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ZGEGV(3F)							     ZGEGV(3F)

NAME
     ZGEGV - compute for a pair of N-by-N complex nonsymmetric matrices A and
     B, the generalized eigenvalues (alpha, beta), and optionally,

SYNOPSIS
     SUBROUTINE ZGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA, VL, LDVL,
		       VR, LDVR, WORK, LWORK, RWORK, INFO )

	 CHARACTER     JOBVL, JOBVR

	 INTEGER       INFO, LDA, LDB, LDVL, LDVR, LWORK, N

	 DOUBLE	       PRECISION RWORK( * )

	 COMPLEX*16    A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ), VL(
		       LDVL, * ), VR( LDVR, * ), WORK( * )

PURPOSE
     ZGEGV computes for a pair of N-by-N complex nonsymmetric matrices A and
     B, the generalized eigenvalues (alpha, beta), and optionally, the left
     and/or right generalized eigenvectors (VL and VR).

     A generalized eigenvalue for a pair of matrices (A,B) is, roughly
     speaking, a scalar w or a ratio  alpha/beta = w, such that	 A - w*B is
     singular.	It is usually represented as the pair (alpha,beta), as there
     is a reasonable interpretation for beta=0, and even for both being zero.
     A good beginning reference is the book, "Matrix Computations", by G.
     Golub & C. van Loan (Johns Hopkins U. Press)

     A right generalized eigenvector corresponding to a generalized eigenvalue
     w	for a pair of matrices (A,B) is a vector  r  such that	(A - w B) r =
     0 .  A left generalized eigenvector is a vector l such that l**H * (A - w
     B) = 0, where l**H is the
     conjugate-transpose of l.

     Note: this routine performs "full balancing" on A and B -- see "Further
     Details", below.

ARGUMENTS
     JOBVL   (input) CHARACTER*1
	     = 'N':  do not compute the left generalized eigenvectors;
	     = 'V':  compute the left generalized eigenvectors.

     JOBVR   (input) CHARACTER*1
	     = 'N':  do not compute the right generalized eigenvectors;
	     = 'V':  compute the right generalized eigenvectors.

     N	     (input) INTEGER
	     The order of the matrices A, B, VL, and VR.  N >= 0.

									Page 1

ZGEGV(3F)							     ZGEGV(3F)

     A	     (input/output) COMPLEX*16 array, dimension (LDA, N)
	     On entry, the first of the pair of matrices whose generalized
	     eigenvalues and (optionally) generalized eigenvectors are to be
	     computed.	On exit, the contents will have been destroyed.	 (For
	     a description of the contents of A on exit, see "Further
	     Details", below.)

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

     B	     (input/output) COMPLEX*16 array, dimension (LDB, N)
	     On entry, the second of the pair of matrices whose generalized
	     eigenvalues and (optionally) generalized eigenvectors are to be
	     computed.	On exit, the contents will have been destroyed.	 (For
	     a description of the contents of B on exit, see "Further
	     Details", below.)

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

     ALPHA   (output) COMPLEX*16 array, dimension (N)
	     BETA    (output) COMPLEX*16 array, dimension (N) On exit,
	     ALPHA(j)/BETA(j), j=1,...,N, will be the generalized eigenvalues.

	     Note: the quotients ALPHA(j)/BETA(j) may easily over- or
	     underflow, and BETA(j) may even be zero.  Thus, the user should
	     avoid naively computing the ratio alpha/beta.  However, ALPHA
	     will be always less than and usually comparable with norm(A) in
	     magnitude, and BETA always less than and usually comparable with
	     norm(B).

     VL	     (output) COMPLEX*16 array, dimension (LDVL,N)
	     If JOBVL = 'V', the left generalized eigenvectors.	 (See
	     "Purpose", above.)	 Each eigenvector will be scaled so the
	     largest component will have abs(real part) + abs(imag. part) = 1,
	     *except* that for eigenvalues with alpha=beta=0, a zero vector
	     will be returned as the corresponding eigenvector.	 Not
	     referenced if JOBVL = 'N'.

     LDVL    (input) INTEGER
	     The leading dimension of the matrix VL. LDVL >= 1, and if JOBVL =
	     'V', LDVL >= N.

     VR	     (output) COMPLEX*16 array, dimension (LDVR,N)
	     If JOBVL = 'V', the right generalized eigenvectors.  (See
	     "Purpose", above.)	 Each eigenvector will be scaled so the
	     largest component will have abs(real part) + abs(imag. part) = 1,
	     *except* that for eigenvalues with alpha=beta=0, a zero vector
	     will be returned as the corresponding eigenvector.	 Not
	     referenced if JOBVR = 'N'.

									Page 2

ZGEGV(3F)							     ZGEGV(3F)

     LDVR    (input) INTEGER
	     The leading dimension of the matrix VR. LDVR >= 1, and if JOBVR =
	     'V', LDVR >= N.

     WORK    (workspace/output) COMPLEX*16 array, dimension (LWORK)
	     On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

     LWORK   (input) INTEGER
	     The dimension of the array WORK.  LWORK >= max(1,2*N).  For good
	     performance, LWORK must generally be larger.  To compute the
	     optimal value of LWORK, call ILAENV to get blocksizes (for
	     ZGEQRF, ZUNMQR, and CUNGQR.)  Then compute:  NB  -- MAX of the
	     blocksizes for ZGEQRF, ZUNMQR, and CUNGQR; The optimal LWORK is
	     MAX( 2*N, N*(NB+1) ).

     RWORK   (workspace/output) DOUBLE PRECISION array, dimension (8*N)

     INFO    (output) INTEGER
	     = 0:  successful exit
	     < 0:  if INFO = -i, the i-th argument had an illegal value.
	     =1,...,N:	The QZ iteration failed.  No eigenvectors have been
	     calculated, but ALPHA(j) and BETA(j) should be correct for
	     j=INFO+1,...,N.  > N:  errors that usually indicate LAPACK
	     problems:
	     =N+1: error return from ZGGBAL
	     =N+2: error return from ZGEQRF
	     =N+3: error return from ZUNMQR
	     =N+4: error return from ZUNGQR
	     =N+5: error return from ZGGHRD
	     =N+6: error return from ZHGEQZ (other than failed iteration)
	     =N+7: error return from ZTGEVC
	     =N+8: error return from ZGGBAK (computing VL)
	     =N+9: error return from ZGGBAK (computing VR)
	     =N+10: error return from ZLASCL (various calls)

FURTHER DETAILS
     Balancing
     ---------

     This driver calls ZGGBAL to both permute and scale rows and columns of A
     and B.  The permutations PL and PR are chosen so that PL*A*PR and PL*B*R
     will be upper triangular except for the diagonal blocks A(i:j,i:j) and
     B(i:j,i:j), with i and j as close together as possible.  The diagonal
     scaling matrices DL and DR are chosen so that the pair  DL*PL*A*PR*DR,
     DL*PL*B*PR*DR have elements close to one (except for the elements that
     start out zero.)

     After the eigenvalues and eigenvectors of the balanced matrices have been
     computed, ZGGBAK transforms the eigenvectors back to what they would have
     been (in perfect arithmetic) if they had not been balanced.

     Contents of A and B on Exit

									Page 3

ZGEGV(3F)							     ZGEGV(3F)

     -------- -- - --- - -- ----

     If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or both),
     then on exit the arrays A and B will contain the complex Schur form[*] of
     the "balanced" versions of A and B.  If no eigenvectors are computed,
     then only the diagonal blocks will be correct.

     [*] In other words, upper triangular form.

									Page 4

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