ZSTEGR - compute selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix T

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, * )

ZSTEGR 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 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 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) 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 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 ZLARRV.

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