cstemr.f (3) - Linux Manuals


cstemr.f -




Function/Subroutine Documentation

subroutine cstemr (characterJOBZ, characterRANGE, integerN, real, dimension( * )D, real, dimension( * )E, realVL, realVU, integerIL, integerIU, integerM, real, dimension( * )W, complex, dimension( ldz, * )Z, integerLDZ, integerNZC, integer, dimension( * )ISUPPZ, logicalTRYRAC, real, dimension( * )WORK, integerLWORK, integer, dimension( * )IWORK, integerLIWORK, integerINFO)



 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

 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
hy_1}, ... , e^{i
hy_{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.




          JOBZ is CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.


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


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


          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 is REAL


          VU is REAL

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


          IL is INTEGER


          IU is INTEGER

          If RANGE='I', the indices (in ascending order) of the
          smallest and largest eigenvalues to be returned.
          1 <= IL <= IU <= N, if N > 0.
          Not referenced if RANGE = 'A' or 'V'.


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


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


          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 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 is LOGICAL
          If TRYRAC.EQ..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.EQ..FALSE., the code is not required to guarantee
          relatively accurate eigenvalues and can use the fastest possible
          On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix
          does not define its eigenvalues to high relative accuracy.


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



Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.


September 2012


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 

Definition at line 328 of file cstemr.f.


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