slar1v.f

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*> \brief \b SLAR1V computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of the tridiagonal matrix LDLT - λI.
*
*  =========== DOCUMENTATION ===========
*
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*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE SLAR1V( N, B1, BN, LAMBDA, D, L, LD, LLD,
*                  PIVMIN, GAPTOL, Z, WANTNC, NEGCNT, ZTZ, MINGMA,
*                  R, ISUPPZ, NRMINV, RESID, RQCORR, WORK )
*
*       .. Scalar Arguments ..
*       LOGICAL            WANTNC
*       INTEGER   B1, BN, N, NEGCNT, R
*       REAL               GAPTOL, LAMBDA, MINGMA, NRMINV, PIVMIN, RESID,
*      $                   RQCORR, ZTZ
*       ..
*       .. Array Arguments ..
*       INTEGER            ISUPPZ( * )
*       REAL               D( * ), L( * ), LD( * ), LLD( * ),
*      $                  WORK( * )
*       REAL             Z( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> SLAR1V 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.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>           The order of the matrix L D L**T.
*> \endverbatim
*>
*> \param[in] B1
*> \verbatim
*>          B1 is INTEGER
*>           First index of the submatrix of L D L**T.
*> \endverbatim
*>
*> \param[in] BN
*> \verbatim
*>          BN is INTEGER
*>           Last index of the submatrix of L D L**T.
*> \endverbatim
*>
*> \param[in] LAMBDA
*> \verbatim
*>          LAMBDA is REAL
*>           The shift. In order to compute an accurate eigenvector,
*>           LAMBDA should be a good approximation to an eigenvalue
*>           of L D L**T.
*> \endverbatim
*>
*> \param[in] L
*> \verbatim
*>          L is REAL array, dimension (N-1)
*>           The (n-1) subdiagonal elements of the unit bidiagonal matrix
*>           L, in elements 1 to N-1.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is REAL array, dimension (N)
*>           The n diagonal elements of the diagonal matrix D.
*> \endverbatim
*>
*> \param[in] LD
*> \verbatim
*>          LD is REAL array, dimension (N-1)
*>           The n-1 elements L(i)*D(i).
*> \endverbatim
*>
*> \param[in] LLD
*> \verbatim
*>          LLD is REAL array, dimension (N-1)
*>           The n-1 elements L(i)*L(i)*D(i).
*> \endverbatim
*>
*> \param[in] PIVMIN
*> \verbatim
*>          PIVMIN is REAL
*>           The minimum pivot in the Sturm sequence.
*> \endverbatim
*>
*> \param[in] GAPTOL
*> \verbatim
*>          GAPTOL is REAL
*>           Tolerance that indicates when eigenvector entries are negligible
*>           w.r.t. their contribution to the residual.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*>          Z is REAL 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.
*> \endverbatim
*>
*> \param[in] WANTNC
*> \verbatim
*>          WANTNC is LOGICAL
*>           Specifies whether NEGCNT has to be computed.
*> \endverbatim
*>
*> \param[out] NEGCNT
*> \verbatim
*>          NEGCNT is INTEGER
*>           If WANTNC is .TRUE. then NEGCNT = the number of pivots < pivmin
*>           in the  matrix factorization L D L**T, and NEGCNT = -1 otherwise.
*> \endverbatim
*>
*> \param[out] ZTZ
*> \verbatim
*>          ZTZ is REAL
*>           The square of the 2-norm of Z.
*> \endverbatim
*>
*> \param[out] MINGMA
*> \verbatim
*>          MINGMA is REAL
*>           The reciprocal of the largest (in magnitude) diagonal
*>           element of the inverse of L D L**T - sigma I.
*> \endverbatim
*>
*> \param[in,out] R
*> \verbatim
*>          R is 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.
*> \endverbatim
*>
*> \param[out] ISUPPZ
*> \verbatim
*>          ISUPPZ is 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 ).
*> \endverbatim
*>
*> \param[out] NRMINV
*> \verbatim
*>          NRMINV is REAL
*>           NRMINV = 1/SQRT( ZTZ )
*> \endverbatim
*>
*> \param[out] RESID
*> \verbatim
*>          RESID is REAL
*>           The residual of the FP vector.
*>           RESID = ABS( MINGMA )/SQRT( ZTZ )
*> \endverbatim
*>
*> \param[out] RQCORR
*> \verbatim
*>          RQCORR is REAL
*>           The Rayleigh Quotient correction to LAMBDA.
*>           RQCORR = MINGMA*TMP
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is REAL array, dimension (4*N)
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup realOTHERauxiliary
*
*> \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 slar1v( N, B1, BN, LAMBDA, D, L, LD, LLD,
     $           PIVMIN, GAPTOL, Z, WANTNC, NEGCNT, ZTZ, MINGMA,
     $           R, ISUPPZ, NRMINV, RESID, RQCORR, WORK )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      LOGICAL            WANTNC
      INTEGER   B1, BN, N, NEGCNT, R
      REAL               GAPTOL, LAMBDA, MINGMA, NRMINV, PIVMIN, RESID,
     $                   rqcorr, ztz
*     ..
*     .. Array Arguments ..
      INTEGER            ISUPPZ( * )
      REAL               D( * ), L( * ), LD( * ), LLD( * ),
     $                  work( * )
      REAL             Z( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ZERO, ONE
      PARAMETER          ( ZERO = 0.0e0, one = 1.0e0 )
 
