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

NAME

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

SYNOPSIS

       SUBROUTINE DSYEVR( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL,  M,  W,  Z,  LDZ,
                          ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO )

           CHARACTER      JOBZ, RANGE, UPLO

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

           DOUBLE         PRECISION ABSTOL, VL, VU

           INTEGER        ISUPPZ( * ), IWORK( * )

           DOUBLE         PRECISION A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * )

PURPOSE

       DSYEVR  computes  selected  eigenvalues  and, optionally, eigenvectors of a real symmetric
       matrix A.  Eigenvalues and eigenvectors can be
        selected by specifying either a range of values or a range of
        indices for the desired eigenvalues.
        DSYEVR first reduces the matrix A to tridiagonal form T with a call
        to DSYTRD.  Then, whenever possible, DSYEVR 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 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.
        The desired accuracy of the output can be specified by the input
        parameter ABSTOL.
        For more details, see DSTEMR's documentation and:
        - 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.
        Note 1 : DSYEVR calls DSTEMR when the full spectrum is requested
        on machines which conform to the ieee-754 floating point standard.
        DSYEVR calls DSTEBZ and SSTEIN 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.

        UPLO    (input) CHARACTER*1
                = 'U':  Upper triangle of A is stored;
                = 'L':  Lower triangle of A is stored.

        N       (input) INTEGER
                The order of the matrix A.  N >= 0.

        A       (input/output) DOUBLE PRECISION array, dimension (LDA, N)
                On entry, the symmetric matrix A.  If UPLO = 'U', the
                leading N-by-N upper triangular part of A contains the
                upper triangular part of the matrix A.  If UPLO = 'L',
                the leading N-by-N lower triangular part of A contains
                the lower triangular part of the matrix A.
                On exit, the lower triangle (if UPLO='L') or the upper
                triangle (if UPLO='U') of A, including the diagonal, is
                destroyed.

        LDA     (input) INTEGER
                The leading dimension of the array A.  LDA >= max(1,N).

        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).
                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 an upper bound must be used.
                Supplying N columns is always safe.

        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 ).

        WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
                On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

        LWORK   (input) INTEGER
                The dimension of the array WORK.  LWORK >= max(1,26*N).
                For optimal efficiency, LWORK >= (NB+6)*N,
                where NB is the max of the blocksize for DSYTRD and DORMTR
                returned by ILAENV.
                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 (MAX(1,LIWORK))
                On exit, if INFO = 0, IWORK(1) returns the optimal LWORK.

        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 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
                = 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
           Jason Riedy, Computer Science Division, University of
             California at Berkeley, USA

 LAPACK driver routine (version 3.2.2)      April 2011                            DSYEVR(3lapack)