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# dspgv

```
NAME
DSPGV - compute all the eigenvalues and, optionally, the
eigenvectors of a real generalized symmetric-definite eigen-
problem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or
B*A*x=(lambda)*x

SYNOPSIS
SUBROUTINE DSPGV( ITYPE, JOBZ, UPLO, N, AP, BP, W, Z, LDZ,
WORK, INFO )

CHARACTER     JOBZ, UPLO

INTEGER       INFO, ITYPE, LDZ, N

DOUBLE        PRECISION AP( * ), BP( * ), W( * ), WORK(
* ), Z( LDZ, * )

PURPOSE
DSPGV computes all the eigenvalues and, optionally, the
eigenvectors of a real generalized symmetric-definite eigen-
problem, of the form A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or
B*A*x=(lambda)*x.  Here A and B are assumed to be symmetric,
stored in packed format, and B is also positive definite.

ARGUMENTS
ITYPE   (input) INTEGER
Specifies the problem type to be solved:
= 1:  A*x = (lambda)*B*x
= 2:  A*B*x = (lambda)*x
= 3:  B*A*x = (lambda)*x

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 matrix A.  N >= 0.

AP      (input/workspace) DOUBLE PRECISION array, dimension
(N*(N+1)/2) On entry, the upper or lower triangle of
the symmetric matrix A, packed columnwise in a
linear array.  The j-th column of A is stored in the
array AP as follows: if UPLO = 'U', AP(i + (j-
1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i +
(j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

On exit, the contents of AP are destroyed.

(N*(N+1)/2)
BP      (input/output) DOUBLE PRECISION array, dimension
On entry, the upper or lower triangle of the sym-
metric matrix B, packed columnwise in a linear
array.  The j-th column of B is stored in the array
BP as follows: if UPLO = 'U', BP(i + (j-1)*j/2) =
B(i,j) for 1<=i<=j; if UPLO = 'L', BP(i + (j-
1)*(2n-j)/2) = B(i,j) for j<=i<=n.

On exit, the triangular factor U or L from the
Cholesky factorization B = U**T*U or B = L*L**T, in
the same storage format as B.

W       (output) DOUBLE PRECISION array, dimension (N)
If INFO = 0, the eigenvalues in ascending order.

Z       (output) DOUBLE PRECISION array, dimension (LDZ, N)
If JOBZ = 'V', then if INFO = 0, Z contains the
matrix Z of eigenvectors.  The eigenvectors are nor-
malized as follows: if ITYPE = 1 or 2, Z**T*B*Z = I;
if ITYPE = 3, Z**T*inv(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) DOUBLE PRECISION array, dimension (3*N)

INFO    (output) INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal
value
> 0:  DPPTRF or DSPEV returned an error code:
<= N:  if INFO = i, DSPEV 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 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.
```