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CGGEVX(3S)							    CGGEVX(3S)

NAME
     CGGEVX - compute for a pair of N-by-N complex nonsymmetric matrices (A,B)
     the generalized eigenvalues, and optionally, the left and/or right
     generalized eigenvectors

SYNOPSIS
     SUBROUTINE CGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB, ALPHA,
			BETA, VL, LDVL, VR, LDVR, ILO, IHI, LSCALE, RSCALE,
			ABNRM, BBNRM, RCONDE, RCONDV, WORK, LWORK, RWORK,
			IWORK, BWORK, INFO )

	 CHARACTER	BALANC, JOBVL, JOBVR, SENSE

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

	 REAL		ABNRM, BBNRM

	 LOGICAL	BWORK( * )

	 INTEGER	IWORK( * )

	 REAL		LSCALE( * ), RCONDE( * ), RCONDV( * ), RSCALE( * ),
			RWORK( * )

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

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
     CGGEVX computes for a pair of N-by-N complex nonsymmetric matrices (A,B)
     the generalized eigenvalues, and optionally, the left and/or right
     generalized eigenvectors. Optionally, it also computes a balancing
     transformation to improve the conditioning of the eigenvalues and
     eigenvectors (ILO, IHI, LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal
     condition numbers for the eigenvalues (RCONDE), and reciprocal condition
     numbers for the right eigenvectors (RCONDV).

     A generalized eigenvalue for a pair of matrices (A,B) is a scalar lambda
     or a ratio alpha/beta = lambda, such that A - lambda*B is singular. It is
     usually represented as the pair (alpha,beta), as there is a reasonable

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CGGEVX(3S)							    CGGEVX(3S)

     interpretation for beta=0, and even for both being zero.

     The right eigenvector v(j) corresponding to the eigenvalue lambda(j) of
     (A,B) satisfies
		      A * v(j) = lambda(j) * B * v(j) .
     The left eigenvector u(j) corresponding to the eigenvalue lambda(j) of
     (A,B) satisfies
		      u(j)**H * A  = lambda(j) * u(j)**H * B.
     where u(j)**H is the conjugate-transpose of u(j).

ARGUMENTS
     BALANC  (input) CHARACTER*1
	     Specifies the balance option to be performed:
	     = 'N':  do not diagonally scale or permute;
	     = 'P':  permute only;
	     = 'S':  scale only;
	     = 'B':  both permute and scale.  Computed reciprocal condition
	     numbers will be for the matrices after permuting and/or
	     balancing. Permuting does not change condition numbers (in exact
	     arithmetic), but balancing does.

     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.

     SENSE   (input) CHARACTER*1
	     Determines which reciprocal condition numbers are computed.  =
	     'N': none are computed;
	     = 'E': computed for eigenvalues only;
	     = 'V': computed for eigenvectors only;
	     = 'B': computed for eigenvalues and eigenvectors.

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

     A	     (input/output) COMPLEX array, dimension (LDA, N)
	     On entry, the matrix A in the pair (A,B).	On exit, A has been
	     overwritten. If JOBVL='V' or JOBVR='V' or both, then A contains
	     the first part of the complex Schur form of the "balanced"
	     versions of the input A and B.

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

									Page 2

CGGEVX(3S)							    CGGEVX(3S)

     B	     (input/output) COMPLEX array, dimension (LDB, N)
	     On entry, the matrix B in the pair (A,B).	On exit, B has been
	     overwritten. If JOBVL='V' or JOBVR='V' or both, then B contains
	     the second part of the complex Schur form of the "balanced"
	     versions of the input A and B.

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

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

	     Note: the quotient 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 array, dimension (LDVL,N)
	     If JOBVL = 'V', the left generalized eigenvectors u(j) are stored
	     one after another in the columns of VL, in the same order as
	     their eigenvalues.	 Each eigenvector will be scaled so the
	     largest component will have abs(real part) + abs(imag. part) = 1.
	     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 array, dimension (LDVR,N)
	     If JOBVR = 'V', the right generalized eigenvectors v(j) are
	     stored one after another in the columns of VR, in the same order
	     as their eigenvalues.  Each eigenvector will be scaled so the
	     largest component will have abs(real part) + abs(imag. part) = 1.
	     Not referenced if JOBVR = 'N'.

