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DGEGS(3F)							     DGEGS(3F)


NAME    [Toc]    [Back]

     DGEGS - compute for a pair	of N-by-N real nonsymmetric matrices A,	B

SYNOPSIS    [Toc]    [Back]

     SUBROUTINE	DGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHAR, ALPHAI,
		       BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, INFO )

	 CHARACTER     JOBVSL, JOBVSR

	 INTEGER       INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N

	 DOUBLE	       PRECISION A( LDA, * ), ALPHAI( *	), ALPHAR( * ),	B(
		       LDB, * ), BETA( * ), VSL( LDVSL,	* ), VSR( LDVSR, * ),
		       WORK( * )

PURPOSE    [Toc]    [Back]

     DGEGS computes for	a pair of N-by-N real nonsymmetric matrices A, B:  the
     generalized eigenvalues (alphar +/- alphai*i, beta), the real Schur form
     (A, B), and optionally left and/or	right Schur vectors (VSL and VSR).

     (If only the generalized eigenvalues are needed, use the driver DGEGV
     instead.)

     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)

     The (generalized) Schur form of a pair of matrices	is the result of
     multiplying both matrices on the left by one orthogonal matrix and	both
     on	the right by another orthogonal	matrix,	these two orthogonal matrices
     being chosen so as	to bring the pair of matrices into (real) Schur	form.

     A pair of matrices	A, B is	in generalized real Schur form if B is upper
     triangular	with non-negative diagonal and A is block upper	triangular
     with 1-by-1 and 2-by-2 blocks.  1-by-1 blocks correspond to real
     generalized eigenvalues, while 2-by-2 blocks of A will be "standardized"
     by	making the corresponding elements of B have the	form:
	     [	a  0  ]
	     [	0  b  ]

     and the pair of corresponding 2-by-2 blocks in A and B will have a
     complex conjugate pair of generalized eigenvalues.

     The left and right	Schur vectors are the columns of VSL and VSR,
     respectively, where VSL and VSR are the orthogonal	matrices which reduce
     A and B to	Schur form:

     Schur form	of (A,B) = ( (VSL)**T A	(VSR), (VSL)**T	B (VSR)	)




									Page 1






DGEGS(3F)							     DGEGS(3F)


ARGUMENTS    [Toc]    [Back]

     JOBVSL  (input) CHARACTER*1
	     = 'N':  do	not compute the	left Schur vectors;
	     = 'V':  compute the left Schur vectors.

     JOBVSR  (input) CHARACTER*1
	     = 'N':  do	not compute the	right Schur vectors;
	     = 'V':  compute the right Schur vectors.

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

     A	     (input/output) DOUBLE PRECISION array, dimension (LDA, N)
	     On	entry, the first of the	pair of	matrices whose generalized
	     eigenvalues and (optionally) Schur	vectors	are to be computed.
	     On	exit, the generalized Schur form of A.	Note: to avoid
	     overflow, the Frobenius norm of the matrix	A should be less than
	     the overflow threshold.

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

     B	     (input/output) DOUBLE PRECISION array, dimension (LDB, N)
	     On	entry, the second of the pair of matrices whose	generalized
	     eigenvalues and (optionally) Schur	vectors	are to be computed.
	     On	exit, the generalized Schur form of B.	Note: to avoid
	     overflow, the Frobenius norm of the matrix	B should be less than
	     the overflow threshold.

     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.  ALPHAR(j) + ALPHAI(j)*i,	j=1,...,N  and
	     BETA(j),j=1,...,N	are the	diagonals of the complex Schur form
	     (A,B) that	would result if	the 2-by-2 diagonal blocks of the real
	     Schur form	of (A,B) were further reduced to triangular form using
	     2-by-2 complex unitary transformations.  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).




									Page 2






DGEGS(3F)							     DGEGS(3F)



     VSL     (output) DOUBLE PRECISION array, dimension	(LDVSL,N)
	     If	JOBVSL = 'V', VSL will contain the left	Schur vectors.	(See
	     "Purpose",	above.)	 Not referenced	if JOBVSL = 'N'.

