heevr(3)

NAME
heevr - {he,sy}evr: eig, MRRR

SYNOPSIS

  Functions

subroutine cheevr (jobz, range, uplo, n, a, lda, vl, vu, il, iu, abstol, m, w, z, ldz, isuppz, work, lwork, rwork, lrwork, iwork, liwork, info)
CHEEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices
subroutine dsyevr (jobz, range, uplo, n, a, lda, vl, vu, il, iu, abstol, m, w, z, ldz, isuppz, work, lwork, iwork, liwork, info)
DSYEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices
subroutine ssyevr (jobz, range, uplo, n, a, lda, vl, vu, il, iu, abstol, m, w, z, ldz, isuppz, work, lwork, iwork, liwork, info)
SSYEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices
subroutine zheevr (jobz, range, uplo, n, a, lda, vl, vu, il, iu, abstol, m, w, z, ldz, isuppz, work, lwork, rwork, lrwork, iwork, liwork, info)
ZHEEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices

Detailed Description

Function Documentation

  subroutine cheevr (character jobz, character range, character uplo, integer n, complex, dimension( lda, * ) a, integer lda, real vl, real vu, integer il, integer iu, real abstol, integer m, real, dimension( * ) w, complex, dimension( ldz, * ) z, integer ldz, integer, dimension( * ) isuppz, complex, dimension( * ) work, integer lwork, real, dimension( * ) rwork, integer lrwork, integer, dimension( * ) iwork, integer liwork, integer info)

CHEEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices

Purpose:

    !>
    !> CHEEVR computes selected eigenvalues and, optionally, eigenvectors
    !> of a complex Hermitian matrix A. Eigenvalues and eigenvectors can be
    !> selected by specifying either a range of values or a range of indices
    !> for the desired eigenvalues. Invocations with different choices for
    !> these parameters may result in the computation of slightly different
    !> eigenvalues and/or eigenvectors for the same matrix. The reason for
    !> this behavior is that there exists a variety of algorithms (each
    !> performing best for a particular set of options) with CHEEVR
    !> attempting to select the best based on the various parameters. In all
    !> cases, the computed values are accurate within the limits of finite
    !> precision arithmetic.
    !>
    !> CHEEVR first reduces the matrix A to tridiagonal form T with a call
    !> to CHETRD.  Then, whenever possible, CHEEVR calls CSTEMR to compute
    !> the eigenspectrum using Relatively Robust Representations.  CSTEMR
    !> 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 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.
    !>
    !> The desired accuracy of the output can be specified by the input
    !> parameter ABSTOL.
    !>
    !> For more details, see CSTEMR's documentation and:
    !> - Inderjit S. Dhillon and Beresford N. Parlett:
    !>   Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
    !> - Inderjit Dhillon and Beresford Parlett:  SIAM Journal on Matrix Analysis and Applications, Vol. 25,
    !>   2004.  Also LAPACK Working Note 154.
    !> - Inderjit Dhillon: ,
    !>   Computer Science Division Technical Report No. UCB/CSD-97-971,
    !>   UC Berkeley, May 1997.
    !>
    !>
    !> Note 1 : CHEEVR calls CSTEMR when the full spectrum is requested
    !> on machines which conform to the ieee-754 floating point standard.
    !> CHEEVR calls SSTEBZ and CSTEIN on non-ieee machines and
    !> when partial spectrum requests are made.
    !>
    !> Normal execution of CSTEMR 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.
    !>
    !>          This parameter influences the choice of the algorithm and
    !>          may alter the computed values.
    !>

    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
    !>          CSTEIN are called
    !>
    !>          This parameter influences the choice of the algorithm and
    !>          may alter the computed values.
    !>

    UPLO

    !>          UPLO is CHARACTER*1
    !>          = 'U':  Upper triangle of A is stored;
    !>          = 'L':  Lower triangle of A is stored.
    !>

