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  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,  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 , 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  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,
    !>          , 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

  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  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,  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 , 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  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,
    !>          , 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

Author

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