dstevr.f

          SUBROUTINE DSTEVR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL,
         $                   M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK,
         $                   LIWORK, INFO )
    *
    *  -- LAPACK driver routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          CHARACTER          JOBZ, RANGE
          INTEGER            IL, INFO, IU, LDZ, LIWORK, LWORK, M, N
          DOUBLE PRECISION   ABSTOL, VL, VU
    *     ..
    *     .. Array Arguments ..
          INTEGER            ISUPPZ( * ), IWORK( * )
          DOUBLE PRECISION   D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  DSTEVR computes selected eigenvalues and, optionally, eigenvectors
    *  of a real symmetric tridiagonal matrix T.  Eigenvalues and
    *  eigenvectors can be selected by specifying either a range of values
    *  or a range of indices for the desired eigenvalues.
    *
    *  Whenever possible, DSTEVR calls DSTEMR to compute the
    *  eigenspectrum using Relatively Robust Representations.  DSTEMR
    *  computes eigenvalues by the dqds algorithm, while orthogonal
    *  eigenvectors are computed from various "good" L D L^T representations
    *  (also known as Relatively Robust Representations). Gram-Schmidt
    *  orthogonalization is avoided as far as possible. More specifically,
    *  the various steps of the algorithm are as follows. For the i-th
    *  unreduced block of T,
    *     (a) Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T
    *          is a relatively robust representation,
    *     (b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high
    *         relative accuracy by the dqds algorithm,
    *     (c) If there is a cluster of close eigenvalues, "choose" sigma_i
    *         close to the cluster, and go to step (a),
    *     (d) Given the approximate eigenvalue lambda_j of L_i D_i L_i^T,
    *         compute the corresponding eigenvector by forming a
    *         rank-revealing twisted factorization.
    *  The desired accuracy of the output can be specified by the input
    *  parameter ABSTOL.
    *
    *  For more details, see "A new O(n^2) algorithm for the symmetric
    *  tridiagonal eigenvalue/eigenvector problem", by Inderjit Dhillon,
    *  Computer Science Division Technical Report No. UCB//CSD-97-971,
    *  UC Berkeley, May 1997.
    *
    *
    *  Note 1 : DSTEVR calls DSTEMR when the full spectrum is requested
    *  on machines which conform to the ieee-754 floating point standard.
    *  DSTEVR calls DSTEBZ and DSTEIN on non-ieee machines and
    *  when partial spectrum requests are made.
    *
    *  Normal execution of DSTEMR may create NaNs and infinities and
    *  hence may abort due to a floating point exception in environments
    *  which do not handle NaNs and infinities in the ieee standard default
    *  manner.
    *
    *  Arguments
    *  =========
    *
    *  JOBZ    (input) CHARACTER*1
    *          = 'N':  Compute eigenvalues only;
    *          = 'V':  Compute eigenvalues and eigenvectors.
    *
    *  RANGE   (input) CHARACTER*1
    *          = 'A': all eigenvalues will be found.
    *          = 'V': all eigenvalues in the half-open interval (VL,VU]
    *                 will be found.
    *          = 'I': the IL-th through IU-th eigenvalues will be found.
    ********** For RANGE = 'V' or 'I' and IU - IL < N - 1, DSTEBZ and
    ********** DSTEIN are called
    *
    *  N       (input) INTEGER
    *          The order of the matrix.  N >= 0.
    *
    *  D       (input/output) DOUBLE PRECISION array, dimension (N)
    *          On entry, the n diagonal elements of the tridiagonal matrix
    *          A.
    *          On exit, D may be multiplied by a constant factor chosen
    *          to avoid over/underflow in computing the eigenvalues.
    *
    *  E       (input/output) DOUBLE PRECISION array, dimension (max(1,N-1))
    *          On entry, the (n-1) subdiagonal elements of the tridiagonal
    *          matrix A in elements 1 to N-1 of E.
    *          On exit, E may be multiplied by a constant factor chosen
    *          to avoid over/underflow in computing the eigenvalues.
    *
    *  VL      (input) DOUBLE PRECISION
    *  VU      (input) DOUBLE PRECISION
    *          If RANGE='V', the lower and upper bounds of the interval to
    *          be searched for eigenvalues. VL < VU.
    *          Not referenced if RANGE = 'A' or 'I'.
    *
    *  IL      (input) INTEGER
    *  IU      (input) INTEGER
    *          If RANGE='I', the indices (in ascending order) of the
    *          smallest and largest eigenvalues to be returned.
