*> \brief \b DLARRF finds a new relatively robust representation such that at least one of the eigenvalues is relatively isolated.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARRF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarrf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarrf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarrf.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLARRF( N, D, L, LD, CLSTRT, CLEND,
* W, WGAP, WERR,
* SPDIAM, CLGAPL, CLGAPR, PIVMIN, SIGMA,
* DPLUS, LPLUS, WORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER CLSTRT, CLEND, INFO, N
* DOUBLE PRECISION CLGAPL, CLGAPR, PIVMIN, SIGMA, SPDIAM
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * ), DPLUS( * ), L( * ), LD( * ),
* $ LPLUS( * ), W( * ), WGAP( * ), WERR( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> Given the initial representation L D L^T and its cluster of close
*> eigenvalues (in a relative measure), W( CLSTRT ), W( CLSTRT+1 ), ...
*> W( CLEND ), DLARRF finds a new relatively robust representation
*> L D L^T - SIGMA I = L(+) D(+) L(+)^T such that at least one of the
*> eigenvalues of L(+) D(+) L(+)^T is relatively isolated.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix (subblock, if the matrix split).
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The N diagonal elements of the diagonal matrix D.
*> \endverbatim
*>
*> \param[in] L
*> \verbatim
*> L is DOUBLE PRECISION array, dimension (N-1)
*> The (N-1) subdiagonal elements of the unit bidiagonal
*> matrix L.
*> \endverbatim
*>
*> \param[in] LD
*> \verbatim
*> LD is DOUBLE PRECISION array, dimension (N-1)
*> The (N-1) elements L(i)*D(i).
*> \endverbatim
*>
*> \param[in] CLSTRT
*> \verbatim
*> CLSTRT is INTEGER
*> The index of the first eigenvalue in the cluster.
*> \endverbatim
*>
*> \param[in] CLEND
*> \verbatim
*> CLEND is INTEGER
*> The index of the last eigenvalue in the cluster.
*> \endverbatim
*>
*> \param[in] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension
*> dimension is >= (CLEND-CLSTRT+1)
*> The eigenvalue APPROXIMATIONS of L D L^T in ascending order.
*> W( CLSTRT ) through W( CLEND ) form the cluster of relatively
*> close eigenalues.
*> \endverbatim
*>
*> \param[in,out] WGAP
*> \verbatim
*> WGAP is DOUBLE PRECISION array, dimension
*> dimension is >= (CLEND-CLSTRT+1)
*> The separation from the right neighbor eigenvalue in W.
*> \endverbatim
*>
*> \param[in] WERR
*> \verbatim
*> WERR is DOUBLE PRECISION array, dimension
*> dimension is >= (CLEND-CLSTRT+1)
*> WERR contain the semiwidth of the uncertainty
*> interval of the corresponding eigenvalue APPROXIMATION in W
*> \endverbatim
*>
*> \param[in] SPDIAM
*> \verbatim
*> SPDIAM is DOUBLE PRECISION
*> estimate of the spectral diameter obtained from the
*> Gerschgorin intervals
*> \endverbatim
*>
*> \param[in] CLGAPL
*> \verbatim
*> CLGAPL is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in] CLGAPR
*> \verbatim
*> CLGAPR is DOUBLE PRECISION
*> absolute gap on each end of the cluster.
*> Set by the calling routine to protect against shifts too close
*> to eigenvalues outside the cluster.
*> \endverbatim
*>
*> \param[in] PIVMIN
*> \verbatim
*> PIVMIN is DOUBLE PRECISION
*> The minimum pivot allowed in the Sturm sequence.
*> \endverbatim
*>
*> \param[out] SIGMA
*> \verbatim
*> SIGMA is DOUBLE PRECISION
*> The shift used to form L(+) D(+) L(+)^T.
*> \endverbatim
*>
*> \param[out] DPLUS
*> \verbatim
*> DPLUS is DOUBLE PRECISION array, dimension (N)
*> The N diagonal elements of the diagonal matrix D(+).
