SYNOPSIS

SUBROUTINE ZSTEMR(

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

    

IMPLICIT NONE

    

CHARACTER JOBZ, RANGE

    

LOGICAL TRYRAC

    

INTEGER IL, INFO, IU, LDZ, NZC, LIWORK, LWORK, M, N

    

DOUBLE PRECISION VL, VU

    

INTEGER ISUPPZ( * ), IWORK( * )

    

DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * )

    

COMPLEX*16 Z( LDZ, * )

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
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.

ARGUMENTS

JOBZ (input) CHARACTER*1

= 'N': Compute eigenvalues only;
= 'V': Compute eigenvalues and eigenvectors.

RANGE (input) 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 (input) INTEGER

The order of the matrix. N >= 0.

D (input/output) DOUBLE PRECISION array, dimension (N)

On entry, the N diagonal elements of the tridiagonal matrix T. On exit, D is overwritten.

E (input/output) 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 (input) DOUBLE PRECISION

VU (input) 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 (input) INTEGER

IU (input) 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 (output) INTEGER

The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

W (output) DOUBLE PRECISION array, dimension (N)

The first M elements contain the selected eigenvalues in ascending order.

Z (output) 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 (input) INTEGER

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

NZC (input) 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 (output) 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 (input/output) 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 (workspace/output) DOUBLE PRECISION array, dimension (LWORK)

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

LWORK (input) 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 (workspace/output) INTEGER array, dimension (LIWORK)

On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

LIWORK (input) 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 (output) 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.

FURTHER DETAILS

Based on contributions by

   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