    *          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
    *          Not referenced if RANGE = 'A' or 'V'.
    *
    *  ABSTOL  (input) DOUBLE PRECISION
    *          The absolute error tolerance for the eigenvalues.
    *          An approximate eigenvalue is accepted as converged
    *          when it is determined to lie in an interval [a,b]
    *          of width less than or equal to
    *
    *                  ABSTOL + EPS *   max( |a|,|b| ) ,
    *
    *          where EPS is the machine precision.  If ABSTOL is less than
    *          or equal to zero, then  EPS*|T|  will be used in its place,
    *          where |T| is the 1-norm of the tridiagonal matrix obtained
    *          by reducing A to tridiagonal form.
    *
    *          See "Computing Small Singular Values of Bidiagonal Matrices
    *          with Guaranteed High Relative Accuracy," by Demmel and
    *          Kahan, LAPACK Working Note #3.
    *
    *          If high relative accuracy is important, set ABSTOL to
    *          DLAMCH( 'Safe minimum' ).  Doing so will guarantee that
    *          eigenvalues are computed to high relative accuracy when
    *          possible in future releases.  The current code does not
    *          make any guarantees about high relative accuracy, but
    *          future releases will. See J. Barlow and J. Demmel,
    *          "Computing Accurate Eigensystems of Scaled Diagonally
    *          Dominant Matrices", LAPACK Working Note #7, for a discussion
    *          of which matrices define their eigenvalues to high relative
    *          accuracy.
    *
    *  M       (output) INTEGER
    *          The total number of eigenvalues found.  0 <= M <= N.
    *          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
    *
    *  W       (output) DOUBLE PRECISION array, dimension (N)
    *          The first M elements contain the selected eigenvalues in
    *          ascending order.
    *
    *  Z       (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
    *          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
    *          contain the orthonormal eigenvectors of the matrix A
    *          corresponding to the selected eigenvalues, with the i-th
    *          column of Z holding the eigenvector associated with W(i).
    *          Note: the user must ensure that at least max(1,M) columns are
    *          supplied in the array Z; if RANGE = 'V', the exact value of M
    *          is not known in advance and an upper bound must be used.
    *
    *  LDZ     (input) INTEGER
    *          The leading dimension of the array Z.  LDZ >= 1, and if
    *          JOBZ = 'V', LDZ >= max(1,N).
    *
    *  ISUPPZ  (output) INTEGER array, dimension ( 2*max(1,M) )
    *          The support of the eigenvectors in Z, i.e., the indices
    *          indicating the nonzero elements in Z. The i-th eigenvector
    *          is nonzero only in elements ISUPPZ( 2*i-1 ) through
    *          ISUPPZ( 2*i ).
    ********** Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1
    *
    *  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
    *          On exit, if INFO = 0, WORK(1) returns the optimal (and
    *          minimal) LWORK.
    *
    *  LWORK   (input) INTEGER
    *          The dimension of the array WORK.  LWORK >= max(1,20*N).
    *
    *          If LWORK = -1, then a workspace query is assumed; the routine
    *          only calculates the optimal sizes of the WORK and IWORK
    *          arrays, returns these values as the first entries of the WORK
    *          and IWORK arrays, and no error message related to LWORK or
    *          LIWORK is issued by XERBLA.
    *
    *  IWORK   (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
    *          On exit, if INFO = 0, IWORK(1) returns the optimal (and
    *          minimal) LIWORK.
    *
    *  LIWORK  (input) INTEGER
    *          The dimension of the array IWORK.  LIWORK >= max(1,10*N).
    *
    *          If LIWORK = -1, then a workspace query is assumed; the
    *          routine only calculates the optimal sizes of the WORK and
    *          IWORK arrays, returns these values as the first entries of
    *          the WORK and IWORK arrays, and no error message related to
    *          LWORK or LIWORK is issued by XERBLA.
    *
    *  INFO    (output) INTEGER
    *          = 0:  successful exit
    *          < 0:  if INFO = -i, the i-th argument had an illegal value
    *          > 0:  Internal error
    *
    *  Further Details
    *  ===============
    *
    *  Based on contributions by
    *     Inderjit Dhillon, IBM Almaden, USA
    *     Osni Marques, LBNL/NERSC, USA
    *     Ken Stanley, Computer Science Division, University of
    *       California at Berkeley, USA
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          DOUBLE PRECISION   ZERO, ONE, TWO
          PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 )
    *     ..
