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



NAME
       SSTEVR

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



PURPOSE
            SSTEVR 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, SSTEVR calls SSTEMR to compute the
            eigenspectrum using Relatively Robust Representations.  SSTEMR
            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.



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

            Normal execution of SSTEMR 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
                     SSTEIN are called

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

           D         (input/output)
                     D is REAL 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)
                     E is REAL 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)
                     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.

                     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
                     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
                     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)
                     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 REAL 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)
                     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 REAL array, dimension (MAX(1,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 >= 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     (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 >= 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)
                     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                      SSTEVR(3)