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CHPGVX(3S)							    CHPGVX(3S)

NAME
     CHPGVX - compute 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 CHPGVX( ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU, IL, IU,
			ABSTOL, M, W, Z, LDZ, WORK, RWORK, IWORK, IFAIL, INFO
			)

	 CHARACTER	JOBZ, RANGE, UPLO

	 INTEGER	IL, INFO, ITYPE, IU, LDZ, M, N

	 REAL		ABSTOL, VL, VU

	 INTEGER	IFAIL( * ), IWORK( * )

	 REAL		RWORK( * ), W( * )

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

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

PURPOSE
     CHPGVX 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, stored in packed format, and B is also positive
     definite.	Eigenvalues 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

									Page 1

CHPGVX(3S)							    CHPGVX(3S)

     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.

     AP	     (input/output) COMPLEX 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 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.

     VL	     (input) REAL
	     VU	     (input) REAL 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) REAL
	     The absolute error tolerance for the eigenvalues.	An approximate
	     eigenvalue is accepted as converged when it is determined to lie
	     in an interval [a,b] of width less than or equal to

									Page 2

CHPGVX(3S)							    CHPGVX(3S)

	     ABSTOL + EPS *   max( |a|,|b| ) ,

	     where EPS is the machine precision.  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
	     AP to tridiagonal form.

	     Eigenvalues will be computed most accurately when ABSTOL is set
	     to twice the underflow threshold 2*SLAMCH('S'), not zero.	If
	     this routine returns with INFO>0, indicating that some
	     eigenvectors did not converge, try setting ABSTOL to
	     2*SLAMCH('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) REAL array, dimension (N)
	     On normal exit, the first M elements contain the selected
	     eigenvalues in ascending order.

     Z	     (output) COMPLEX array, dimension (LDZ, N)
	     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
	     eigenvalues, with the i-th column of Z holding the eigenvector
	     associated with W(i).  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 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) columns 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) COMPLEX array, dimension (2*N)

     RWORK   (workspace) REAL 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.

									Page 3

CHPGVX(3S)							    CHPGVX(3S)

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

SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

     This man page is available only online.

									Page 4

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