    N

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

    A

    !>          A is COMPLEX array, dimension (LDA, N)
    !>          On entry, the Hermitian matrix A.  If UPLO = 'U', the
    !>          leading N-by-N upper triangular part of A contains the
    !>          upper triangular part of the matrix A.  If UPLO = 'L',
    !>          the leading N-by-N lower triangular part of A contains
    !>          the lower triangular part of the matrix A.
    !>          On exit, the lower triangle (if UPLO='L') or the upper
    !>          triangle (if UPLO='U') of A, including the diagonal, is
    !>          destroyed.
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of the array A.  LDA >= max(1,N).
    !>

    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 COMPLEX 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).
    !>          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 an upper bound must be used.
    !>          Supplying N columns is always safe.
    !>

    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 ). This is an output of CSTEMR (tridiagonal
    !>          matrix). The support of the eigenvectors of A is typically
    !>          1:N because of the unitary transformations applied by CUNMTR.
    !>          Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1
    !>

    WORK

    !>          WORK is COMPLEX array, dimension (MAX(1,LWORK))
    !>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The length of the array WORK.
    !>          If N <= 1, LWORK >= 1, else LWORK >= 2*N.
    !>          For optimal efficiency, LWORK >= (NB+1)*N,
    !>          where NB is the max of the blocksize for CHETRD and for
    !>          CUNMTR as returned by ILAENV.
    !>
    !>          If LWORK = -1, then a workspace query is assumed; the routine
    !>          only calculates the optimal sizes of the WORK, RWORK and
    !>          IWORK arrays, returns these values as the first entries of
    !>          the WORK, RWORK and IWORK arrays, and no error message
    !>          related to LWORK or LRWORK or LIWORK is issued by XERBLA.
    !>

    RWORK

    !>          RWORK is REAL array, dimension (MAX(1,LRWORK))
    !>          On exit, if INFO = 0, RWORK(1) returns the optimal
    !>          (and minimal) LRWORK.
    !>

    LRWORK

    !>          LRWORK is INTEGER
    !>          The length of the array RWORK.
    !>          If N <= 1, LRWORK >= 1, else LRWORK >= 24*N.
    !>
    !>          If LRWORK = -1, then a workspace query is assumed; the
    !>          routine only calculates the optimal sizes of the WORK, RWORK
    !>          and IWORK arrays, returns these values as the first entries
    !>          of the WORK, RWORK and IWORK arrays, and no error message
    !>          related to LWORK or LRWORK 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.
    !>          If N <= 1, LIWORK >= 1, else LIWORK >= 10*N.
    !>
    !>          If LIWORK = -1, then a workspace query is assumed; the
    !>          routine only calculates the optimal sizes of the WORK, RWORK
    !>          and IWORK arrays, returns these values as the first entries
    !>          of the WORK, RWORK and IWORK arrays, and no error message
    !>          related to LWORK or LRWORK 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

  subroutine dsyevr (character jobz, character range, character uplo, integer n, double precision, dimension( lda, * ) a, integer lda, 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)

DSYEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices

Purpose:

    !>
    !> DSYEVR computes selected eigenvalues and, optionally, eigenvectors
    !> of a real symmetric matrix A. Eigenvalues and eigenvectors can be
    !> selected by specifying either a range of values or a range of indices
    !> for the desired eigenvalues. Invocations with different choices for
    !> these parameters may result in the computation of slightly different
    !> eigenvalues and/or eigenvectors for the same matrix. The reason for
    !> this behavior is that there exists a variety of algorithms (each
    !> performing best for a particular set of options) with DSYEVR
    !> attempting to select the best based on the various parameters. In all
    !> cases, the computed values are accurate within the limits of finite
    !> precision arithmetic.
    !>
    !> DSYEVR first reduces the matrix A to tridiagonal form T with a call
    !> to DSYTRD.  Then, whenever possible, DSYEVR 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 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.
    !>
    !> The desired accuracy of the output can be specified by the input
    !> parameter ABSTOL.
    !>
    !> For more details, see DSTEMR's documentation and:
    !> - Inderjit S. Dhillon and Beresford N. Parlett:
    !>   Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
    !> - Inderjit Dhillon and Beresford Parlett:  SIAM Journal on Matrix Analysis and Applications, Vol. 25,
    !>   2004.  Also LAPACK Working Note 154.
    !> - Inderjit Dhillon: ,
    !>   Computer Science Division Technical Report No. UCB/CSD-97-971,
    !>   UC Berkeley, May 1997.
    !>
    !>
    !> Note 1 : DSYEVR calls DSTEMR when the full spectrum is requested
    !> on machines which conform to the ieee-754 floating point standard.
    !> DSYEVR 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.
    !>
    !>          This parameter influences the choice of the algorithm and
    !>          may alter the computed values.
    !>

