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GNU/Linux |
CentOS 4.8 |
i386 |
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dstegr(l) |
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DSTEGR - compute selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix T
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SUBROUTINE DSTEGR( |
JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO ) | ||
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CHARACTER |
JOBZ, RANGE | ||
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INTEGER |
IL, INFO, IU, LDZ, LIWORK, LWORK, M, N | ||
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DOUBLE |
PRECISION ABSTOL, VL, VU | ||
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INTEGER |
ISUPPZ( * ), IWORK( * ) | ||
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DOUBLE |
PRECISION D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * ) |
DSTEGR computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix T. Eigenvalues and
(a) Compute T -
sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T
is a relatively robust representation,
(b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to
high
relative accuracy by the dqds algorithm,
(c) If there is a cluster of close eigenvalues,
"choose" sigma_i
close to the cluster, and go to step (a),
(d) Given the approximate eigenvalue lambda_j of L_i D_i
L_i^T,
compute the corresponding eigenvector by forming a
rank-revealing twisted factorization.
The desired accuracy of the output can be specified by the
input parameter ABSTOL.
For more details, see "A new O(n^2) algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem", by Inderjit Dhillon, Computer Science Division Technical Report No. UCB/CSD-97-971, UC Berkeley, May 1997.
Note 1 :
Currently DSTEGR is only set up to find ALL the n
eigenvalues and eigenvectors of T in O(n^2) time
Note 2 : Currently the routine DSTEIN is called when an
appropriate sigma_i cannot be chosen in step (c) above.
DSTEIN invokes modified Gram-Schmidt when eigenvalues are
close.
Note 3 : DSTEGR works only on machines which follow ieee-754
floating-point standard in their handling of infinities and
NaNs. Normal execution of DSTEGR may create NaNs and
infinities and hence may abort due to a floating point
exception in environments which do not conform to the ieee
standard.
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 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; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = ’A’ or ’V’.
ABSTOL (input) DOUBLE PRECISION
The absolute error tolerance for the eigenvalues/eigenvectors. IF JOBZ = ’V’, the eigenvalues and eigenvectors output have residual norms bounded by ABSTOL, and the dot products between different eigenvectors are bounded by ABSTOL. If ABSTOL is less than N*EPS*|T|, then N*EPS*|T| will be used in its place, where EPS is the machine precision and |T| is the 1-norm of the tridiagonal matrix. The eigenvalues are computed to an accuracy of EPS*|T| irrespective of ABSTOL. If high relative accuracy is important, set ABSTOL to DLAMCH( ’Safe minimum’ ). See Barlow and Demmel "Computing Accurate Eigensystems of Scaled Diagonally Dominant Matrices", LAPACK Working Note #7 for a discussion of which matrices define their eigenvalues to high relative accuracy.
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) DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
If JOBZ = ’V’, then if INFO = 0, 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 an upper bound must be used.
LDZ (input) INTEGER
The leading dimension of the array Z. LDZ >= 1, and if JOBZ = ’V’, LDZ >= max(1,N).
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 eigenvector is nonzero only in elements ISUPPZ( 2*i-1 ) through ISUPPZ( 2*i ).
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 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 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
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = 1, internal error in DLARRE, if INFO = 2,
internal error in DLARRV.
Based on
contributions by
Inderjit Dhillon, IBM Almaden, USA
Osni Marques, LBNL/NERSC, USA
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dstegr(l) | |