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# cheevx

```
NAME
CHEEVX - compute selected eigenvalues and, optionally,
eigenvectors of a complex Hermitian matrix A

SYNOPSIS
SUBROUTINE CHEEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL,
IU, ABSTOL, M, W, Z, LDZ, WORK, LWORK,
RWORK, IWORK, IFAIL, INFO )

CHARACTER      JOBZ, RANGE, UPLO

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

REAL           ABSTOL, VL, VU

INTEGER        IFAIL( * ), IWORK( * )

REAL           RWORK( * ), W( * )

COMPLEX        A( LDA, * ), WORK( * ), Z( LDZ, * )

PURPOSE
CHEEVX computes selected eigenvalues and, optionally, eigen-
vectors of a complex Hermitian 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.

UPLO    (input) CHARACTER*1
= 'U':  Upper triangle of A is stored;
= 'L':  Lower triangle of A is stored.

N       (input) INTEGER
The order of the matrix A.  N >= 0.

A       (input/workspace) COMPLEX array, dimension (LDA, N)
On entry, the Hermitian matrix A.  If UPLO = 'U',
the leading N-by-N upper triangular part of A con-
tains the upper triangular part of the matrix A.  If
UPLO = 'L', the leading N-by-N lower triangular part
of A contains the lower triangular part of the

matrix A.  On exit, the lower triangle (if UPLO='L')
or the upper triangle (if UPLO='U') of A, including
the diagonal, is destroyed.

LDA     (input) INTEGER
The leading dimension of the array A.  LDA >=
max(1,N).

VL      (input) REAL
If RANGE='V', the lower bound of the interval to be
searched for eigenvalues.  Not referenced if RANGE =
'A' or 'I'.

VU      (input) REAL
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.  min(IL,N)
<= IU <= N.  Not referenced if RANGE = 'A' or 'V'.

ABSTOL  (input) REAL
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 obtained by reducing A to tridi-
agonal form.

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) REAL array, dimension (N)

On normal exit, the first M entries contain the
selected eigenvalues in ascending order.

Z       (output) COMPLEX 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 corresponding to the selected eigenvalues.
If an eigenvector fails to converge, 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) COMPLEX array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal
LWORK.

LWORK   (input) INTEGER
The length of the array WORK.  LWORK >= max(1,2*N-
1).  For optimal efficiency, LWORK >= (NB+1)*N,
where NB is the blocksize for CHETRD returned by
ILAENV.

RWORK   (workspace) REAL array, dimension (7*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.
```