    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
    !>
    !>          This parameter influences the choice of the algorithm and
    !>          may alter the computed values.
    !>

    UPLO

    !>          UPLO is CHARACTER*1
    !>          = 'U':  Upper triangle of A is stored;
    !>          = 'L':  Lower triangle of A is stored.
    !>

    N

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

    A

    !>          A is DOUBLE PRECISION array, dimension (LDA, N)
    !>          On entry, the symmetric matrix A.  If UPLO = 'U', the
    !>          leading N-by-N upper triangular part of A contains the
    !>          upper triangular part of the matrix A.  If UPLO = 'L',
    !>          the leading N-by-N lower triangular part of A contains
    !>          the lower triangular part of the matrix A.
    !>          On exit, the lower triangle (if UPLO='L') or the upper
    !>          triangle (if UPLO='U') of A, including the diagonal, is
    !>          destroyed.
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of the array A.  LDA >= max(1,N).
    !>

    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).
    !>          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 an upper bound must be used.
    !>          Supplying N columns is always safe.
    !>

    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 ). This is an output of DSTEMR (tridiagonal
    !>          matrix). The support of the eigenvectors of A is typically
    !>          1:N because of the orthogonal transformations applied by DORMTR.
    !>          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 LWORK.
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The dimension of the array WORK.
    !>          If N <= 1, LWORK >= 1, else LWORK >= 26*N.
    !>          For optimal efficiency, LWORK >= (NB+6)*N,
    !>          where NB is the max of the blocksize for DSYTRD and DORMTR
    !>          returned by ILAENV.
    !>
    !>          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

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

    LIWORK

    !>          LIWORK is INTEGER
    !>          The dimension of the array IWORK.
    !>          If N <= 1, LIWORK >= 1, else LIWORK >= 10*N.
    !>
    !>          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

    !>          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

  subroutine ssyevr (character jobz, character range, character uplo, integer n, real, dimension( lda, * ) a, integer lda, 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)

SSYEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices

Purpose:

    !>
    !> SSYEVR computes selected eigenvalues and, optionally, eigenvectors
    !> of a real symmetric matrix A. Eigenvalues and eigenvectors can be
    !> selected by specifying either a range of values or a range of indices
    !> for the desired eigenvalues. Invocations with different choices for
    !> these parameters may result in the computation of slightly different
    !> eigenvalues and/or eigenvectors for the same matrix. The reason for
    !> this behavior is that there exists a variety of algorithms (each
    !> performing best for a particular set of options) with SSYEVR
    !> attempting to select the best based on the various parameters. In all
    !> cases, the computed values are accurate within the limits of finite
    !> precision arithmetic.
    !>
    !> SSYEVR first reduces the matrix A to tridiagonal form T with a call
    !> to SSYTRD.  Then, whenever possible, SSYEVR 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 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.
    !>
    !> The desired accuracy of the output can be specified by the input
    !> parameter ABSTOL.
    !>
    !> For more details, see SSTEMR's documentation and:
    !> - Inderjit S. Dhillon and Beresford N. Parlett:
    !>   Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
    !> - Inderjit Dhillon and Beresford Parlett:  SIAM Journal on Matrix Analysis and Applications, Vol. 25,
    !>   2004.  Also LAPACK Working Note 154.
    !> - Inderjit Dhillon: ,
    !>   Computer Science Division Technical Report No. UCB/CSD-97-971,
    !>   UC Berkeley, May 1997.
    !>
    !>
    !> Note 1 : SSYEVR calls SSTEMR when the full spectrum is requested
    !> on machines which conform to the ieee-754 floating point standard.
    !> SSYEVR 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.
    !>
    !>          This parameter influences the choice of the algorithm and
    !>          may alter the computed values.
    !>

