dsygvx man page on Scientific

```DSYGVX(1)	      LAPACK driver routine (version 3.2)	     DSYGVX(1)

NAME
DSYGVX - computes selected eigenvalues, and optionally, eigenvectors of
a  real	generalized  symmetric-definite	 eigenproblem,	of  the	  form
A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x

SYNOPSIS
SUBROUTINE DSYGVX( ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB, VL, VU,
IL, IU, ABSTOL, M, W, Z, LDZ,	 WORK,	LWORK,	IWORK,
IFAIL, INFO )

CHARACTER	  JOBZ, RANGE, UPLO

INTEGER	  IL, INFO, ITYPE, IU, LDA, LDB, LDZ, LWORK, M, N

DOUBLE	  PRECISION ABSTOL, VL, VU

INTEGER	  IFAIL( * ), IWORK( * )

DOUBLE	  PRECISION  A( LDA, * ), B( LDB, * ), W( * ), WORK( *
), Z( LDZ, * )

PURPOSE
DSYGVX computes selected eigenvalues, and optionally, eigenvectors of a
real   generalized   symmetric-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 symmetric and B is also positive definite.  Eigenval‐
ues and eigenvectors can be selected by specifying either  a  range  of
values or a range of indices for the desired eigenvalues.

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.

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 triangle of A and B are stored;
= 'L':  Lower triangle of A and B are stored.

N       (input) INTEGER
The order of the matrix pencil (A,B).  N >= 0.

A       (input/output) DOUBLE PRECISION array, dimension (LDA, N)
On  entry,  the symmetric 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,  the lower triangle (if UPLO='L') or the
upper triangle (if UPLO='U') of A, including the	 diagonal,  is
destroyed.

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

B       (input/output) DOUBLE PRECISION array, dimension (LDA, N)
On  entry,  the symmetric 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**T*U or B = L*L**T.

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

VL      (input) DOUBLE PRECISION
VU	(input)	 DOUBLE	 PRECISION 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) DOUBLE PRECISION
The absolute error tolerance for the eigenvalues.  An  approxi‐
mate  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 pre‐
cision.	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*DLAMCH('S'), not  zero.   If
this  routine  returns with INFO>0, indicating that some eigen‐
vectors did not converge, try setting ABSTOL to 2*DLAMCH('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) DOUBLE PRECISION array, dimension (N)
On  normal  exit, the first M elements contain the selected ei‐
genvalues in ascending order.

Z       (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M))
If JOBZ = 'N', then Z is not referenced.	 If JOBZ =  'V',  then
if  INFO	 = 0, the first M columns of Z contain the orthonormal
eigenvectors of the matrix A corresponding to the selected  ei‐
genvalues,  with	 the  i-th column of Z holding the eigenvector
associated with W(i).  The eigenvectors are normalized as  fol‐
lows:  if  ITYPE	 =  1  or  2,  Z**T*B*Z	 =  I;	if  ITYPE = 3,
Z**T*inv(B)*Z = I.  If an eigenvector fails to  converge,  then
that  column  of	 Z  contains  the  latest approximation to the
eigenvector, and the index of the eigenvector  is  returned  in
IFAIL.	Note: the user must ensure that at least max(1,M) col‐
umns are supplied in the array Z; if RANGE  =  'V',  the	 exact
value  of  M is not known in advance and an upper bound must be
used.

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

WORK	  (workspace/output)   DOUBLE	PRECISION   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(1,8*N).  For	 opti‐
mal  efficiency,	 LWORK	>= (NB+3)*N, where NB is the blocksize
for DSYTRD 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.

IWORK   (workspace) INTEGER array, dimension (5*N)

IFAIL   (output) INTEGER array, dimension (N)
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
eigenvectors 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
> 0:  DPOTRF or DSYEVX returned an error code:
<=  N:	if INFO = i, DSYEVX failed to converge; i eigenvectors
failed to converge.  Their indices are stored in	 array	IFAIL.
> 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 3November 2008			     DSYGVX(1)
```
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