Provided by: liblapack-doc_3.12.0-3build1.1_all bug

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.

            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 '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 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: 'Multiple representations
              to compute orthogonal eigenvectors of symmetric tridiagonal matrices,'
              Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
            - Inderjit Dhillon and Beresford Parlett: 'Orthogonal Eigenvectors and
              Relative Gaps,' SIAM Journal on Matrix Analysis and Applications, Vol. 25,
              2004.  Also LAPACK Working Note 154.
            - Inderjit Dhillon: 'A new O(n^2) algorithm for the symmetric
              tridiagonal eigenvalue/eigenvector problem',
              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.

           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

           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 '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 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.

           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.  LWORK >= max(1,2*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.  LRWORK >= max(1,24*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.  LIWORK >= max(1,10*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.

            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 '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 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: 'Multiple representations
              to compute orthogonal eigenvectors of symmetric tridiagonal matrices,'
              Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
            - Inderjit Dhillon and Beresford Parlett: 'Orthogonal Eigenvectors and
              Relative Gaps,' SIAM Journal on Matrix Analysis and Applications, Vol. 25,
              2004.  Also LAPACK Working Note 154.
            - Inderjit Dhillon: 'A new O(n^2) algorithm for the symmetric
              tridiagonal eigenvalue/eigenvector problem',
              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.

           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

           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 '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).
                     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.  LWORK >= max(1,26*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 LWORK.

           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 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.

            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 '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 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: 'Multiple representations
              to compute orthogonal eigenvectors of symmetric tridiagonal matrices,'
              Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
            - Inderjit Dhillon and Beresford Parlett: 'Orthogonal Eigenvectors and
              Relative Gaps,' SIAM Journal on Matrix Analysis and Applications, Vol. 25,
              2004.  Also LAPACK Working Note 154.
            - Inderjit Dhillon: 'A new O(n^2) algorithm for the symmetric
              tridiagonal eigenvalue/eigenvector problem',
              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.

           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

           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 '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).
                     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.  LWORK >= max(1,26*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.  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
            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.

            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 '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 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: 'Multiple representations
              to compute orthogonal eigenvectors of symmetric tridiagonal matrices,'
              Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
            - Inderjit Dhillon and Beresford Parlett: 'Orthogonal Eigenvectors and
              Relative Gaps,' SIAM Journal on Matrix Analysis and Applications, Vol. 25,
              2004.  Also LAPACK Working Note 154.
            - Inderjit Dhillon: 'A new O(n^2) algorithm for the symmetric
              tridiagonal eigenvalue/eigenvector problem',
              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.

           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

           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 '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 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.

           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.  LWORK >= max(1,2*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.  LRWORK >= max(1,24*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.  LIWORK >= max(1,10*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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