Provided by: liblapack-doc_3.3.1-1_all bug

NAME

       LAPACK-3 - computes selected eigenvalues and, optionally, eigenvectors of a real symmetric
       tridiagonal matrix T

SYNOPSIS

       SUBROUTINE DSTEMR( JOBZ, RANGE, N, D, E, VL, VU, IL,  IU,  M,  W,  Z,  LDZ,  NZC,  ISUPPZ,
                          TRYRAC, WORK, LWORK, IWORK, LIWORK, INFO )

           IMPLICIT       NONE

           CHARACTER      JOBZ, RANGE

           LOGICAL        TRYRAC

           INTEGER        IL, INFO, IU, LDZ, NZC, LIWORK, LWORK, M, N

           DOUBLE         PRECISION VL, VU

           INTEGER        ISUPPZ( * ), IWORK( * )

           DOUBLE         PRECISION D( * ), E( * ), W( * ), WORK( * )

           DOUBLE         PRECISION Z( LDZ, * )

PURPOSE

       DSTEMR  computes  selected  eigenvalues  and, optionally, eigenvectors of a real symmetric
       tridiagonal matrix T. Any such unreduced matrix has
        a well defined set of pairwise different real eigenvalues, the corresponding
        real eigenvectors are pairwise orthogonal.
        The spectrum may be computed either completely or partially by specifying
        either an interval (VL,VU] or a range of indices IL:IU for the desired
        eigenvalues.
        Depending on the number of desired eigenvalues, these are computed either
        by bisection or the dqds algorithm. Numerically orthogonal eigenvectors are
        computed by the use of various suitable L D L^T factorizations near clusters
        of close eigenvalues (referred to as RRRs, Relatively Robust
        Representations). An informal sketch of the algorithm follows.
        For each unreduced block (submatrix) of T,
           (a) Compute T - sigma I  = L D L^T, so that L and D
               define all the wanted eigenvalues to high relative accuracy.
               This means that small relative changes in the entries of D and L
               cause only small relative changes in the eigenvalues and
               eigenvectors. The standard (unfactored) representation of the
               tridiagonal matrix T does not have this property in general.
           (b) Compute the eigenvalues to suitable accuracy.
               If the eigenvectors are desired, the algorithm attains full
               accuracy of the computed eigenvalues only right before
               the corresponding vectors have to be computed, see steps c) and d).
           (c) For each cluster of close eigenvalues, select a new
               shift close to the cluster, find a new factorization, and refine
               the shifted eigenvalues to suitable accuracy.
           (d) For each eigenvalue with a large enough relative separation compute
               the corresponding eigenvector by forming a rank revealing twisted
               factorization. Go back to (c) for any clusters that remain.
        For more details, see:
        - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations
          to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"
          Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
        - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and
          Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,
          2004.  Also LAPACK Working Note 154.
        - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
          tridiagonal eigenvalue/eigenvector problem",
          Computer Science Division Technical Report No. UCB/CSD-97-971,
          UC Berkeley, May 1997.
        Further Details
        floating-point standard in their handling of infinities and NaNs.
        This permits the use of efficient inner loops avoiding a check for
        zero divisors.

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.

        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
                T. On exit, D is overwritten.

        E       (input/output) DOUBLE PRECISION array, dimension (N)
                On entry, the (N-1) subdiagonal elements of the tridiagonal
                matrix T in elements 1 to N-1 of E. E(N) need not be set on
                input, but is used internally as workspace.
                On exit, E is overwritten.

        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.
                Not referenced if RANGE = 'A' or 'V'.

        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', and if INFO = 0, then the first M columns of Z
                contain the orthonormal eigenvectors of the matrix T
                corresponding to the selected eigenvalues, with the i-th
                column of Z holding the eigenvector associated with W(i).
                If JOBZ = 'N', then Z is not referenced.
                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 can be computed with a workspace
                query by setting NZC = -1, see below.

        LDZ     (input) INTEGER
                The leading dimension of the array Z.  LDZ >= 1, and if
                JOBZ = 'V', then LDZ >= max(1,N).

        NZC     (input) INTEGER
                The number of eigenvectors to be held in the array Z.
                If RANGE = 'A', then NZC >= max(1,N).
                If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU].
                If RANGE = 'I', then NZC >= IU-IL+1.
                If NZC = -1, then a workspace query is assumed; the
                routine calculates the number of columns of the array Z that
                are needed to hold the eigenvectors.
                This value is returned as the first entry of the Z array, and
                no error message related to NZC is issued by XERBLA.

        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 computed eigenvector
                is nonzero only in elements ISUPPZ( 2*i-1 ) through
                ISUPPZ( 2*i ). This is relevant in the case when the matrix
                is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0.

        TRYRAC  (input/output) LOGICAL
                If TRYRAC.EQ..TRUE., indicates that the code should check whether
                the tridiagonal matrix defines its eigenvalues to high relative
                accuracy.  If so, the code uses relative-accuracy preserving
                algorithms that might be (a bit) slower depending on the matrix.
                If the matrix does not define its eigenvalues to high relative
                accuracy, the code can uses possibly faster algorithms.
                If TRYRAC.EQ..FALSE., the code is not required to guarantee
                relatively accurate eigenvalues and can use the fastest possible
                techniques.
                On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix
                does not define its eigenvalues to high relative accuracy.

        WORK    (workspace/output) DOUBLE PRECISION array, dimension (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(1,18*N)
                if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'.
                If LWORK = -1, then a workspace query is assumed; the routine
                only calculates the optimal size of the WORK array, returns
                this value as the first entry of the WORK array, and no error
                message related to LWORK is issued by XERBLA.

        IWORK   (workspace/output) INTEGER array, dimension (LIWORK)
                On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

        LIWORK  (input) INTEGER
                The dimension of the array IWORK.  LIWORK >= max(1,10*N)
                if the eigenvectors are desired, and LIWORK >= max(1,8*N)
                if only the eigenvalues are to be computed.
                If LIWORK = -1, then a workspace query is assumed; the
                routine only calculates the optimal size of the IWORK array,
                returns this value as the first entry of the IWORK array, and
                no error message related to LIWORK is issued by XERBLA.

        INFO    (output) INTEGER
                On exit, INFO
                = 0:  successful exit
                < 0:  if INFO = -i, the i-th argument had an illegal value
                > 0:  if INFO = 1X, internal error in DLARRE,
                if INFO = 2X, internal error in DLARRV.
                Here, the digit X = ABS( IINFO ) < 10, where IINFO is
                the nonzero error code returned by DLARRE or
                DLARRV, respectively.

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

 LAPACK computational routine (version 3.2.2April 2011                            DSTEMR(3lapack)