Provided by: liblapack-doc_3.3.1-1_all

#### NAME

```       LAPACK-3  -  computes  selected  eigenvalues,  and  optionally,  eigenvectors  of  a  real
generalized symmetric-definite banded eigenproblem, of the form A*x=(lambda)*B*x

```

#### SYNOPSIS

```       SUBROUTINE SSBGVX( JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Q, LDQ, VL,  VU,  IL,
IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO )

CHARACTER      JOBZ, RANGE, UPLO

INTEGER        IL, INFO, IU, KA, KB, LDAB, LDBB, LDQ, LDZ, M, N

REAL           ABSTOL, VL, VU

INTEGER        IFAIL( * ), IWORK( * )

REAL           AB( LDAB, * ), BB( LDBB, * ), Q( LDQ, * ), W( * ), WORK( * ), Z( LDZ, *
)

```

#### PURPOSE

```       SSBGVX computes selected eigenvalues, and optionally, eigenvectors of a  real  generalized
symmetric-definite  banded  eigenproblem,  of the form A*x=(lambda)*B*x.  Here A and B are
assumed to be symmetric
and banded, and B is also positive definite.  Eigenvalues and
eigenvectors can be selected by specifying either all eigenvalues,
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 triangles of A and B are stored;
= 'L':  Lower triangles of A and B are stored.

N       (input) INTEGER
The order of the matrices A and B.  N >= 0.

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

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

AB      (input/output) REAL array, dimension (LDAB, N)
On entry, the upper or lower triangle of the symmetric band
matrix A, stored in the first ka+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(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).
On exit, the contents of AB are destroyed.

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

BB      (input/output) REAL array, dimension (LDBB, N)
On entry, the upper or lower triangle of the symmetric band
matrix B, stored in the first kb+1 rows of the array.  The
j-th column of B is stored in the j-th column of the array BB
as follows:
if UPLO = 'U', BB(ka+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
if UPLO = 'L', BB(1+i-j,j)    = B(i,j) for j<=i<=min(n,j+kb).
On exit, the factor S from the split Cholesky factorization
B = S**T*S, as returned by SPBSTF.

LDBB    (input) INTEGER
The leading dimension of the array BB.  LDBB >= KB+1.

Q       (output) REAL array, dimension (LDQ, N)
If JOBZ = 'V', the n-by-n matrix used in the reduction of
A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x,
and consequently C to tridiagonal form.
If JOBZ = 'N', the array Q is not referenced.

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

VL      (input) REAL
VU      (input) REAL
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) 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 tridiagonal form.
Eigenvalues will be computed most accurately when ABSTOL is
set to twice the underflow threshold 2*SLAMCH('S'), not zero.
If this routine returns with INFO>0, indicating that some
eigenvectors did not converge, try setting ABSTOL to
2*SLAMCH('S').

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)
If INFO = 0, the eigenvalues in ascending order.

Z       (output) REAL array, dimension (LDZ, N)
If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
eigenvectors, with the i-th column of Z holding the
eigenvector associated with W(i).  The eigenvectors are
normalized so Z**T*B*Z = I.
If JOBZ = 'N', then Z is not referenced.

LDZ     (input) INTEGER
The leading dimension of the array Z.  LDZ >= 1, and if
JOBZ = 'V', LDZ >= max(1,N).

WORK    (workspace/output) REAL array, dimension (7N)

IWORK   (workspace/output) INTEGER array, dimension (5N)

IFAIL   (output) INTEGER array, dimension (M)
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 eigenvalues 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
<= N: if INFO = i, then i eigenvectors failed to converge.
Their indices are stored in IFAIL.
> N : SPBSTF returned an error code; i.e.,
if INFO = N + i, for 1 <= i <= N, then the leading
minor of order i of B is not positive definite.
The factorization of B could not be completed and
no eigenvalues or eigenvectors were computed.

```

#### FURTHERDETAILS

```        Based on contributions by
Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

LAPACK driver routine (version 3.2)        April 2011                            SSBGVX(3lapack)
```