stemr(3)

NAME
stemr - stemr: eig, relatively robust representation (RRR)

SYNOPSIS

  Functions
subroutine cstemr (jobz, range, n, d, e, vl, vu, il, iu, m, w, z, ldz, nzc, isuppz, tryrac, work, lwork, iwork, liwork, info)
CSTEMR
subroutine dstemr (jobz, range, n, d, e, vl, vu, il, iu, m, w, z, ldz, nzc, isuppz, tryrac, work, lwork, iwork, liwork, info)
DSTEMR
subroutine sstemr (jobz, range, n, d, e, vl, vu, il, iu, m, w, z, ldz, nzc, isuppz, tryrac, work, lwork, iwork, liwork, info)
SSTEMR
subroutine zstemr (jobz, range, n, d, e, vl, vu, il, iu, m, w, z, ldz, nzc, isuppz, tryrac, work, lwork, iwork, liwork, info)
ZSTEMR

Detailed Description

Function Documentation

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

CSTEMR

Purpose:

     CSTEMR computes selected eigenvalues and, optionally, eigenvectors
     of a real symmetric tridiagonal matrix T. Any such unreduced matrix has
     a well defined set of pairwise different real eigenvalues, the corresponding
     real eigenvectors are pairwise orthogonal.
     The spectrum may be computed either completely or partially by specifying
     either an interval (VL,VU] or a range of indices IL:IU for the desired
     eigenvalues.
     Depending on the number of desired eigenvalues, these are computed either
     by bisection or the dqds algorithm. Numerically orthogonal eigenvectors are
     computed by the use of various suitable L D L^T factorizations near clusters
     of close eigenvalues (referred to as RRRs, Relatively Robust
     Representations). An informal sketch of the algorithm 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.
     For more details, see:
     - 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.
     Further Details
     1.CSTEMR works only on machines which follow IEEE-754
     floating-point standard in their handling of infinities and NaNs.
     This permits the use of efficient inner loops avoiding a check for
     zero divisors.
     2. LAPACK routines can be used to reduce a complex Hermitean matrix to
     real symmetric tridiagonal form.
     (Any complex Hermitean tridiagonal matrix has real values on its diagonal
     and potentially complex numbers on its off-diagonals. By applying a
     similarity transform with an appropriate diagonal matrix
     diag(1,e^{i \phy_1}, ... , e^{i \phy_{n-1}}), the complex Hermitean
     matrix can be transformed into a real symmetric matrix and complex
     arithmetic can be entirely avoided.)
     While the eigenvectors of the real symmetric tridiagonal matrix are real,
     the eigenvectors of original complex Hermitean matrix have complex entries
     in general.
     Since LAPACK drivers overwrite the matrix data with the eigenvectors,
     CSTEMR accepts complex workspace to facilitate interoperability
     with CUNMTR or CUPMTR.

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.

    N

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

    D

              D is REAL array, dimension (N)
              On entry, the N diagonal elements of the tridiagonal matrix
              T. On exit, D is overwritten.

    E

              E is REAL array, dimension (N)
              On entry, the (N-1) subdiagonal elements of the tridiagonal
              matrix T in elements 1 to N-1 of E. E(N) need not be set on
              input, but is used internally as workspace.
              On exit, E is overwritten.

    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.
              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.
              Not referenced if RANGE = 'A' or 'V'.

    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', and if INFO = 0, then the first M columns of Z
              contain the orthonormal eigenvectors of the matrix T
              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 can be computed with a workspace
              query by setting NZC = -1, see below.

    LDZ

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

    NZC

              NZC is INTEGER
              The number of eigenvectors to be held in the array Z.
              If RANGE = 'A', then NZC >= max(1,N).
              If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU].
              If RANGE = 'I', then NZC >= IU-IL+1.
              If NZC = -1, then a workspace query is assumed; the
              routine calculates the number of columns of the array Z that
              are needed to hold the eigenvectors.
              This value is returned as the first entry of the Z array, and
              no error message related to NZC is issued by XERBLA.

    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 computed eigenvector
              is nonzero only in elements ISUPPZ( 2*i-1 ) through
              ISUPPZ( 2*i ). This is relevant in the case when the matrix
              is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0.

    TRYRAC

              TRYRAC is LOGICAL
              If TRYRAC = .TRUE., indicates that the code should check whether
              the tridiagonal matrix defines its eigenvalues to high relative
              accuracy.  If so, the code uses relative-accuracy preserving
              algorithms that might be (a bit) slower depending on the matrix.
              If the matrix does not define its eigenvalues to high relative
              accuracy, the code can uses possibly faster algorithms.
              If TRYRAC = .FALSE., the code is not required to guarantee
              relatively accurate eigenvalues and can use the fastest possible
              techniques.
              On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix
              does not define its eigenvalues to high relative accuracy.

