zstegr(3)

NAME

ZSTEGR - compute selected eigenvalues and, optionally,

eigenvectors of a real symmetric tridiagonal matrix T

SYNOPSIS

SUBROUTINE  ZSTEGR(  JOBZ, RANGE, N, D, E, VL, VU, IL, IU,
ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO )
    CHARACTER      JOBZ, RANGE
    INTEGER        IL, INFO, IU, LDZ, LIWORK, LWORK, M, N
    DOUBLE         PRECISION ABSTOL, VL, VU
    INTEGER        ISUPPZ( * ), IWORK( * )
    DOUBLE         PRECISION D( * ), E( * ), W( * ), WORK(
* )
    COMPLEX*16     Z( LDZ, * )

PURPOSE

ZSTEGR computes selected eigenvalues and, optionally,

eigenvectors of a real symmetric tridiagonal matrix T. Eigenval

ues 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 sym

metric tridiagonal eigenvalue/eigenvector problem", by Inderjit

Dhillon, Computer Science Division Technical Report No.

UCB/CSD-97-971, UC Berkeley, May 1997.

Note 1 : Currently ZSTEGR is only set up to find ALL the n

eigenvalues and eigenvectors of T in O(n^2) time
Note 2 : Currently the routine ZSTEIN is called when an

appropriate sigma_i cannot be chosen in step (c) above. ZSTEIN

invokes modified Gram-Schmidt when eigenvalues are close.
Note 3 : ZSTEGR works only on machines which follow

ieee-754 floating-point standard in their handling of infinities

and NaNs. Normal execution of ZSTEGR may create NaNs and infini

ties and hence may abort due to a floating point exception in en

vironments which do not conform to the ieee standard.

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 eigenval

ues 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 tridiago

nal 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 eigen

values. 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 eigenval

ues/eigenvectors. IF JOBZ = 'V', the eigenvalues and eigenvectors

output have residual norms bounded by ABSTOL, and the dot prod

ucts between different eigenvectors are bounded by ABSTOL. If AB

STOL 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 ac

curacy of EPS*|T| irrespective of ABSTOL. If high relative accu

racy is important, set ABSTOL to DLAMCH( 'Safe minimum' ). See

Barlow and Demmel "Computing Accurate Eigensystems of Scaled Di

agonally Dominant Matrices", LAPACK Working Note #7 for a discus

sion 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 eigen

values in ascending order.

Z (output) COMPLEX*16 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 eigenvec

tor is nonzero only in elements ISUPPZ( 2*i-1 ) through ISUPPZ(

2*i ).

WORK (workspace/output) DOUBLE PRECISION array, dimen

sion (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 (LI

WORK)

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 ille

gal value
> 0: if INFO = 1, internal error in DLARRE, if

INFO = 2, internal error in ZLARRV.

FURTHER DETAILS

Based on contributions by

Inderjit Dhillon, IBM Almaden, USA
Osni Marques, LBNL/NERSC, USA
Ken Stanley, Computer Science Division, University of

California at Berkeley, USA

LAPACK computational version 3.0 15 June 2000

ZSTEGR(3)