CHEEVR(3)      LAPACK routine of NEC Numeric Library Collection      CHEEVR(3)



NAME
       CHEEVR

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



PURPOSE
            CHEEVR computes selected eigenvalues and, optionally, eigenvectors
            of a complex Hermitian matrix A.  Eigenvalues and eigenvectors can
            be selected by specifying either a range of values or a range of
            indices for the desired eigenvalues.

            CHEEVR first reduces the matrix A to tridiagonal form T with a call
            to CHETRD.  Then, whenever possible, CHEEVR calls CSTEMR to compute
            the eigenspectrum using Relatively Robust Representations.  CSTEMR
            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.



            Note 1 : CHEEVR calls CSTEMR when the full spectrum is requested
            on machines which conform to the ieee-754 floating point standard.
            CHEEVR calls SSTEBZ and CSTEIN on non-ieee machines and
            when partial spectrum requests are made.

            Normal execution of CSTEMR 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)
                     JOBZ is CHARACTER*1
                     = 'N':  Compute eigenvalues only;
                     = 'V':  Compute eigenvalues and eigenvectors.

           RANGE     (input)
                     RANGE is 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, SSTEBZ and
                     CSTEIN are called

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

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

           A         (input/output)
                     A is COMPLEX array, dimension (LDA, N)
                     On entry, the Hermitian 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)
                     LDA is INTEGER
                     The leading dimension of the array A.  LDA >= max(1,N).

           VL        (input)
                     VL is REAL

           VU        (input)
                     VU is REAL
                     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)
                     IL is INTEGER

           IU        (input)
                     IU is 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)
                     ABSTOL is REAL
                     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.

                     If high relative accuracy is important, set ABSTOL to
                     SLAMCH( '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
                     furutre 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)
                     M is 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)
                     W is REAL array, dimension (N)
                     The first M elements contain the selected eigenvalues in
                     ascending order.

           Z         (output)
                     Z is COMPLEX 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.

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

           ISUPPZ    (output)
                     ISUPPZ is 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      (output)
                     WORK is COMPLEX array, dimension (MAX(1,LWORK))
                     On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

           LWORK     (input)
                     LWORK is INTEGER
                     The length of the array WORK.  LWORK >= max(1,2*N).
                     For optimal efficiency, LWORK >= (NB+1)*N,
                     where NB is the max of the blocksize for CHETRD and for
                     CUNMTR as returned by ILAENV.

                     If LWORK = -1, then a workspace query is assumed; the routine
                     only calculates the optimal sizes of the WORK, RWORK and
                     IWORK arrays, returns these values as the first entries of
                     the WORK, RWORK and IWORK arrays, and no error message
                     related to LWORK or LRWORK or LIWORK is issued by XERBLA.

           RWORK     (output)
                     RWORK is REAL array, dimension (MAX(1,LRWORK))
                     On exit, if INFO = 0, RWORK(1) returns the optimal
                     (and minimal) LRWORK.

           LRWORK    (input)
                     LRWORK is INTEGER
                     The length of the array RWORK.  LRWORK >= max(1,24*N).

                     If LRWORK = -1, then a workspace query is assumed; the
                     routine only calculates the optimal sizes of the WORK, RWORK
                     and IWORK arrays, returns these values as the first entries
                     of the WORK, RWORK and IWORK arrays, and no error message
                     related to LWORK or LRWORK or LIWORK is issued by XERBLA.

           IWORK     (output)
                     IWORK is INTEGER array, dimension (MAX(1,LIWORK))
                     On exit, if INFO = 0, IWORK(1) returns the optimal
                     (and minimal) LIWORK.

           LIWORK    (input)
                     LIWORK is 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, RWORK
                     and IWORK arrays, returns these values as the first entries
                     of the WORK, RWORK and IWORK arrays, and no error message
                     related to LWORK or LRWORK or LIWORK is issued by XERBLA.

           INFO      (output)
                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  Internal error



LAPACK routine                  31 October 2017                      CHEEVR(3)