     LDVSL   (input) INTEGER
	     The leading dimension of the matrix VSL. LDVSL >=1, and if	JOBVSL
	     = 'V', LDVSL >= N.

     VSR     (output) DOUBLE PRECISION array, dimension	(LDVSR,N)
	     If	JOBVSR = 'V', VSR will contain the right Schur vectors.	 (See
	     "Purpose",	above.)	 Not referenced	if JOBVSR = 'N'.

     LDVSR   (input) INTEGER
	     The leading dimension of the matrix VSR. LDVSR >= 1, and if
	     JOBVSR = 'V', LDVSR >= N.

     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(1,4*N).  For good
	     performance, LWORK	must generally be larger.  To compute the
	     optimal value of LWORK, call ILAENV to get	blocksizes (for
	     DGEQRF, DORMQR, and DORGQR.)  Then	compute:  NB  -- MAX of	the
	     blocksizes	for DGEQRF, DORMQR, and	DORGQR The optimal LWORK is
	     2*N + N*(NB+1).

     INFO    (output) INTEGER
	     = 0:  successful exit
	     < 0:  if INFO = -i, the i-th argument had an illegal value.
	     = 1,...,N:	 The QZ	iteration failed.  (A,B) are not in Schur
	     form, but ALPHAR(j), ALPHAI(j), and BETA(j) should	be correct for
	     j=INFO+1,...,N.  >	N:  errors that	usually	indicate LAPACK
	     problems:
	     =N+1: error return	from DGGBAL
	     =N+2: error return	from DGEQRF
	     =N+3: error return	from DORMQR
	     =N+4: error return	from DORGQR
	     =N+5: error return	from DGGHRD
	     =N+6: error return	from DHGEQZ (other than	failed iteration)
	     =N+7: error return	from DGGBAK (computing VSL)
	     =N+8: error return	from DGGBAK (computing VSR)
	     =N+9: error return	from DLASCL (various places)
DGEGS(3F)							     DGEGS(3F)


NAME    [Toc]    [Back]

     DGEGS - compute for a pair	of N-by-N real nonsymmetric matrices A,	B

SYNOPSIS    [Toc]    [Back]

     SUBROUTINE	DGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHAR, ALPHAI,
		       BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, INFO )

	 CHARACTER     JOBVSL, JOBVSR

	 INTEGER       INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N

	 DOUBLE	       PRECISION A( LDA, * ), ALPHAI( *	), ALPHAR( * ),	B(
		       LDB, * ), BETA( * ), VSL( LDVSL,	* ), VSR( LDVSR, * ),
		       WORK( * )

PURPOSE    [Toc]    [Back]

     DGEGS computes for	a pair of N-by-N real nonsymmetric matrices A, B:  the
     generalized eigenvalues (alphar +/- alphai*i, beta), the real Schur form
     (A, B), and optionally left and/or	right Schur vectors (VSL and VSR).

     (If only the generalized eigenvalues are needed, use the driver DGEGV
     instead.)

     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)

     The (generalized) Schur form of a pair of matrices	is the result of
     multiplying both matrices on the left by one orthogonal matrix and	both
     on	the right by another orthogonal	matrix,	these two orthogonal matrices
     being chosen so as	to bring the pair of matrices into (real) Schur	form.

     A pair of matrices	A, B is	in generalized real Schur form if B is upper
     triangular	with non-negative diagonal and A is block upper	triangular
     with 1-by-1 and 2-by-2 blocks.  1-by-1 blocks correspond to real
     generalized eigenvalues, while 2-by-2 blocks of A will be "standardized"
     by	making the corresponding elements of B have the	form:
	     [	a  0  ]
	     [	0  b  ]

     and the pair of corresponding 2-by-2 blocks in A and B will have a
     complex conjugate pair of generalized eigenvalues.

