zhegvx man page on Scientific

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ZHEGVX(1)	      LAPACK driver routine (version 3.2)	     ZHEGVX(1)

NAME
       ZHEGVX - computes selected eigenvalues, and optionally, 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 ZHEGVX( ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB, VL, VU,
			  IL, IU, ABSTOL, M, W, Z, LDZ,	 WORK,	LWORK,	RWORK,
			  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 RWORK( * ), W( * )

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

PURPOSE
       ZHEGVX computes selected eigenvalues, and optionally, 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.  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 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,  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) COMPLEX*16 array, dimension (LDB, N)
	       On entry, the Hermitian 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).

       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)
	       The first  M  elements  contain	the  selected  eigenvalues  in
	       ascending order.

       Z       (output) COMPLEX*16 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) 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).  For	 opti‐
	       mal  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 (7*N)

       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:  ZPOTRF or ZHEEVX returned an error code:
	       <=  N:	if INFO = i, ZHEEVX 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			     ZHEGVX(1)
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