NITER = ITER, MAXIT
*
IF( ORGATI ) THEN
TEMP1 = DSCALE( 2 ) - TAU
TEMP2 = DSCALE( 3 ) - TAU
ELSE
TEMP1 = DSCALE( 1 ) - TAU
TEMP2 = DSCALE( 2 ) - TAU
END IF
A = ( TEMP1+TEMP2 )*F - TEMP1*TEMP2*DF
B = TEMP1*TEMP2*F
C = F - ( TEMP1+TEMP2 )*DF + TEMP1*TEMP2*DDF
TEMP = MAX( ABS( A ), ABS( B ), ABS( C ) )
A = A / TEMP
B = B / TEMP
C = C / TEMP
IF( C.EQ.ZERO ) THEN
ETA = B / A
ELSE IF( A.LE.ZERO ) THEN
ETA = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
ELSE
ETA = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) )
END IF
IF( F*ETA.GE.ZERO ) THEN
ETA = -F / DF
END IF
*
TAU = TAU + ETA
IF( TAU .LT. LBD .OR. TAU .GT. UBD )
$ TAU = ( LBD + UBD )/TWO
*
FC = ZERO
ERRETM = ZERO
DF = ZERO
DDF = ZERO
DO 40 I = 1, 3
IF ( ( DSCALE( I )-TAU ).NE.ZERO ) THEN
TEMP = ONE / ( DSCALE( I )-TAU )
TEMP1 = ZSCALE( I )*TEMP
TEMP2 = TEMP1*TEMP
TEMP3 = TEMP2*TEMP
TEMP4 = TEMP1 / DSCALE( I )
FC = FC + TEMP4
ERRETM = ERRETM + ABS( TEMP4 )
DF = DF + TEMP2
DDF = DDF + TEMP3
ELSE
GO TO 60
END IF
40 CONTINUE
F = FINIT + TAU*FC
ERRETM = EIGHT*( ABS( FINIT )+ABS( TAU )*ERRETM ) +
$ ABS( TAU )*DF
IF( ABS( F ).LE.EPS*ERRETM )
$ GO TO 60
IF( F .LE. ZERO )THEN
LBD = TAU
ELSE
UBD = TAU
END IF
50 CONTINUE
INFO = 1
60 CONTINUE
*
* Undo scaling
*
IF( SCALE )
$ TAU = TAU*SCLINV
RETURN
*
* End of DLAED6
*
END
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