SUBROUTINE DLAR1V( N, B1, BN, LAMBDA, D, L, LD, LLD,
$ PIVMIN, GAPTOL, Z, WANTNC, NEGCNT, ZTZ, MINGMA,
$ R, ISUPPZ, NRMINV, RESID, RQCORR, WORK )
*
* -- LAPACK auxiliary routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
LOGICAL WANTNC
INTEGER B1, BN, N, NEGCNT, R
DOUBLE PRECISION GAPTOL, LAMBDA, MINGMA, NRMINV, PIVMIN, RESID,
$ RQCORR, ZTZ
* ..
* .. Array Arguments ..
INTEGER ISUPPZ( * )
DOUBLE PRECISION D( * ), L( * ), LD( * ), LLD( * ),
$ WORK( * )
DOUBLE PRECISION Z( * )
* ..
*
* Purpose
* =======
*
* DLAR1V computes the (scaled) r-th column of the inverse of
* the sumbmatrix in rows B1 through BN of the tridiagonal matrix
* L D L^T - sigma I. When sigma is close to an eigenvalue, the
* computed vector is an accurate eigenvector. Usually, r corresponds
* to the index where the eigenvector is largest in magnitude.
* The following steps accomplish this computation :
* (a) Stationary qd transform, L D L^T - sigma I = L(+) D(+) L(+)^T,
* (b) Progressive qd transform, L D L^T - sigma I = U(-) D(-) U(-)^T,
* (c) Computation of the diagonal elements of the inverse of
* L D L^T - sigma I by combining the above transforms, and choosing
* r as the index where the diagonal of the inverse is (one of the)
* largest in magnitude.
* (d) Computation of the (scaled) r-th column of the inverse using the
* twisted factorization obtained by combining the top part of the
* the stationary and the bottom part of the progressive transform.
*
* Arguments
* =========
*
* N (input) INTEGER
* The order of the matrix L D L^T.
*
* B1 (input) INTEGER
* First index of the submatrix of L D L^T.
*
* BN (input) INTEGER
* Last index of the submatrix of L D L^T.
*
* LAMBDA (input) DOUBLE PRECISION
* The shift. In order to compute an accurate eigenvector,
* LAMBDA should be a good approximation to an eigenvalue
* of L D L^T.
*
* L (input) DOUBLE PRECISION array, dimension (N-1)
* The (n-1) subdiagonal elements of the unit bidiagonal matrix
* L, in elements 1 to N-1.
*
* D (input) DOUBLE PRECISION array, dimension (N)
* The n diagonal elements of the diagonal matrix D.
*
* LD (input) DOUBLE PRECISION array, dimension (N-1)
* The n-1 elements L(i)*D(i).
*
* LLD (input) DOUBLE PRECISION array, dimension (N-1)
* The n-1 elements L(i)*L(i)*D(i).
*
* PIVMIN (input) DOUBLE PRECISION
* The minimum pivot in the Sturm sequence.
*
* GAPTOL (input) DOUBLE PRECISION
* Tolerance that indicates when eigenvector entries are negligible
* w.r.t. their contribution to the residual.
*
* Z (input/output) DOUBLE PRECISION array, dimension (N)
* On input, all entries of Z must be set to 0.
* On output, Z contains the (scaled) r-th column of the
* inverse. The scaling is such that Z(R) equals 1.
*
* WANTNC (input) LOGICAL
* Specifies whether NEGCNT has to be computed.
*
* NEGCNT (output) INTEGER
* If WANTNC is .TRUE. then NEGCNT = the number of pivots < pivmin
* in the matrix factorization L D L^T, and NEGCNT = -1 otherwise.
*
* ZTZ (output) DOUBLE PRECISION
* The square of the 2-norm of Z.
*
* MINGMA (output) DOUBLE PRECISION
* The reciprocal of the largest (in magnitude) diagonal
* element of the inverse of L D L^T - sigma I.
*
* R (input/output) INTEGER
* The twist index for the twisted factorization used to
* compute Z.
* On input, 0 <= R <= N. If R is input as 0, R is set to
* the index where (L D L^T - sigma I)^{-1} is largest
* in magnitude. If 1 <= R <= N, R is unchanged.
* On output, R contains the twist index used to compute Z.
* Ideally, R designates the position of the maximum entry in the
* eigenvector.
*
* ISUPPZ (output) INTEGER array, dimension (2)
* The support of the vector in Z, i.e., the vector Z is
* nonzero only in elements ISUPPZ(1) through ISUPPZ( 2 ).
*
* NRMINV (output) DOUBLE PRECISION
* NRMINV = 1/SQRT( ZTZ )
*
* RESID (output) DOUBLE PRECISION
* The residual of the FP vector.