*> \endverbatim
*>
*> \param[out] LPLUS
*> \verbatim
*> LPLUS is DOUBLE PRECISION array, dimension (N-1)
*> The first (N-1) elements of LPLUS contain the subdiagonal
*> elements of the unit bidiagonal matrix L(+).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (2*N)
*> Workspace.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> Signals processing OK (=0) or failure (=1)
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date June 2016
*
*> \ingroup OTHERauxiliary
*
*> \par Contributors:
* ==================
*>
*> Beresford Parlett, University of California, Berkeley, USA \n
*> Jim Demmel, University of California, Berkeley, USA \n
*> Inderjit Dhillon, University of Texas, Austin, USA \n
*> Osni Marques, LBNL/NERSC, USA \n
*> Christof Voemel, University of California, Berkeley, USA
*
* =====================================================================
SUBROUTINE DLARRF( N, D, L, LD, CLSTRT, CLEND,
$ W, WGAP, WERR,
$ SPDIAM, CLGAPL, CLGAPR, PIVMIN, SIGMA,
$ DPLUS, LPLUS, WORK, INFO )
*
* -- LAPACK auxiliary routine (version 3.7.1) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* June 2016
*
* .. Scalar Arguments ..
INTEGER CLSTRT, CLEND, INFO, N
DOUBLE PRECISION CLGAPL, CLGAPR, PIVMIN, SIGMA, SPDIAM
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), DPLUS( * ), L( * ), LD( * ),
$ LPLUS( * ), W( * ), WGAP( * ), WERR( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION FOUR, MAXGROWTH1, MAXGROWTH2, ONE, QUART, TWO
PARAMETER ( ONE = 1.0D0, TWO = 2.0D0, FOUR = 4.0D0,
$ QUART = 0.25D0,
$ MAXGROWTH1 = 8.D0,
$ MAXGROWTH2 = 8.D0 )
* ..
* .. Local Scalars ..
LOGICAL DORRR1, FORCER, NOFAIL, SAWNAN1, SAWNAN2, TRYRRR1
INTEGER I, INDX, KTRY, KTRYMAX, SLEFT, SRIGHT, SHIFT
PARAMETER ( KTRYMAX = 1, SLEFT = 1, SRIGHT = 2 )
DOUBLE PRECISION AVGAP, BESTSHIFT, CLWDTH, EPS, FACT, FAIL,
$ FAIL2, GROWTHBOUND, LDELTA, LDMAX, LSIGMA,
$ MAX1, MAX2, MINGAP, OLDP, PROD, RDELTA, RDMAX,
$ RRR1, RRR2, RSIGMA, S, SMLGROWTH, TMP, ZNM2
* ..
* .. External Functions ..
LOGICAL DISNAN
DOUBLE PRECISION DLAMCH
EXTERNAL DISNAN, DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DCOPY
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS
* ..
* .. Executable Statements ..
*
INFO = 0
*
* Quick return if possible
*
IF( N.LE.0 ) THEN
RETURN
END IF
*
FACT = DBLE(2**KTRYMAX)
EPS = DLAMCH( 'Precision' )
SHIFT = 0
FORCER = .FALSE.
* Note that we cannot guarantee that for any of the shifts tried,
* the factorization has a small or even moderate element growth.
* There could be Ritz values at both ends of the cluster and despite
* backing off, there are examples where all factorizations tried
* (in IEEE mode, allowing zero pivots & infinities) have INFINITE
* element growth.
* For this reason, we should use PIVMIN in this subroutine so that at
* least the L D L^T factorization exists. It can be checked afterwards
* whether the element growth caused bad residuals/orthogonality.
* Decide whether the code should accept the best among all
* representations despite large element growth or signal INFO=1
* Setting NOFAIL to .FALSE. for quick fix for bug 113
NOFAIL = .FALSE.
