stevr(3)

NAME
stevr - stevr: eig, MRRR

SYNOPSIS

  Functions
subroutine dstevr (jobz, range, n, d, e, vl, vu, il, iu, abstol, m, w, z, ldz, isuppz, work, lwork, iwork, liwork, info)
DSTEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
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

Detailed Description

Function Documentation

  subroutine dstevr (character jobz, character range, integer n, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision vl, double precision vu, integer il, integer iu, double precision abstol, integer m, double precision, dimension( * ) w, double precision, dimension( ldz, * ) z, integer ldz, integer, dimension( * ) isuppz, double precision, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, integer info)

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

Purpose:

     DSTEVR 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, DSTEVR 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 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 : DSTEVR calls DSTEMR when the full spectrum is requested
     on machines which conform to the ieee-754 floating point standard.
     DSTEVR calls DSTEBZ and DSTEIN 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.

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, DSTEBZ and
              DSTEIN are called

    N

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

    D

              D is DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION
              If RANGE='V', the lower bound of the interval to
              be searched for eigenvalues. VL < VU.
              Not referenced if RANGE = 'A' or 'I'.

    VU

              VU is DOUBLE PRECISION
              If RANGE='V', the upper bound of the interval to
              be searched for eigenvalues. VL < VU.
              Not referenced if RANGE = 'A' or 'I'.

    IL

              IL is INTEGER
              If RANGE='I', the index of the
              smallest eigenvalue 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'.

    IU

              IU is INTEGER
              If RANGE='I', the index of the
              largest eigenvalue 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 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

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

    Z

              Z is 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).
              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 DOUBLE PRECISION 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 >= max(1,20*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 >= max(1,10*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.

Contributors:
    Inderjit Dhillon, IBM Almaden, USA
     Osni Marques, LBNL/NERSC, USA
     Ken Stanley, Computer Science Division, University of California at Berkeley, USA

Definition at line 301 of file dstevr.f.

  subroutine sstevr (character jobz, character range, integer n, real, dimension( * ) d, real, dimension( * ) e, real vl, real vu, integer il, integer iu, real abstol, integer m, real, dimension( * ) w, real, dimension( ldz, * ) z, integer ldz, integer, dimension( * ) isuppz, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, integer info)

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
              If RANGE='V', the lower bound of the interval to
              be searched for eigenvalues. VL < VU.
              Not referenced if RANGE = 'A' or 'I'.

    VU

              VU is REAL
              If RANGE='V', the upper bound of the interval to
              be searched for eigenvalues. VL < VU.
              Not referenced if RANGE = 'A' or 'I'.

    IL

              IL is INTEGER
              If RANGE='I', the index of the
              smallest eigenvalue 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'.

    IU

              IU is INTEGER
              If RANGE='I', the index of the
              largest eigenvalue 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.

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 303 of file sstevr.f.

Author

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