    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
    !>
    !>          This parameter influences the choice of the algorithm and
    !>          may alter the computed values.
    !>

    UPLO

    !>          UPLO is CHARACTER*1
    !>          = 'U':  Upper triangle of A is stored;
    !>          = 'L':  Lower triangle of A is stored.
    !>

    N

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

    A

    !>          A is REAL array, dimension (LDA, N)
    !>          On entry, the symmetric matrix A.  If UPLO = 'U', the
    !>          leading N-by-N upper triangular part of A contains the
    !>          upper triangular part of the matrix A.  If UPLO = 'L',
    !>          the leading N-by-N lower triangular part of A contains
    !>          the lower triangular part of the matrix A.
    !>          On exit, the lower triangle (if UPLO='L') or the upper
    !>          triangle (if UPLO='U') of A, including the diagonal, is
    !>          destroyed.
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of the array A.  LDA >= max(1,N).
    !>

    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).
    !>          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 an upper bound must be used.
    !>          Supplying N columns is always safe.
    !>

    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 ). This is an output of SSTEMR (tridiagonal
    !>          matrix). The support of the eigenvectors of A is typically
    !>          1:N because of the orthogonal transformations applied by SORMTR.
    !>          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 LWORK.
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The dimension of the array WORK.
    !>          If N <= 1, LWORK >= 1, else LWORK >= 26*N.
    !>          For optimal efficiency, LWORK >= (NB+6)*N,
    !>          where NB is the max of the blocksize for SSYTRD and SORMTR
    !>          returned by ILAENV.
    !>
    !>          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 LWORK.
    !>

    LIWORK

    !>          LIWORK is INTEGER
    !>          The dimension of the array IWORK.
    !>          If N <= 1, LIWORK >= 1, else 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

  subroutine zheevr (character jobz, character range, character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, double precision vl, double precision vu, integer il, integer iu, double precision abstol, integer m, double precision, dimension( * ) w, complex*16, dimension( ldz, * ) z, integer ldz, integer, dimension( * ) isuppz, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer lrwork, integer, dimension( * ) iwork, integer liwork, integer info)

ZHEEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices

Purpose:

    !>
    !> ZHEEVR computes selected eigenvalues and, optionally, eigenvectors
    !> of a complex Hermitian matrix A. Eigenvalues and eigenvectors can be
    !> selected by specifying either a range of values or a range of indices
    !> for the desired eigenvalues. Invocations with different choices for
    !> these parameters may result in the computation of slightly different
    !> eigenvalues and/or eigenvectors for the same matrix. The reason for
    !> this behavior is that there exists a variety of algorithms (each
    !> performing best for a particular set of options) with ZHEEVR
    !> attempting to select the best based on the various parameters. In all
    !> cases, the computed values are accurate within the limits of finite
    !> precision arithmetic.
    !>
    !> ZHEEVR first reduces the matrix A to tridiagonal form T with a call
    !> to ZHETRD.  Then, whenever possible, ZHEEVR calls ZSTEMR to compute
    !> eigenspectrum using Relatively Robust Representations.  ZSTEMR
    !> 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 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.
    !>
    !> The desired accuracy of the output can be specified by the input
    !> parameter ABSTOL.
    !>
    !> For more details, see ZSTEMR's documentation and:
    !> - Inderjit S. Dhillon and Beresford N. Parlett:
    !>   Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
    !> - Inderjit Dhillon and Beresford Parlett:  SIAM Journal on Matrix Analysis and Applications, Vol. 25,
    !>   2004.  Also LAPACK Working Note 154.
    !> - Inderjit Dhillon: ,
    !>   Computer Science Division Technical Report No. UCB/CSD-97-971,
    !>   UC Berkeley, May 1997.
    !>
    !>
    !> Note 1 : ZHEEVR calls ZSTEMR when the full spectrum is requested
    !> on machines which conform to the ieee-754 floating point standard.
    !> ZHEEVR calls DSTEBZ and ZSTEIN on non-ieee machines and
    !> when partial spectrum requests are made.
    !>
    !> Normal execution of ZSTEMR 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.
    !>
    !>          This parameter influences the choice of the algorithm and
    !>          may alter the computed values.
    !>

