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# LAPACK-3

## man page of LAPACK-3

### LAPACK-3: computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite

```NAME
LAPACK-3   -   computes   all  the  eigenvalues,  and  optionally,  the
eigenvectors of a complex generalized Hermitian-definite  eigenproblem,
of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x

SYNOPSIS
SUBROUTINE ZHEGV( ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W, WORK, LWORK,
RWORK, INFO )

CHARACTER     JOBZ, UPLO

INTEGER       INFO, ITYPE, LDA, LDB, LWORK, N

DOUBLE        PRECISION RWORK( * ), W( * )

COMPLEX*16    A( LDA, * ), B( LDB, * ), WORK( * )

PURPOSE
ZHEGV computes all the eigenvalues, and optionally, the eigenvectors of
a  complex  generalized  Hermitian-definite  eigenproblem,  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 Hermitian 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 matrices A and B.  N >= 0.

A       (input/output) COMPLEX*16 array, dimension (LDA, N)
On entry, the Hermitian matrix A.  If UPLO = 'U', the
leading N-by-N upper triangular part of A contains 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, if JOBZ = 'V', then if INFO = 0, A contains the
matrix Z of eigenvectors.  The eigenvectors are normalized
as follows:
if ITYPE = 1 or 2, Z**H*B*Z = I;
if ITYPE = 3, Z**H*inv(B)*Z = I.
If JOBZ = 'N', then on exit the upper triangle (if UPLO='U')
or the lower triangle (if UPLO='L') of A, including the
diagonal, is destroyed.

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

B       (input/output) COMPLEX*16 array, dimension (LDB, N)
On entry, the Hermitian positive definite matrix B.
If UPLO = 'U', the leading N-by-N upper triangular part of B
contains the upper triangular part of the matrix B.
If UPLO = 'L', the leading N-by-N lower triangular part of B
contains the lower triangular part of the matrix B.
On exit, if INFO <= N, the part of B containing the matrix is
overwritten by the triangular factor U or L from the Cholesky
factorization B = U**H*U or B = L*L**H.

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

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

WORK    (workspace/output) COMPLEX*16 array, dimension (MAX(1,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 ZHETRD returned by ILAENV.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.

RWORK   (workspace) DOUBLE PRECISION array, dimension (max(1, 3*N-2))

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

LAPACK driver routine (version 3.March 2011                       LAPACK-3(3)
```

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