    WORK

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

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK. LWORK >= max(1,18*N)
              if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'.
              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 (LIWORK)
              On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

    LIWORK

              LIWORK is INTEGER
              The dimension of the array IWORK.  LIWORK >= max(1,10*N)
              if the eigenvectors are desired, and LIWORK >= max(1,8*N)
              if only the eigenvalues are to be computed.
              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
              On exit, INFO
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  if INFO = 1X, internal error in SLARRE,
                    if INFO = 2X, internal error in CLARRV.
                    Here, the digit X = ABS( IINFO ) < 10, where IINFO is
                    the nonzero error code returned by SLARRE or
                    CLARRV, respectively.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Beresford Parlett, University of California, Berkeley, USA
     Jim Demmel, University of California, Berkeley, USA
     Inderjit Dhillon, University of Texas, Austin, USA
     Osni Marques, LBNL/NERSC, USA
     Christof Voemel, University of California, Berkeley, USA
     Aravindh Krishnamoorthy, FAU, Erlangen, Germany

Definition at line 336 of file cstemr.f.

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

DSTEMR

Purpose:

     DSTEMR computes selected eigenvalues and, optionally, eigenvectors
     of a real symmetric tridiagonal matrix T. Any such unreduced matrix has
     a well defined set of pairwise different real eigenvalues, the corresponding
     real eigenvectors are pairwise orthogonal.
     The spectrum may be computed either completely or partially by specifying
     either an interval (VL,VU] or a range of indices IL:IU for the desired
     eigenvalues.
     Depending on the number of desired eigenvalues, these are computed either
     by bisection or the dqds algorithm. Numerically orthogonal eigenvectors are
     computed by the use of various suitable L D L^T factorizations near clusters
     of close eigenvalues (referred to as RRRs, Relatively Robust
     Representations). An informal sketch of the algorithm 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.
     For more details, see:
     - 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.
     Further Details
     1.DSTEMR works only on machines which follow IEEE-754
     floating-point standard in their handling of infinities and NaNs.
     This permits the use of efficient inner loops avoiding a check for
     zero divisors.

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.

    N

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

    D

              D is DOUBLE PRECISION array, dimension (N)
              On entry, the N diagonal elements of the tridiagonal matrix
              T. On exit, D is overwritten.

    E

              E is DOUBLE PRECISION array, dimension (N)
              On entry, the (N-1) subdiagonal elements of the tridiagonal
              matrix T in elements 1 to N-1 of E. E(N) need not be set on
              input, but is used internally as workspace.
              On exit, E is overwritten.

    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.
              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.
              Not referenced if RANGE = 'A' or 'V'.

    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', and if INFO = 0, then the first M columns of Z
              contain the orthonormal eigenvectors of the matrix T
              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 can be computed with a workspace
              query by setting NZC = -1, see below.

    LDZ

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

    NZC

              NZC is INTEGER
              The number of eigenvectors to be held in the array Z.
              If RANGE = 'A', then NZC >= max(1,N).
              If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU].
              If RANGE = 'I', then NZC >= IU-IL+1.
              If NZC = -1, then a workspace query is assumed; the
              routine calculates the number of columns of the array Z that
              are needed to hold the eigenvectors.
              This value is returned as the first entry of the Z array, and
              no error message related to NZC is issued by XERBLA.

    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 computed eigenvector
              is nonzero only in elements ISUPPZ( 2*i-1 ) through
              ISUPPZ( 2*i ). This is relevant in the case when the matrix
              is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0.

    TRYRAC

              TRYRAC is LOGICAL
              If TRYRAC = .TRUE., indicates that the code should check whether
              the tridiagonal matrix defines its eigenvalues to high relative
              accuracy.  If so, the code uses relative-accuracy preserving
              algorithms that might be (a bit) slower depending on the matrix.
              If the matrix does not define its eigenvalues to high relative
              accuracy, the code can uses possibly faster algorithms.
              If TRYRAC = .FALSE., the code is not required to guarantee
              relatively accurate eigenvalues and can use the fastest possible
              techniques.
              On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix
              does not define its eigenvalues to high relative accuracy.

    WORK

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

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK. LWORK >= max(1,18*N)
              if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'.
              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 (LIWORK)
              On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

    LIWORK

              LIWORK is INTEGER
              The dimension of the array IWORK.  LIWORK >= max(1,10*N)
              if the eigenvectors are desired, and LIWORK >= max(1,8*N)
              if only the eigenvalues are to be computed.
              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
              On exit, INFO
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  if INFO = 1X, internal error in DLARRE,
                    if INFO = 2X, internal error in DLARRV.
                    Here, the digit X = ABS( IINFO ) < 10, where IINFO is
                    the nonzero error code returned by DLARRE or
                    DLARRV, respectively.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Beresford Parlett, University of California, Berkeley, USA
     Jim Demmel, University of California, Berkeley, USA
     Inderjit Dhillon, University of Texas, Austin, USA
     Osni Marques, LBNL/NERSC, USA
     Christof Voemel, University of California, Berkeley, USA
     Aravindh Krishnamoorthy, FAU, Erlangen, Germany

Definition at line 319 of file dstemr.f.