    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
    !>          ZSTEIN are called
    !>
    !>          This parameter influences the choice of the algorithm and
    !>          may alter the computed values.
    !>

    UPLO

    !>          UPLO is CHARACTER*1
    !>          = 'U':  Upper triangle of A is stored;
    !>          = 'L':  Lower triangle of A is stored.
    !>

    N

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

    A

    !>          A is COMPLEX*16 array, dimension (LDA, N)
    !>          On entry, the Hermitian matrix A.  If UPLO = 'U', the
    !>          leading N-by-N upper triangular part of A contains the
    !>          upper triangular part of the matrix A.  If UPLO = 'L',
    !>          the leading N-by-N lower triangular part of A contains
    !>          the lower triangular part of the matrix A.
    !>          On exit, the lower triangle (if UPLO='L') or the upper
    !>          triangle (if UPLO='U') of A, including the diagonal, is
    !>          destroyed.
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of the array A.  LDA >= max(1,N).
    !>

    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 COMPLEX*16 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).
    !>          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 an upper bound must be used.
    !>          Supplying N columns is always safe.
    !>

    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 ). This is an output of ZSTEMR (tridiagonal
    !>          matrix). The support of the eigenvectors of A is typically
    !>          1:N because of the unitary transformations applied by ZUNMTR.
    !>          Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1
    !>

    WORK

    !>          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
    !>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The length of the array WORK.
    !>          If N <= 1, LWORK >= 1, else LWORK >= 2*N.
    !>          For optimal efficiency, LWORK >= (NB+1)*N,
    !>          where NB is the max of the blocksize for ZHETRD and for
    !>          ZUNMTR as returned by ILAENV.
    !>
    !>          If LWORK = -1, then a workspace query is assumed; the routine
    !>          only calculates the optimal sizes of the WORK, RWORK and
    !>          IWORK arrays, returns these values as the first entries of
    !>          the WORK, RWORK and IWORK arrays, and no error message
    !>          related to LWORK or LRWORK or LIWORK is issued by XERBLA.
    !>

    RWORK

    !>          RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK))
    !>          On exit, if INFO = 0, RWORK(1) returns the optimal
    !>          (and minimal) LRWORK.
    !>

    LRWORK

    !>          LRWORK is INTEGER
    !>          The length of the array RWORK.
    !>          If N <= 1, LRWORK >= 1, else LRWORK >= 24*N.
    !>
    !>          If LRWORK = -1, then a workspace query is assumed; the
    !>          routine only calculates the optimal sizes of the WORK, RWORK
    !>          and IWORK arrays, returns these values as the first entries
    !>          of the WORK, RWORK and IWORK arrays, and no error message
    !>          related to LWORK or LRWORK 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.
    !>          If N <= 1, LIWORK >= 1, else LIWORK >= 10*N.
    !>
    !>          If LIWORK = -1, then a workspace query is assumed; the
    !>          routine only calculates the optimal sizes of the WORK, RWORK
    !>          and IWORK arrays, returns these values as the first entries
    !>          of the WORK, RWORK and IWORK arrays, and no error message
    !>          related to LWORK or LRWORK 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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