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# dsbevx

```
NAME
DSBEVX - compute selected eigenvalues and, optionally,
eigenvectors of a real symmetric band matrix A

SYNOPSIS
SUBROUTINE DSBEVX( JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q,
LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z,
LDZ, WORK, IWORK, IFAIL, INFO )

CHARACTER      JOBZ, RANGE, UPLO

INTEGER        IL, INFO, IU, KD, LDAB, LDQ, LDZ, M, N

DOUBLE         PRECISION ABSTOL, VL, VU

INTEGER        IFAIL( * ), IWORK( * )

DOUBLE         PRECISION AB( LDAB, * ), Q( LDQ, * ), W(
* ), WORK( * ), Z( LDZ, * )

PURPOSE
DSBEVX computes selected eigenvalues and, optionally, eigen-
vectors of a real symmetric band 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.

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.

KD      (input) INTEGER
The number of superdiagonals of the matrix A if UPLO
= 'U', or the number of subdiagonals if UPLO = 'L'.
KD >= 0.

N)

AB      (input/output) DOUBLE PRECISION array, dimension (LDAB,
On entry, the upper or lower triangle of the sym-
metric band matrix A, stored in the first KD+1 rows
of the array.  The j-th column of A is stored in the
j-th column of the array AB as follows: if UPLO =
'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for
j<=i<=min(n,j+kd).

On exit, AB is overwritten by values generated dur-
ing the reduction to tridiagonal form.  If UPLO =
'U', the first superdiagonal and the diagonal of the
tridiagonal matrix T are returned in rows KD and
KD+1 of AB, and if UPLO = 'L', the diagonal and
first subdiagonal of T are returned in the first two
rows of AB.

LDAB    (input) INTEGER
The leading dimension of the array AB.  LDAB >= KD +
1.

Q       (output) DOUBLE PRECISION array, dimension (LDQ, N)
If JOBZ = 'V', the N-by-N orthogonal matrix used in
the reduction to tridiagonal form.  If JOBZ = 'N',
the array Q is not referenced.

LDQ     (input) INTEGER
The leading dimension of the array Q.  If JOBZ =
'V', then LDQ >= max(1,N).

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.  min(IL,N)
<= 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 obtained by reducing AB 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) DOUBLE PRECISION array, dimension (N)
The first M elements contain the selected eigen-
values in ascending order.

max(1,M))
Z       (output) DOUBLE PRECISION array, dimension (LDZ,
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) DOUBLE PRECISION 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.
```