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DGGEVX(3S)							    DGGEVX(3S)

NAME
     DGGEVX - compute for a pair of N-by-N real nonsymmetric matrices (A,B)

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

	 CHARACTER	BALANC, JOBVL, JOBVR, SENSE

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

	 DOUBLE		PRECISION ABNRM, BBNRM

	 LOGICAL	BWORK( * )

	 INTEGER	IWORK( * )

	 DOUBLE		PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B(
			LDB, * ), BETA( * ), LSCALE( * ), RCONDE( * ), RCONDV(
			* ), RSCALE( * ), 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
     DGGEVX computes for a pair of N-by-N real nonsymmetric matrices (A,B) the
     generalized eigenvalues, and optionally, the left and/or right
     generalized eigenvectors.

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

									Page 1

DGGEVX(3S)							    DGGEVX(3S)

     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) DOUBLE PRECISION 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 real 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

DGGEVX(3S)							    DGGEVX(3S)

     B	     (input/output) DOUBLE PRECISION 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 real Schur form of the "balanced" versions
	     of the input A and B.

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

     ALPHAR  (output) DOUBLE PRECISION array, dimension (N)
	     ALPHAI  (output) DOUBLE PRECISION array, dimension (N) BETA
	     (output) DOUBLE PRECISION array, dimension (N) On exit,
	     (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will be the
	     generalized eigenvalues.  If ALPHAI(j) is zero, then the j-th
	     eigenvalue is real; if positive, then the j-th and (j+1)-st
	     eigenvalues are a complex conjugate pair, with ALPHAI(j+1)
	     negative.

	     Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(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, ALPHAR and ALPHAI 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) DOUBLE PRECISION array, dimension (LDVL,N)
	     If JOBVL = 'V', the left eigenvectors u(j) are stored one after
	     another in the columns of VL, in the same order as their
	     eigenvalues. If the j-th eigenvalue is real, then u(j) = VL(:,j),
	     the j-th column of VL. If the j-th and (j+1)-th eigenvalues form
	     a complex conjugate pair, then u(j) = VL(:,j)+i*VL(:,j+1) and
	     u(j+1) = VL(:,j)-i*VL(:,j+1).  Each eigenvector will be scaled so
	     the largest component 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) DOUBLE PRECISION array, dimension (LDVR,N)
	     If JOBVR = 'V', the right eigenvectors v(j) are stored one after
	     another in the columns of VR, in the same order as their
	     eigenvalues. If the j-th eigenvalue is real, then v(j) = VR(:,j),
	     the j-th column of VR. If the j-th and (j+1)-th eigenvalues form
	     a complex conjugate pair, then v(j) = VR(:,j)+i*VR(:,j+1) and
	     v(j+1) = VR(:,j)-i*VR(:,j+1).  Each eigenvector will be scaled so
	     the largest component 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.

									Page 3

DGGEVX(3S)							    DGGEVX(3S)

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

     RSCALE  (output) DOUBLE PRECISION 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) DOUBLE PRECISION
	     The one-norm of the balanced matrix A.

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

     RCONDE  (output) DOUBLE PRECISION array, dimension (N)
	     If SENSE = 'E' or 'B', the reciprocal condition numbers of the
	     selected eigenvalues, stored in consecutive elements of the
	     array. For a complex conjugate pair of eigenvalues two
	     consecutive elements of RCONDE are set to the same value.	Thus
	     RCONDE(j), RCONDV(j), and the j-th columns of VL and VR all
	     correspond to the same eigenpair (but not in general the j-th
	     eigenpair, unless all eigenpairs are selected).  If SENSE = 'V',
	     RCONDE is not referenced.

     RCONDV  (output) DOUBLE PRECISION array, dimension (N)
	     If SENSE = 'V' or 'B', the estimated reciprocal condition numbers
	     of the selected eigenvectors, stored in consecutive elements of
	     the array. For a complex eigenvector two consecutive elements of
	     RCONDV are set to the same value. 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.

     WORK    (workspace/output) DOUBLE PRECISION 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(1,6*N).  If SENSE =
	     'E', LWORK >= 12*N.  If SENSE = 'V' or 'B', LWORK >=

									Page 4

DGGEVX(3S)							    DGGEVX(3S)

	     2*N*N+12*N+16.

	     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 (N+6)
	     If SENSE = 'E', IWORK is not referenced.

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

     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 ALPHAR(j), ALPHAI(j), and BETA(j) should be
	     correct for j=INFO+1,...,N.  > N:	=N+1: other than QZ iteration
	     failed in DHGEQZ.
	     =N+2: error return from DTGEVC.

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)

									Page 5

DGGEVX(3S)							    DGGEVX(3S)

     This man page is available only online.

									Page 6

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