CSTEMR(1) LAPACK computational routine (version 3.2) CSTEMR(1)NAME
CSTEMR - computes selected eigenvalues and, optionally, eigenvectors of
a real symmetric tridiagonal matrix T
SYNOPSIS
SUBROUTINE CSTEMR( 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
REAL VL, VU
INTEGER ISUPPZ( * ), IWORK( * )
REAL D( * ), E( * ), W( * ), WORK( * )
COMPLEX Z( LDZ, * )
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 corre‐
sponding real eigenvectors are pairwise orthogonal.
The spectrum may be computed either completely or partially by specify‐
ing 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 fac‐
torizations 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 com‐
pute
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 representa‐
tions
to compute orthogonal eigenvectors of symmetric tridiagonal matri‐
ces,"
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 diago‐
nal 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 com‐
plex entries in general.
Since LAPACK drivers overwrite the matrix data with the eigenvectors,
CSTEMR accepts complex workspace to facilitate interoperability with
CUNMTR or CUPMTR.
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) REAL array, dimension (N)
On entry, the N diagonal elements of the tridiagonal matrix T.
On exit, D is overwritten.
E (input/output) 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 (input) REAL
VU (input) 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 (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) REAL array, dimension (N)
The first M elements contain the selected eigenvalues in
ascending order.
Z (output) 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 corre‐
sponding 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 indi‐
cating 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 pre‐
serving 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 algo‐
rithms. 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) REAL 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 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.
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
LAPACK computational routine (veNovember22008 CSTEMR(1)