*
* Compute the average gap length of the cluster
CLWDTH = ABS(W(CLEND)-W(CLSTRT)) + WERR(CLEND) + WERR(CLSTRT)
AVGAP = CLWDTH / DBLE(CLEND-CLSTRT)
MINGAP = MIN(CLGAPL, CLGAPR)
* Initial values for shifts to both ends of cluster
LSIGMA = MIN(W( CLSTRT ),W( CLEND )) - WERR( CLSTRT )
RSIGMA = MAX(W( CLSTRT ),W( CLEND )) + WERR( CLEND )
* Use a small fudge to make sure that we really shift to the outside
LSIGMA = LSIGMA - ABS(LSIGMA)* FOUR * EPS
RSIGMA = RSIGMA + ABS(RSIGMA)* FOUR * EPS
* Compute upper bounds for how much to back off the initial shifts
LDMAX = QUART * MINGAP + TWO * PIVMIN
RDMAX = QUART * MINGAP + TWO * PIVMIN
LDELTA = MAX(AVGAP,WGAP( CLSTRT ))/FACT
RDELTA = MAX(AVGAP,WGAP( CLEND-1 ))/FACT
*
* Initialize the record of the best representation found
*
S = DLAMCH( 'S' )
SMLGROWTH = ONE / S
FAIL = DBLE(N-1)*MINGAP/(SPDIAM*EPS)
FAIL2 = DBLE(N-1)*MINGAP/(SPDIAM*SQRT(EPS))
BESTSHIFT = LSIGMA
*
* while (KTRY <= KTRYMAX)
KTRY = 0
GROWTHBOUND = MAXGROWTH1*SPDIAM
5 CONTINUE
SAWNAN1 = .FALSE.
SAWNAN2 = .FALSE.
* Ensure that we do not back off too much of the initial shifts
LDELTA = MIN(LDMAX,LDELTA)
RDELTA = MIN(RDMAX,RDELTA)
* Compute the element growth when shifting to both ends of the cluster
* accept the shift if there is no element growth at one of the two ends
* Left end
S = -LSIGMA
DPLUS( 1 ) = D( 1 ) + S
IF(ABS(DPLUS(1)).LT.PIVMIN) THEN
DPLUS(1) = -PIVMIN
* Need to set SAWNAN1 because refined RRR test should not be used
* in this case
SAWNAN1 = .TRUE.
ENDIF
MAX1 = ABS( DPLUS( 1 ) )
DO 6 I = 1, N - 1
LPLUS( I ) = LD( I ) / DPLUS( I )
S = S*LPLUS( I )*L( I ) - LSIGMA
DPLUS( I+1 ) = D( I+1 ) + S
IF(ABS(DPLUS(I+1)).LT.PIVMIN) THEN
DPLUS(I+1) = -PIVMIN
* Need to set SAWNAN1 because refined RRR test should not be used
* in this case
SAWNAN1 = .TRUE.
ENDIF
MAX1 = MAX( MAX1,ABS(DPLUS(I+1)) )
6 CONTINUE
SAWNAN1 = SAWNAN1 .OR. DISNAN( MAX1 )
IF( FORCER .OR.
$ (MAX1.LE.GROWTHBOUND .AND. .NOT.SAWNAN1 ) ) THEN
SIGMA = LSIGMA
SHIFT = SLEFT
GOTO 100
ENDIF
* Right end
S = -RSIGMA
WORK( 1 ) = D( 1 ) + S
IF(ABS(WORK(1)).LT.PIVMIN) THEN
WORK(1) = -PIVMIN
* Need to set SAWNAN2 because refined RRR test should not be used
* in this case
SAWNAN2 = .TRUE.
ENDIF
MAX2 = ABS( WORK( 1 ) )
DO 7 I = 1, N - 1
WORK( N+I ) = LD( I ) / WORK( I )
S = S*WORK( N+I )*L( I ) - RSIGMA
WORK( I+1 ) = D( I+1 ) + S
IF(ABS(WORK(I+1)).LT.PIVMIN) THEN
WORK(I+1) = -PIVMIN
* Need to set SAWNAN2 because refined RRR test should not be used
* in this case
SAWNAN2 = .TRUE.
ENDIF
MAX2 = MAX( MAX2,ABS(WORK(I+1)) )
7 CONTINUE
SAWNAN2 = SAWNAN2 .OR. DISNAN( MAX2 )
IF( FORCER .OR.