* RESID = ABS( MINGMA )/SQRT( ZTZ )
*
* RQCORR (output) DOUBLE PRECISION
* The Rayleigh Quotient correction to LAMBDA.
* RQCORR = MINGMA*TMP
*
* WORK (workspace) DOUBLE PRECISION array, dimension (4*N)
*
* Further Details
* ===============
*
* Based on contributions by
* Beresford Parlett, University of California, Berkeley, USA
* Jim Demmel, University of California, Berkeley, USA
* Inderjit Dhillon, University of Texas, Austin, USA
* Osni Marques, LBNL/NERSC, USA
* Christof Voemel, University of California, Berkeley, USA
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
LOGICAL SAWNAN1, SAWNAN2
INTEGER I, INDLPL, INDP, INDS, INDUMN, NEG1, NEG2, R1,
$ R2
DOUBLE PRECISION DMINUS, DPLUS, EPS, S, TMP
* ..
* .. External Functions ..
LOGICAL DISNAN
DOUBLE PRECISION DLAMCH
EXTERNAL DISNAN, DLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS
* ..
* .. Executable Statements ..
*
EPS = DLAMCH( 'Precision' )
IF( R.EQ.0 ) THEN
R1 = B1
R2 = BN
ELSE
R1 = R
R2 = R
END IF
* Storage for LPLUS
INDLPL = 0
* Storage for UMINUS
INDUMN = N
INDS = 2*N + 1
INDP = 3*N + 1
IF( B1.EQ.1 ) THEN
WORK( INDS ) = ZERO
ELSE
WORK( INDS+B1-1 ) = LLD( B1-1 )
END IF
*
* Compute the stationary transform (using the differential form)
* until the index R2.
*
SAWNAN1 = .FALSE.
NEG1 = 0
S = WORK( INDS+B1-1 ) - LAMBDA
DO 50 I = B1, R1 - 1
DPLUS = D( I ) + S
WORK( INDLPL+I ) = LD( I ) / DPLUS
IF(DPLUS.LT.ZERO) NEG1 = NEG1 + 1
WORK( INDS+I ) = S*WORK( INDLPL+I )*L( I )
S = WORK( INDS+I ) - LAMBDA
50 CONTINUE
SAWNAN1 = DISNAN( S )
IF( SAWNAN1 ) GOTO 60
DO 51 I = R1, R2 - 1
DPLUS = D( I ) + S
WORK( INDLPL+I ) = LD( I ) / DPLUS
WORK( INDS+I ) = S*WORK( INDLPL+I )*L( I )
S = WORK( INDS+I ) - LAMBDA
51 CONTINUE
SAWNAN1 = DISNAN( S )
*
60 CONTINUE
IF( SAWNAN1 ) THEN
* Runs a slower version of the above loop if a NaN is detected
NEG1 = 0
S = WORK( INDS+B1-1 ) - LAMBDA
DO 70 I = B1, R1 - 1
DPLUS = D( I ) + S
IF(ABS(DPLUS).LT.PIVMIN) DPLUS = -PIVMIN
WORK( INDLPL+I ) = LD( I ) / DPLUS
IF(DPLUS.LT.ZERO) NEG1 = NEG1 + 1
WORK( INDS+I ) = S*WORK( INDLPL+I )*L( I )
IF( WORK( INDLPL+I ).EQ.ZERO )
$ WORK( INDS+I ) = LLD( I )
S = WORK( INDS+I ) - LAMBDA
70 CONTINUE
DO 71 I = R1, R2 - 1
DPLUS = D( I ) + S
IF(ABS(DPLUS).LT.PIVMIN) DPLUS = -PIVMIN
WORK( INDLPL+I ) = LD( I ) / DPLUS
WORK( INDS+I ) = S*WORK( INDLPL+I )*L( I )
IF( WORK( INDLPL+I ).EQ.ZERO )
$ WORK( INDS+I ) = LLD( I )
S = WORK( INDS+I ) - LAMBDA
71 CONTINUE
END IF
*
* Compute the progressive transform (using the differential form)
* until the index R1
*
SAWNAN2 = .FALSE.