     The left and right	Schur vectors are the columns of VSL and VSR,
     respectively, where VSL and VSR are the orthogonal	matrices which reduce
     A and B to	Schur form:

     Schur form	of (A,B) = ( (VSL)**T A	(VSR), (VSL)**T	B (VSR)	)




									Page 1






DGEGS(3F)							     DGEGS(3F)


ARGUMENTS    [Toc]    [Back]

     JOBVSL  (input) CHARACTER*1
	     = 'N':  do	not compute the	left Schur vectors;
	     = 'V':  compute the left Schur vectors.

     JOBVSR  (input) CHARACTER*1
	     = 'N':  do	not compute the	right Schur vectors;
	     = 'V':  compute the right Schur vectors.

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

     A	     (input/output) DOUBLE PRECISION array, dimension (LDA, N)
	     On	entry, the first of the	pair of	matrices whose generalized
	     eigenvalues and (optionally) Schur	vectors	are to be computed.
	     On	exit, the generalized Schur form of A.	Note: to avoid
	     overflow, the Frobenius norm of the matrix	A should be less than
	     the overflow threshold.

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

     B	     (input/output) DOUBLE PRECISION array, dimension (LDB, N)
	     On	entry, the second of the pair of matrices whose	generalized
	     eigenvalues and (optionally) Schur	vectors	are to be computed.
	     On	exit, the generalized Schur form of B.	Note: to avoid
	     overflow, the Frobenius norm of the matrix	B should be less than
	     the overflow threshold.

     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.  ALPHAR(j) + ALPHAI(j)*i,	j=1,...,N  and
	     BETA(j),j=1,...,N	are the	diagonals of the complex Schur form
	     (A,B) that	would result if	the 2-by-2 diagonal blocks of the real
	     Schur form	of (A,B) were further reduced to triangular form using
	     2-by-2 complex unitary transformations.  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).




									Page 2






DGEGS(3F)							     DGEGS(3F)



     VSL     (output) DOUBLE PRECISION array, dimension	(LDVSL,N)
	     If	JOBVSL = 'V', VSL will contain the left	Schur vectors.	(See
	     "Purpose",	above.)	 Not referenced	if JOBVSL = 'N'.

     LDVSL   (input) INTEGER
	     The leading dimension of the matrix VSL. LDVSL >=1, and if	JOBVSL
	     = 'V', LDVSL >= N.

     VSR     (output) DOUBLE PRECISION array, dimension	(LDVSR,N)
	     If	JOBVSR = 'V', VSR will contain the right Schur vectors.	 (See
	     "Purpose",	above.)	 Not referenced	if JOBVSR = 'N'.

     LDVSR   (input) INTEGER
	     The leading dimension of the matrix VSR. LDVSR >= 1, and if
	     JOBVSR = 'V', LDVSR >= N.

     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(1,4*N).  For good
	     performance, LWORK	must generally be larger.  To compute the
	     optimal value of LWORK, call ILAENV to get	blocksizes (for
	     DGEQRF, DORMQR, and DORGQR.)  Then	compute:  NB  -- MAX of	the
	     blocksizes	for DGEQRF, DORMQR, and	DORGQR The optimal LWORK is
	     2*N + N*(NB+1).

     INFO    (output) INTEGER
	     = 0:  successful exit
	     < 0:  if INFO = -i, the i-th argument had an illegal value.
	     = 1,...,N:	 The QZ	iteration failed.  (A,B) are not in Schur
	     form, but ALPHAR(j), ALPHAI(j), and BETA(j) should	be correct for
	     j=INFO+1,...,N.  >	N:  errors that	usually	indicate LAPACK
	     problems:
	     =N+1: error return	from DGGBAL
	     =N+2: error return	from DGEQRF
	     =N+3: error return	from DORMQR
	     =N+4: error return	from DORGQR
	     =N+5: error return	from DGGHRD
	     =N+6: error return	from DHGEQZ (other than	failed iteration)
	     =N+7: error return	from DGGBAK (computing VSL)
	     =N+8: error return	from DGGBAK (computing VSR)
	     =N+9: error return	from DLASCL (various places)


									PPPPaaaaggggeeee 3333
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