$ (MAX2.LE.GROWTHBOUND .AND. .NOT.SAWNAN2 ) ) THEN
SIGMA = RSIGMA
SHIFT = SRIGHT
GOTO 100
ENDIF
* If we are at this point, both shifts led to too much element growth
* Record the better of the two shifts (provided it didn't lead to NaN)
IF(SAWNAN1.AND.SAWNAN2) THEN
* both MAX1 and MAX2 are NaN
GOTO 50
ELSE
IF( .NOT.SAWNAN1 ) THEN
INDX = 1
IF(MAX1.LE.SMLGROWTH) THEN
SMLGROWTH = MAX1
BESTSHIFT = LSIGMA
ENDIF
ENDIF
IF( .NOT.SAWNAN2 ) THEN
IF(SAWNAN1 .OR. MAX2.LE.MAX1) INDX = 2
IF(MAX2.LE.SMLGROWTH) THEN
SMLGROWTH = MAX2
BESTSHIFT = RSIGMA
ENDIF
ENDIF
ENDIF
* If we are here, both the left and the right shift led to
* element growth. If the element growth is moderate, then
* we may still accept the representation, if it passes a
* refined test for RRR. This test supposes that no NaN occurred.
* Moreover, we use the refined RRR test only for isolated clusters.
IF((CLWDTH.LT.MINGAP/DBLE(128)) .AND.
$ (MIN(MAX1,MAX2).LT.FAIL2)
$ .AND.(.NOT.SAWNAN1).AND.(.NOT.SAWNAN2)) THEN
DORRR1 = .TRUE.
ELSE
DORRR1 = .FALSE.
ENDIF
TRYRRR1 = .TRUE.
IF( TRYRRR1 .AND. DORRR1 ) THEN
IF(INDX.EQ.1) THEN
TMP = ABS( DPLUS( N ) )
ZNM2 = ONE
PROD = ONE
OLDP = ONE
DO 15 I = N-1, 1, -1
IF( PROD .LE. EPS ) THEN
PROD =
$ ((DPLUS(I+1)*WORK(N+I+1))/(DPLUS(I)*WORK(N+I)))*OLDP
ELSE
PROD = PROD*ABS(WORK(N+I))
END IF
OLDP = PROD
ZNM2 = ZNM2 + PROD**2
TMP = MAX( TMP, ABS( DPLUS( I ) * PROD ))
15 CONTINUE
RRR1 = TMP/( SPDIAM * SQRT( ZNM2 ) )
IF (RRR1.LE.MAXGROWTH2) THEN
SIGMA = LSIGMA
SHIFT = SLEFT
GOTO 100
ENDIF
ELSE IF(INDX.EQ.2) THEN
TMP = ABS( WORK( N ) )
ZNM2 = ONE
PROD = ONE
OLDP = ONE
DO 16 I = N-1, 1, -1
IF( PROD .LE. EPS ) THEN
PROD = ((WORK(I+1)*LPLUS(I+1))/(WORK(I)*LPLUS(I)))*OLDP
ELSE
PROD = PROD*ABS(LPLUS(I))
END IF
OLDP = PROD
ZNM2 = ZNM2 + PROD**2
TMP = MAX( TMP, ABS( WORK( I ) * PROD ))
16 CONTINUE
RRR2 = TMP/( SPDIAM * SQRT( ZNM2 ) )
IF (RRR2.LE.MAXGROWTH2) THEN
SIGMA = RSIGMA
SHIFT = SRIGHT
GOTO 100
ENDIF
END IF
ENDIF
50 CONTINUE
IF (KTRY.LT.KTRYMAX) THEN
* If we are here, both shifts failed also the RRR test.
* Back off to the outside
LSIGMA = MAX( LSIGMA - LDELTA,
$ LSIGMA - LDMAX)
RSIGMA = MIN( RSIGMA + RDELTA,
$ RSIGMA + RDMAX )
LDELTA = TWO * LDELTA
RDELTA = TWO * RDELTA
KTRY = KTRY + 1
GOTO 5
ELSE
* None of the representations investigated satisfied our
* criteria. Take the best one we found.
IF((SMLGROWTH.LT.FAIL).OR.NOFAIL) THEN
LSIGMA = BESTSHIFT
RSIGMA = BESTSHIFT
FORCER = .TRUE.
GOTO 5
ELSE
INFO = 1
RETURN
ENDIF
END IF
100 CONTINUE
IF (SHIFT.EQ.SLEFT) THEN
ELSEIF (SHIFT.EQ.SRIGHT) THEN
* store new L and D back into DPLUS, LPLUS
CALL DCOPY( N, WORK, 1, DPLUS, 1 )
CALL DCOPY( N-1, WORK(N+1), 1, LPLUS, 1 )
ENDIF
RETURN
*
* End of DLARRF
*
END