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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 ZHPGVD( ITYPE,  JOBZ,  UPLO,  N,  AP,  BP,  W, Z, LDZ, WORK,
LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO )

CHARACTER      JOBZ, UPLO

INTEGER        INFO, ITYPE, LDZ, LIWORK, LRWORK, LWORK, N

INTEGER        IWORK( * )

DOUBLE         PRECISION RWORK( * ), W( * )

COMPLEX*16     AP( * ), BP( * ), WORK( * ), Z( LDZ, * )

PURPOSE
ZHPGVD 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, stored in packed format, 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
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.

AP      (input/output) COMPLEX*16 array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the Hermitian 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)*(2*n-j)/2) = A(i,j) for j<=i<=n.
On exit, the contents of AP are destroyed.

BP      (input/output) COMPLEX*16 array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the Hermitian 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)*(2*n-j)/2) = B(i,j) for j<=i<=n.
On exit, the triangular factor U or L from the Cholesky
factorization B = U**H*U or B = L*L**H, in the same storage
format as B.

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

Z       (output) COMPLEX*16 array, dimension (LDZ, N)
If JOBZ = 'V', then if INFO = 0, Z 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 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) COMPLEX*16 array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the required LWORK.

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

RWORK   (workspace) DOUBLE PRECISION array, dimension (MAX(1,LRWORK))
On exit, if INFO = 0, RWORK(1) returns the required LRWORK.

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

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

LIWORK  (input) INTEGER
The dimension of 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 required sizes of the WORK, RWORK
and IWORK arrays, returns these values as the first entries
of the WORK, RWORK and IWORK arrays, and no error message
related to LWORK or LRWORK 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:  ZPPTRF or ZHPEVD returned an error code:
<= N:  if INFO = i, ZHPEVD failed to converge;
i off-diagonal elements of an intermediate
tridiagonal form did not convergeto 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.

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

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