    *     .. Local Scalars ..
          LOGICAL            ALLEIG, INDEIG, TEST, LQUERY, VALEIG, WANTZ,
         $                   TRYRAC
          CHARACTER          ORDER
          INTEGER            I, IEEEOK, IMAX, INDIBL, INDIFL, INDISP,
         $                   INDIWO, ISCALE, ITMP1, J, JJ, LIWMIN, LWMIN,
         $                   NSPLIT
          DOUBLE PRECISION   BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, SMLNUM,
         $                   TMP1, TNRM, VLL, VUU
    *     ..
    *     .. External Functions ..
          LOGICAL            LSAME
          INTEGER            ILAENV
          DOUBLE PRECISION   DLAMCH, DLANST
          EXTERNAL           LSAME, ILAENV, DLAMCH, DLANST
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           DCOPY, DSCAL, DSTEBZ, DSTEMR, DSTEIN, DSTERF,
         $                   DSWAP, XERBLA
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          MAX, MIN, SQRT
    *     ..
    *     .. Executable Statements ..
    *
    *
    *     Test the input parameters.
    *
          IEEEOK = ILAENV( 10, 'DSTEVR', 'N', 1, 2, 3, 4 )
    *
          WANTZ = LSAME( JOBZ, 'V' )
          ALLEIG = LSAME( RANGE, 'A' )
          VALEIG = LSAME( RANGE, 'V' )
          INDEIG = LSAME( RANGE, 'I' )
    *
          LQUERY = ( ( LWORK.EQ.-1 ) .OR. ( LIWORK.EQ.-1 ) )
          LWMIN = MAX( 1, 20*N )
          LIWMIN = MAX( 1, 10*N )
    *
    *
          INFO = 0
          IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
             INFO = -1
          ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
             INFO = -2
          ELSE IF( N.LT.0 ) THEN
             INFO = -3
          ELSE
             IF( VALEIG ) THEN
                IF( N.GT.0 .AND. VU.LE.VL )
         $         INFO = -7
             ELSE IF( INDEIG ) THEN
                IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
                   INFO = -8
                ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
                   INFO = -9
                END IF
             END IF
          END IF
          IF( INFO.EQ.0 ) THEN
             IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
                INFO = -14
             END IF
          END IF
    *
          IF( INFO.EQ.0 ) THEN
             WORK( 1 ) = LWMIN
             IWORK( 1 ) = LIWMIN
    *
             IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
                INFO = -17
             ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
                INFO = -19
             END IF
          END IF
    *
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'DSTEVR', -INFO )
             RETURN
          ELSE IF( LQUERY ) THEN
             RETURN
          END IF
    *
    *     Quick return if possible
    *
          M = 0
          IF( N.EQ.0 )
         $   RETURN
    *
          IF( N.EQ.1 ) THEN
             IF( ALLEIG .OR. INDEIG ) THEN
                M = 1
                W( 1 ) = D( 1 )
             ELSE
                IF( VL.LT.D( 1 ) .AND. VU.GE.D( 1 ) ) THEN
                   M = 1
                   W( 1 ) = D( 1 )
                END IF
             END IF
             IF( WANTZ )
         $      Z( 1, 1 ) = ONE
             RETURN
          END IF
    *
    *     Get machine constants.
    *
          SAFMIN = DLAMCH( 'Safe minimum' )
          EPS = DLAMCH( 'Precision' )
          SMLNUM = SAFMIN / EPS
          BIGNUM = ONE / SMLNUM
          RMIN = SQRT( SMLNUM )
          RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
    *
    *
    *     Scale matrix to allowable range, if necessary.
    *
          ISCALE = 0
          VLL = VL
          VUU = VU
    *
          TNRM = DLANST( 'M', N, D, E )
          IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN
             ISCALE = 1
             SIGMA = RMIN / TNRM
          ELSE IF( TNRM.GT.RMAX ) THEN
             ISCALE = 1
             SIGMA = RMAX / TNRM
          END IF
          IF( ISCALE.EQ.1 ) THEN
             CALL DSCAL( N, SIGMA, D, 1 )
             CALL DSCAL( N-1, SIGMA, E( 1 ), 1 )
             IF( VALEIG ) THEN
                VLL = VL*SIGMA
                VUU = VU*SIGMA
             END IF
          END IF

    *     Initialize indices into workspaces.  Note: These indices are used only
    *     if DSTERF or DSTEMR fail.

