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zstemr.f(3) LAPACK zstemr.f(3)

zstemr.f -


subroutine zstemr (JOBZ, RANGE, N, D, E, VL, VU, IL, IU, M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK, IWORK, LIWORK, INFO)
 
ZSTEMR

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 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, 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
VU
          VU is DOUBLE PRECISION
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
          IL is INTEGER
IU
          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
          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.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
          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.
Date:
November 2013
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
 
Definition at line 328 of file zstemr.f.

Generated automatically by Doxygen for LAPACK from the source code.
Sat Nov 16 2013 Version 3.4.2

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