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

SSTEMR

Purpose:

     SSTEMR computes selected eigenvalues and, optionally, eigenvectors
     of a real symmetric tridiagonal matrix T. Any such unreduced matrix has
     a well defined set of pairwise different real eigenvalues, the corresponding
     real eigenvectors are pairwise orthogonal.
     The spectrum may be computed either completely or partially by specifying
     either an interval (VL,VU] or a range of indices IL:IU for the desired
     eigenvalues.
     Depending on the number of desired eigenvalues, these are computed either
     by bisection or the dqds algorithm. Numerically orthogonal eigenvectors are
     computed by the use of various suitable L D L^T factorizations near clusters
     of close eigenvalues (referred to as RRRs, Relatively Robust
     Representations). An informal sketch of the algorithm 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.
     For more details, see:
     - 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.
     Further Details
     1.SSTEMR works only on machines which follow IEEE-754
     floating-point standard in their handling of infinities and NaNs.
     This permits the use of efficient inner loops avoiding a check for
     zero divisors.

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.

    N

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

    D

              D is REAL array, dimension (N)
              On entry, the N diagonal elements of the tridiagonal matrix
              T. On exit, D is overwritten.

    E

              E is REAL array, dimension (N)
              On entry, the (N-1) subdiagonal elements of the tridiagonal
              matrix T in elements 1 to N-1 of E. E(N) need not be set on
              input, but is used internally as workspace.
              On exit, E is overwritten.

    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.
              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.
              Not referenced if RANGE = 'A' or 'V'.

    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', and if INFO = 0, then the first M columns of Z
              contain the orthonormal eigenvectors of the matrix T
              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 can be computed with a workspace
              query by setting NZC = -1, see below.

    LDZ

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

    NZC

              NZC is INTEGER
              The number of eigenvectors to be held in the array Z.
              If RANGE = 'A', then NZC >= max(1,N).
              If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU].
              If RANGE = 'I', then NZC >= IU-IL+1.
              If NZC = -1, then a workspace query is assumed; the
              routine calculates the number of columns of the array Z that
              are needed to hold the eigenvectors.
              This value is returned as the first entry of the Z array, and
              no error message related to NZC is issued by XERBLA.

    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 computed eigenvector
              is nonzero only in elements ISUPPZ( 2*i-1 ) through
              ISUPPZ( 2*i ). This is relevant in the case when the matrix
              is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0.

    TRYRAC

              TRYRAC is LOGICAL
              If TRYRAC = .TRUE., indicates that the code should check whether
              the tridiagonal matrix defines its eigenvalues to high relative
              accuracy.  If so, the code uses relative-accuracy preserving
              algorithms that might be (a bit) slower depending on the matrix.
              If the matrix does not define its eigenvalues to high relative
              accuracy, the code can uses possibly faster algorithms.
              If TRYRAC = .FALSE., the code is not required to guarantee
              relatively accurate eigenvalues and can use the fastest possible
              techniques.
              On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix
              does not define its eigenvalues to high relative accuracy.

    WORK

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

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK. LWORK >= max(1,18*N)
              if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'.
              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 (LIWORK)
              On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

    LIWORK

              LIWORK is INTEGER
              The dimension of the array IWORK.  LIWORK >= max(1,10*N)
              if the eigenvectors are desired, and LIWORK >= max(1,8*N)
              if only the eigenvalues are to be computed.
              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
              On exit, INFO
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  if INFO = 1X, internal error in SLARRE,
                    if INFO = 2X, internal error in SLARRV.
                    Here, the digit X = ABS( IINFO ) < 10, where IINFO is
                    the nonzero error code returned by SLARRE or
                    SLARRV, respectively.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Beresford Parlett, University of California, Berkeley, USA
     Jim Demmel, University of California, Berkeley, USA
     Inderjit Dhillon, University of Texas, Austin, USA
     Osni Marques, LBNL/NERSC, USA
     Christof Voemel, University of California, Berkeley, USA
     Aravindh Krishnamoorthy, FAU, Erlangen, Germany

Definition at line 319 of file sstemr.f.

  subroutine zstemr (character jobz, character range, integer n, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision vl, double precision vu, integer il, integer iu, integer m, double precision, dimension( * ) w, complex*16, dimension( ldz, * ) z, integer ldz, integer nzc, integer, dimension( * ) isuppz, logical tryrac, double precision, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, integer info)

ZSTEMR

Purpose:

     ZSTEMR computes selected eigenvalues and, optionally, eigenvectors
     of a real symmetric tridiagonal matrix T. Any such unreduced matrix has
     a well defined set of pairwise different real eigenvalues, the corresponding
     real eigenvectors are pairwise orthogonal.
     The spectrum may be computed either completely or partially by specifying
     either an interval (VL,VU] or a range of indices IL:IU for the desired
     eigenvalues.
     Depending on the number of desired eigenvalues, these are computed either
     by bisection or the dqds algorithm. Numerically orthogonal eigenvectors are
     computed by the use of various suitable L D L^T factorizations near clusters
     of close eigenvalues (referred to as RRRs, Relatively Robust
     Representations). An informal sketch of the algorithm 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.
     For more details, see:
     - 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.
     Further Details
     1.ZSTEMR works only on machines which follow IEEE-754
     floating-point standard in their handling of infinities and NaNs.
     This permits the use of efficient inner loops avoiding a check for
     zero divisors.
     2. LAPACK routines can be used to reduce a complex Hermitean matrix to
     real symmetric tridiagonal form.
     (Any complex Hermitean tridiagonal matrix has real values on its diagonal
     and potentially complex numbers on its off-diagonals. By applying a
     similarity transform with an appropriate diagonal matrix
     diag(1,e^{i \phy_1}, ... , e^{i \phy_{n-1}}), the complex Hermitean
     matrix can be transformed into a real symmetric matrix and complex
     arithmetic can be entirely avoided.)
     While the eigenvectors of the real symmetric tridiagonal matrix are real,
     the eigenvectors of original complex Hermitean matrix have complex entries
     in general.
     Since LAPACK drivers overwrite the matrix data with the eigenvectors,
     ZSTEMR accepts complex workspace to facilitate interoperability
     with ZUNMTR or ZUPMTR.

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.

    N

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

    D

              D is DOUBLE PRECISION array, dimension (N)
              On entry, the N diagonal elements of the tridiagonal matrix
              T. On exit, D is overwritten.

    E

              E is DOUBLE PRECISION array, dimension (N)
              On entry, the (N-1) subdiagonal elements of the tridiagonal
              matrix T in elements 1 to N-1 of E. E(N) need not be set on
              input, but is used internally as workspace.
              On exit, E is overwritten.

    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.
              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.
              Not referenced if RANGE = 'A' or 'V'.

    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', and if INFO = 0, then the first M columns of Z
              contain the orthonormal eigenvectors of the matrix T
              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 can be computed with a workspace
              query by setting NZC = -1, see below.

    LDZ

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

    NZC

              NZC is INTEGER
              The number of eigenvectors to be held in the array Z.
              If RANGE = 'A', then NZC >= max(1,N).
              If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU].
              If RANGE = 'I', then NZC >= IU-IL+1.
              If NZC = -1, then a workspace query is assumed; the
              routine calculates the number of columns of the array Z that
              are needed to hold the eigenvectors.
              This value is returned as the first entry of the Z array, and
              no error message related to NZC is issued by XERBLA.

    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 computed eigenvector
              is nonzero only in elements ISUPPZ( 2*i-1 ) through
              ISUPPZ( 2*i ). This is relevant in the case when the matrix
              is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0.

    TRYRAC

              TRYRAC is LOGICAL
              If TRYRAC = .TRUE., indicates that the code should check whether
              the tridiagonal matrix defines its eigenvalues to high relative
              accuracy.  If so, the code uses relative-accuracy preserving
              algorithms that might be (a bit) slower depending on the matrix.
              If the matrix does not define its eigenvalues to high relative
              accuracy, the code can uses possibly faster algorithms.
              If TRYRAC = .FALSE., the code is not required to guarantee
              relatively accurate eigenvalues and can use the fastest possible
              techniques.
              On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix
              does not define its eigenvalues to high relative accuracy.

    WORK

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

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK. LWORK >= max(1,18*N)
              if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'.
              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 (LIWORK)
              On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

    LIWORK

              LIWORK is INTEGER
              The dimension of the array IWORK.  LIWORK >= max(1,10*N)
              if the eigenvectors are desired, and LIWORK >= max(1,8*N)
              if only the eigenvalues are to be computed.
              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
              On exit, INFO
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  if INFO = 1X, internal error in DLARRE,
                    if INFO = 2X, internal error in ZLARRV.
                    Here, the digit X = ABS( IINFO ) < 10, where IINFO is
                    the nonzero error code returned by DLARRE or
                    ZLARRV, respectively.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Beresford Parlett, University of California, Berkeley, USA
     Jim Demmel, University of California, Berkeley, USA
     Inderjit Dhillon, University of Texas, Austin, USA
     Osni Marques, LBNL/NERSC, USA
     Christof Voemel, University of California, Berkeley, USA
     Aravindh Krishnamoorthy, FAU, Erlangen, Germany

Definition at line 336 of file zstemr.f.

Author

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