    *     IWORK(INDIBL:INDIBL+M-1) corresponds to IBLOCK in DSTEBZ and
    *     stores the block indices of each of the M<=N eigenvalues.
          INDIBL = 1
    *     IWORK(INDISP:INDISP+NSPLIT-1) corresponds to ISPLIT in DSTEBZ and
    *     stores the starting and finishing indices of each block.
          INDISP = INDIBL + N
    *     IWORK(INDIFL:INDIFL+N-1) stores the indices of eigenvectors
    *     that corresponding to eigenvectors that fail to converge in
    *     DSTEIN.  This information is discarded; if any fail, the driver
    *     returns INFO > 0.
          INDIFL = INDISP + N
    *     INDIWO is the offset of the remaining integer workspace.
          INDIWO = INDISP + N
    *
    *     If all eigenvalues are desired, then
    *     call DSTERF or DSTEMR.  If this fails for some eigenvalue, then
    *     try DSTEBZ.
    *
    *
          TEST = .FALSE.
          IF( INDEIG ) THEN
             IF( IL.EQ.1 .AND. IU.EQ.N ) THEN
                TEST = .TRUE.
             END IF
          END IF
          IF( ( ALLEIG .OR. TEST ) .AND. IEEEOK.EQ.1 ) THEN
             CALL DCOPY( N-1, E( 1 ), 1, WORK( 1 ), 1 )
             IF( .NOT.WANTZ ) THEN
                CALL DCOPY( N, D, 1, W, 1 )
                CALL DSTERF( N, W, WORK, INFO )
             ELSE
                CALL DCOPY( N, D, 1, WORK( N+1 ), 1 )
                IF (ABSTOL .LE. TWO*N*EPS) THEN
                   TRYRAC = .TRUE.
                ELSE
                   TRYRAC = .FALSE.
                END IF
                CALL DSTEMR( JOBZ, 'A', N, WORK( N+1 ), WORK, VL, VU, IL,
         $                   IU, M, W, Z, LDZ, N, ISUPPZ, TRYRAC,
         $                   WORK( 2*N+1 ), LWORK-2*N, IWORK, LIWORK, INFO )
    *
             END IF
             IF( INFO.EQ.0 ) THEN
                M = N
                GO TO 10
             END IF
             INFO = 0
          END IF
    *
    *     Otherwise, call DSTEBZ and, if eigenvectors are desired, DSTEIN.
    *
          IF( WANTZ ) THEN
             ORDER = 'B'
          ELSE
             ORDER = 'E'
          END IF

          CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTOL, D, E, M,
         $             NSPLIT, W, IWORK( INDIBL ), IWORK( INDISP ), WORK,
         $             IWORK( INDIWO ), INFO )
    *
          IF( WANTZ ) THEN
             CALL DSTEIN( N, D, E, M, W, IWORK( INDIBL ), IWORK( INDISP ),
         $                Z, LDZ, WORK, IWORK( INDIWO ), IWORK( INDIFL ),
         $                INFO )
          END IF
    *
    *     If matrix was scaled, then rescale eigenvalues appropriately.
    *
       10 CONTINUE
          IF( ISCALE.EQ.1 ) THEN
             IF( INFO.EQ.0 ) THEN
                IMAX = M
             ELSE
                IMAX = INFO - 1
             END IF
             CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
          END IF
    *
    *     If eigenvalues are not in order, then sort them, along with
    *     eigenvectors.
    *
          IF( WANTZ ) THEN
             DO 30 J = 1, M - 1
                I = 0
                TMP1 = W( J )
                DO 20 JJ = J + 1, M
                   IF( W( JJ ).LT.TMP1 ) THEN
                      I = JJ
                      TMP1 = W( JJ )
                   END IF
       20       CONTINUE
    *
                IF( I.NE.0 ) THEN
                   ITMP1 = IWORK( I )
                   W( I ) = W( J )
                   IWORK( I ) = IWORK( J )
                   W( J ) = TMP1
                   IWORK( J ) = ITMP1
                   CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
                END IF
       30    CONTINUE
          END IF
    *
    *      Causes problems with tests 19 & 20:
    *      IF (wantz .and. INDEIG ) Z( 1,1) = Z(1,1) / 1.002 + .002
    *
    *
          WORK( 1 ) = LWMIN
          IWORK( 1 ) = LIWMIN
          RETURN
    *
    *     End of DSTEVR
    *
          END