*     ..
*     .. Local Scalars ..
      LOGICAL            SAWNAN1, SAWNAN2
      INTEGER            I, INDLPL, INDP, INDS, INDUMN, NEG1, NEG2, R1,
     $                   r2
      REAL               DMINUS, DPLUS, EPS, S, TMP
*     ..
*     .. External Functions ..
      LOGICAL SISNAN
      REAL               SLAMCH
      EXTERNAL           sisnan, slamch
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs
*     ..
*     .. Executable Statements ..
*
      eps = slamch( 'precision' )
 
 
.EQ.      IF( R0 ) 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
 
.EQ.      IF( B11 ) 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
.LT.         IF(DPLUSZERO) NEG1 = NEG1 + 1
         WORK( INDS+I ) = S*WORK( INDLPL+I )*L( I )
         S = WORK( INDS+I ) - LAMBDA
 50   CONTINUE
      SAWNAN1 = SISNAN( 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 = SISNAN( 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
.LT.            IF(ABS(DPLUS)PIVMIN) DPLUS = -PIVMIN
            WORK( INDLPL+I ) = LD( I ) / DPLUS
.LT.            IF(DPLUSZERO) NEG1 = NEG1 + 1
            WORK( INDS+I ) = S*WORK( INDLPL+I )*L( I )
.EQ.            IF( WORK( INDLPL+I )ZERO )
     $                      WORK( INDS+I ) = LLD( I )
            S = WORK( INDS+I ) - LAMBDA
 70      CONTINUE
         DO 71 I = R1, R2 - 1
            DPLUS = D( I ) + S
.LT.            IF(ABS(DPLUS)PIVMIN) DPLUS = -PIVMIN
            WORK( INDLPL+I ) = LD( I ) / DPLUS
            WORK( INDS+I ) = S*WORK( INDLPL+I )*L( I )
.EQ.            IF( WORK( INDLPL+I )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
.LT.         IF(DMINUSZERO) 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 = SISNAN( 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 )
.LT.            IF(ABS(DMINUS)PIVMIN) DMINUS = -PIVMIN
            TMP = D( I ) / DMINUS
.LT.            IF(DMINUSZERO) NEG2 = NEG2 + 1
            WORK( INDUMN+I ) = L( I )*TMP
            WORK( INDP+I-1 ) = WORK( INDP+I )*TMP - LAMBDA
.EQ.            IF( TMPZERO )
     $          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 )
.LT.      IF( MINGMAZERO ) NEG1 = NEG1 + 1
      IF( WANTNC ) THEN
         NEGCNT = NEG1 + NEG2
      ELSE
         NEGCNT = -1
      ENDIF
.EQ.      IF( ABS(MINGMA)ZERO )
     $   MINGMA = EPS*WORK( INDS+R1-1 )
      R = R1
      DO 110 I = R1, R2 - 1
         TMP = WORK( INDS+I ) + WORK( INDP+I )
.EQ.         IF( TMPZERO )
     $      TMP = EPS*WORK( INDS+I )
.LE.         IF( ABS( TMP )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
*
.NOT..AND..NOT.      IF( SAWNAN1  SAWNAN2 ) THEN
         DO 210 I = R-1, B1, -1
            Z( I ) = -( WORK( INDLPL+I )*Z( I+1 ) )
.LT.            IF( (ABS(Z(I))+ABS(Z(I+1)))* ABS(LD(I))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
.EQ.            IF( Z( I+1 )ZERO ) THEN
               Z( I ) = -( LD( I+1 ) / LD( I ) )*Z( I+2 )
            ELSE
               Z( I ) = -( WORK( INDLPL+I )*Z( I+1 ) )
            END IF
.LT.            IF( (ABS(Z(I))+ABS(Z(I+1)))* ABS(LD(I))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
.NOT..AND..NOT.      IF( SAWNAN1  SAWNAN2 ) THEN
         DO 250 I = R, BN-1
            Z( I+1 ) = -( WORK( INDUMN+I )*Z( I ) )
.LT.            IF( (ABS(Z(I))+ABS(Z(I+1)))* ABS(LD(I))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
.EQ.            IF( Z( I )ZERO ) THEN
               Z( I+1 ) = -( LD( I-1 ) / LD( I ) )*Z( I-1 )
            ELSE
               Z( I+1 ) = -( WORK( INDUMN+I )*Z( I ) )
            END IF
.LT.            IF( (ABS(Z(I))+ABS(Z(I+1)))* ABS(LD(I))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 SLAR1V
*
      END