NEG2 = 0
WORK( INDP+BN-1 ) = D( BN ) - LAMBDA
DO 80 I = BN - 1, R1, -1
DMINUS = LLD( I ) + WORK( INDP+I )
TMP = D( I ) / DMINUS
IF(DMINUS.LT.ZERO) NEG2 = NEG2 + 1
WORK( INDUMN+I ) = L( I )*TMP
WORK( INDP+I-1 ) = WORK( INDP+I )*TMP - LAMBDA
80 CONTINUE
TMP = WORK( INDP+R1-1 )
SAWNAN2 = DISNAN( TMP )
IF( SAWNAN2 ) THEN
* Runs a slower version of the above loop if a NaN is detected
NEG2 = 0
DO 100 I = BN-1, R1, -1
DMINUS = LLD( I ) + WORK( INDP+I )
IF(ABS(DMINUS).LT.PIVMIN) DMINUS = -PIVMIN
TMP = D( I ) / DMINUS
IF(DMINUS.LT.ZERO) NEG2 = NEG2 + 1
WORK( INDUMN+I ) = L( I )*TMP
WORK( INDP+I-1 ) = WORK( INDP+I )*TMP - LAMBDA
IF( TMP.EQ.ZERO )
$ WORK( INDP+I-1 ) = D( I ) - LAMBDA
100 CONTINUE
END IF
*
* Find the index (from R1 to R2) of the largest (in magnitude)
* diagonal element of the inverse
*
MINGMA = WORK( INDS+R1-1 ) + WORK( INDP+R1-1 )
IF( MINGMA.LT.ZERO ) NEG1 = NEG1 + 1
IF( WANTNC ) THEN
NEGCNT = NEG1 + NEG2
ELSE
NEGCNT = -1
ENDIF
IF( ABS(MINGMA).EQ.ZERO )
$ MINGMA = EPS*WORK( INDS+R1-1 )
R = R1
DO 110 I = R1, R2 - 1
TMP = WORK( INDS+I ) + WORK( INDP+I )
IF( TMP.EQ.ZERO )
$ TMP = EPS*WORK( INDS+I )
IF( ABS( TMP ).LE.ABS( MINGMA ) ) THEN
MINGMA = TMP
R = I + 1
END IF
110 CONTINUE
*
* Compute the FP vector: solve N^T v = e_r
*
ISUPPZ( 1 ) = B1
ISUPPZ( 2 ) = BN
Z( R ) = ONE
ZTZ = ONE
*
* Compute the FP vector upwards from R
*
IF( .NOT.SAWNAN1 .AND. .NOT.SAWNAN2 ) THEN
DO 210 I = R-1, B1, -1
Z( I ) = -( WORK( INDLPL+I )*Z( I+1 ) )
IF( (ABS(Z(I))+ABS(Z(I+1)))* ABS(LD(I)).LT.GAPTOL )
$ THEN
Z( I ) = ZERO
ISUPPZ( 1 ) = I + 1
GOTO 220
ENDIF
ZTZ = ZTZ + Z( I )*Z( I )
210 CONTINUE
220 CONTINUE
ELSE
* Run slower loop if NaN occurred.
DO 230 I = R - 1, B1, -1
IF( Z( I+1 ).EQ.ZERO ) THEN
Z( I ) = -( LD( I+1 ) / LD( I ) )*Z( I+2 )
ELSE
Z( I ) = -( WORK( INDLPL+I )*Z( I+1 ) )
END IF
IF( (ABS(Z(I))+ABS(Z(I+1)))* ABS(LD(I)).LT.GAPTOL )
$ THEN
Z( I ) = ZERO
ISUPPZ( 1 ) = I + 1
GO TO 240
END IF
ZTZ = ZTZ + Z( I )*Z( I )
230 CONTINUE
240 CONTINUE
ENDIF
* Compute the FP vector downwards from R in blocks of size BLKSIZ
IF( .NOT.SAWNAN1 .AND. .NOT.SAWNAN2 ) THEN
DO 250 I = R, BN-1
Z( I+1 ) = -( WORK( INDUMN+I )*Z( I ) )
IF( (ABS(Z(I))+ABS(Z(I+1)))* ABS(LD(I)).LT.GAPTOL )
$ THEN
Z( I+1 ) = ZERO
ISUPPZ( 2 ) = I
GO TO 260
END IF
ZTZ = ZTZ + Z( I+1 )*Z( I+1 )
250 CONTINUE
260 CONTINUE
ELSE
* Run slower loop if NaN occurred.
DO 270 I = R, BN - 1
IF( Z( I ).EQ.ZERO ) THEN
Z( I+1 ) = -( LD( I-1 ) / LD( I ) )*Z( I-1 )
ELSE
Z( I+1 ) = -( WORK( INDUMN+I )*Z( I ) )
END IF
IF( (ABS(Z(I))+ABS(Z(I+1)))* ABS(LD(I)).LT.GAPTOL )
$ THEN
Z( I+1 ) = ZERO
ISUPPZ( 2 ) = I
GO TO 280
END IF
ZTZ = ZTZ + Z( I+1 )*Z( I+1 )
270 CONTINUE
280 CONTINUE
END IF
*
* Compute quantities for convergence test
*
TMP = ONE / ZTZ
NRMINV = SQRT( TMP )
RESID = ABS( MINGMA )*NRMINV
RQCORR = MINGMA*TMP
*
*
RETURN
*
* End of DLAR1V
*
END