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# dstevx

```
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, eigen-
vectors of a real symmetric tridiagonal matrix A.
Eigenvalues/vectors can be selected by specifying either a
range of values or a range of indices for the desired eigen-
values.

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 con-
stant factor chosen to avoid over/underflow in com-
puting the eigenvalues.

E       (input/output) DOUBLE PRECISION array, dimension (N)
On entry, the (n-1) subdiagonal elements of the tri-
diagonal 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
If RANGE='V', the lower bound of the interval to be
searched for eigenvalues.  Not referenced if RANGE =
'A' or 'I'.

VU      (input) DOUBLE PRECISION
If RANGE='V', the upper bound of the interval to be
searched for eigenvalues.  Not referenced if RANGE =
'A' or 'I'.

IL      (input) INTEGER
If RANGE='I', the index (from smallest to largest)
of the smallest eigenvalue to be returned.  IL >= 1.
Not referenced if RANGE = 'A' or 'V'.

IU      (input) INTEGER
If RANGE='I', the index (from smallest to largest)
of the largest eigenvalue to be returned.  IL <= IU
<= N.  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.

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)
On normal exit, the first M entries 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 ele-
ments 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.
```