Provided by: liblapack-doc_3.3.1-1_all

#### NAME

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

```

#### SYNOPSIS

```       SUBROUTINE SSBGVD( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB,  W,  Z,  LDZ,  WORK,  LWORK,
IWORK, LIWORK, INFO )

CHARACTER      JOBZ, UPLO

INTEGER        INFO, KA, KB, LDAB, LDBB, LDZ, LIWORK, LWORK, N

INTEGER        IWORK( * )

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

```

#### PURPOSE

```       SSBGVD  computes  all  the  eigenvalues,  and  optionally,  the  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.  If eigenvectors are
desired, it uses a divide and conquer algorithm.
The divide and conquer algorithm makes very mild assumptions about
floating point arithmetic. It will work on machines with a guard
digit in add/subtract, or on those binary machines without guard
digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
Cray-2. It could conceivably fail on hexadecimal or decimal machines
without guard digits, but we know of none.

```

#### ARGUMENTS

```        JOBZ    (input) CHARACTER*1
= 'N':  Compute eigenvalues only;
= 'V':  Compute eigenvalues and eigenvectors.

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.

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 (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK   (input) INTEGER
The dimension of the array WORK.
If N <= 1,               LWORK >= 1.
If JOBZ = 'N' and N > 1, LWORK >= 3*N.
If JOBZ = 'V' and N > 1, LWORK >= 1 + 5*N + 2*N**2.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal sizes of the WORK and IWORK
arrays, returns these values as the first entries of the WORK
and IWORK arrays, and no error message related to LWORK or
LIWORK is issued by XERBLA.

IWORK   (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
On exit, if LIWORK > 0, IWORK(1) returns the optimal LIWORK.

LIWORK  (input) INTEGER
The dimension of the array IWORK.
If JOBZ  = 'N' or N <= 1, LIWORK >= 1.
If JOBZ  = 'V' and N > 1, LIWORK >= 3 + 5*N.
If LIWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal sizes of the WORK and
IWORK arrays, returns these values as the first entries of
the WORK and IWORK arrays, and no error message related to
LWORK or LIWORK is issued by XERBLA.

INFO    (output) INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value
> 0:  if INFO = i, and i is:
<= N:  the algorithm failed to converge:
i off-diagonal elements of an intermediate
tridiagonal form did not converge to zero;
> N:   if INFO = N + i, for 1 <= i <= N, then SPBSTF
returned INFO = i: 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                            SSBGVD(3lapack)
```