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

	     ILO,IHI (output) INTEGER ILO and IHI are integer values such that
	     on exit A(i,j) = 0 and B(i,j) = 0 if i > j and j = 1,...,ILO-1 or
	     i = IHI+1,...,N.  If BALANC = 'N' or 'S', ILO = 1 and IHI = N.

     LSCALE  (output) REAL array, dimension (N)
	     Details of the permutations and scaling factors applied to the
	     left side of A and B.  If PL(j) is the index of the row
	     interchanged with row j, and DL(j) is the scaling factor applied
	     to row j, then LSCALE(j) = PL(j)  for j = 1,...,ILO-1 = DL(j)
	     for j = ILO,...,IHI = PL(j)  for j = IHI+1,...,N.	The order in

									Page 3

CGGEVX(3S)							    CGGEVX(3S)

	     which the interchanges are made is N to IHI+1, then 1 to ILO-1.

     RSCALE  (output) REAL array, dimension (N)
	     Details of the permutations and scaling factors applied to the
	     right side of A and B.  If PR(j) is the index of the column
	     interchanged with column j, and DR(j) is the scaling factor
	     applied to column j, then RSCALE(j) = PR(j)  for j = 1,...,ILO-1
	     = DR(j)  for j = ILO,...,IHI = PR(j)  for j = IHI+1,...,N The
	     order in which the interchanges are made is N to IHI+1, then 1 to
	     ILO-1.

     ABNRM   (output) REAL
	     The one-norm of the balanced matrix A.

     BBNRM   (output) REAL
	     The one-norm of the balanced matrix B.

     RCONDE  (output) REAL array, dimension (N)
	     If SENSE = 'E' or 'B', the reciprocal condition numbers of the
	     selected eigenvalues, stored in consecutive elements of the
	     array.  If SENSE = 'V', RCONDE is not referenced.

     RCONDV  (output) REAL array, dimension (N)
	     If JOB = 'V' or 'B', the estimated reciprocal condition numbers
	     of the selected eigenvectors, stored in consecutive elements of
	     the array. If the eigenvalues cannot be reordered to compute
	     RCONDV(j), RCONDV(j) is set to 0; this can only occur when the
	     true value would be very small anyway.  If SENSE = 'E', RCONDV is
	     not referenced.  Not referenced if JOB = 'E'.

     WORK    (workspace/output) COMPLEX 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).  If SENSE =
	     'N' or 'E', LWORK >= 2*N.	If SENSE = 'V' or 'B', LWORK >=
	     2*N*N+2*N.

	     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) REAL array, dimension (6*N)
	     Real workspace.

     IWORK   (workspace) INTEGER array, dimension (N+2)
	     If SENSE = 'E', IWORK is not referenced.

     BWORK   (workspace) LOGICAL array, dimension (N)
	     If SENSE = 'N', BWORK is not referenced.

									Page 4

CGGEVX(3S)							    CGGEVX(3S)

     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:  =N+1: other than QZ iteration failed in
	     CHGEQZ.
	     =N+2: error return from CTGEVC.

FURTHER DETAILS
     Balancing a matrix pair (A,B) includes, first, permuting rows and columns
     to isolate eigenvalues, second, applying diagonal similarity
     transformation to the rows and columns to make the rows and columns as
     close in norm as possible. The computed reciprocal condition numbers
     correspond to the balanced matrix. Permuting rows and columns will not
     change the condition numbers (in exact arithmetic) but diagonal scaling
     will.  For further explanation of balancing, see section 4.11.1.2 of
     LAPACK Users' Guide.

     An approximate error bound on the chordal distance between the i-th
     computed generalized eigenvalue w and the corresponding exact eigenvalue
     lambda is

	  chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I)

     An approximate error bound for the angle between the i-th computed
     eigenvector VL(i) or VR(i) is given by

	  EPS * norm(ABNRM, BBNRM) / DIF(i).

     For further explanation of the reciprocal condition numbers RCONDE and
     RCONDV, see section 4.11 of LAPACK User's Guide.

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

     This man page is available only online.

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