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



NAME
       ZSTEMR

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



PURPOSE
            ZSTEMR 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.


            Further Details
            1.ZSTEMR works only on machines which follow IEEE-754
            floating-point standard in their handling of infinities and NaNs.
            This permits the use of efficient inner loops avoiding a check for
            zero divisors.

            2. LAPACK routines can be used to reduce a complex Hermitean matrix to
            real symmetric tridiagonal form.

            (Any complex Hermitean tridiagonal matrix has real values on its diagonal
            and potentially complex numbers on its off-diagonals. By applying a
            similarity transform with an appropriate diagonal matrix
            diag(1,e^{i hy_1}, ... , e^{i hy_{n-1}}), the complex Hermitean
            matrix can be transformed into a real symmetric matrix and complex
            arithmetic can be entirely avoided.)

            While the eigenvectors of the real symmetric tridiagonal matrix are real,
            the eigenvectors of original complex Hermitean matrix have complex entries
            in general.
            Since LAPACK drivers overwrite the matrix data with the eigenvectors,
            ZSTEMR accepts complex workspace to facilitate interoperability
            with ZUNMTR or ZUPMTR.




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.

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

           D         (input/output)
                     D is DOUBLE PRECISION array, dimension (N)
                     On entry, the N diagonal elements of the tridiagonal matrix
                     T. On exit, D is overwritten.

           E         (input/output)
                     E is 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)
                     VL is DOUBLE PRECISION

           VU        (input)
                     VU is 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)
                     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.
                     Not referenced if RANGE = 'A' or 'V'.

           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 DOUBLE PRECISION array, dimension (N)
                     The first M elements contain the selected eigenvalues in
                     ascending order.

           Z         (output)
                     Z is COMPLEX*16 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)
                     LDZ is INTEGER
                     The leading dimension of the array Z.  LDZ >= 1, and if
                     JOBZ = 'V', then LDZ >= max(1,N).

           NZC       (input)
                     NZC is 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)
                     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 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)
                     TRYRAC is 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      (output)
                     WORK is DOUBLE PRECISION array, dimension (LWORK)
                     On exit, if INFO = 0, WORK(1) returns the optimal
                     (and minimal) LWORK.

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

           LIWORK    (input)
                     LIWORK is 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)
                     INFO is 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 ZLARRV.
                           Here, the digit X = ABS( IINFO ) < 10, where IINFO is
                           the nonzero error code returned by DLARRE or
                           ZLARRV, respectively.



LAPACK routine                  31 October 2017                      ZSTEMR(3)