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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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