sstevr.f man page

sstevr.f —

Synopsis

Functions/Subroutines

subroutine sstevr (JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO)
SSTEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Function/Subroutine Documentation

subroutine sstevr (characterJOBZ, characterRANGE, integerN, real, dimension( * )D, real, dimension( * )E, realVL, realVU, integerIL, integerIU, realABSTOL, integerM, real, dimension( * )W, real, dimension( ldz, * )Z, integerLDZ, integer, dimension( * )ISUPPZ, real, dimension( * )WORK, integerLWORK, integer, dimension( * )IWORK, integerLIWORK, integerINFO)

SSTEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices  

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.

 For more details, see "A new O(n^2) algorithm for the symmetric
 tridiagonal eigenvalue/eigenvector problem", by Inderjit Dhillon,
 Computer Science Division Technical Report No. UCB//CSD-97-971,
 UC Berkeley, May 1997.


 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.
Parameters:

JOBZ

          JOBZ is CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

RANGE

          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

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

D

          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

          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

          VL is REAL

VU

          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

          IL is INTEGER

IU

          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

          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

          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

          W is REAL array, dimension (N)
          The first M elements contain the selected eigenvalues in
          ascending order.

Z

          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

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

ISUPPZ

          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

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

LWORK

          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

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

LIWORK

          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

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

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2011

Contributors:

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

Definition at line 298 of file sstevr.f.

Author

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Referenced By

sstevr(3) is an alias of sstevr.f(3).

Sat Nov 16 2013 Version 3.4.2 LAPACK