NAME

DSTEVX - compute selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix A

SYNOPSIS

SUBROUTINE DSTEVX(
JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO )

CHARACTER JOBZ, RANGE

INTEGER IL, INFO, IU, LDZ, M, N

DOUBLE PRECISION ABSTOL, VL, VU

INTEGER IFAIL( ∗ ), IWORK( ∗ )

DOUBLE PRECISION D( ∗ ), E( ∗ ), W( ∗ ), WORK( ∗ ), Z( LDZ, ∗ )

PURPOSE

DSTEVX computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix A.  Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. 
 

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 A. On exit, D may be multiplied by a constant factor chosen to avoid over/underflow in computing the eigenvalues.

E       (input/output) DOUBLE PRECISION array, dimension (N)
On entry, the (n-1) subdiagonal elements of the tridiagonal matrix A in elements 1 to N-1 of E; E(N) need not be set. On exit, E may be multiplied by a constant factor chosen to avoid over/underflow in computing the eigenvalues.

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. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width less than or equal to
 
ABSTOL + EPS ∗   max( |a|,|b| ) ,
 
where EPS is the machine precision.  If ABSTOL is less than or equal to zero, then  EPS∗|T|  will be used in its place, where |T| is the 1-norm of the tridiagonal matrix.
 
Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2∗DLAMCH(’S’), not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2∗DLAMCH(’S’).
 
See "Computing Small Singular Values of Bidiagonal Matrices with Guaranteed High Relative Accuracy," by Demmel and Kahan, LAPACK Working Note #3.

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 A corresponding to the selected eigenvalues, with the i-th column of Z holding the eigenvector associated with W(i). If an eigenvector fails to converge (INFO > 0), then that column of Z contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in IFAIL.  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).

WORK    (workspace) DOUBLE PRECISION array, dimension (5∗N)

IWORK   (workspace) INTEGER array, dimension (5∗N)

IFAIL   (output) INTEGER array, dimension (N)
If JOBZ = ’V’, then if INFO = 0, the first M elements of IFAIL are zero.  If INFO > 0, then IFAIL contains the indices of the eigenvectors that failed to converge. If JOBZ = ’N’, then IFAIL is not referenced.

INFO    (output) INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value
> 0:  if INFO = i, then i eigenvectors failed to converge. Their indices are stored in array IFAIL.

  —  LAPACK version 2.0  —  08 October 1994