dggev3()

subroutine dggev3	(	character	jobvl,
		character	jobvr,
		integer	n,
		double precision, dimension( lda, * )	a,
		integer	lda,
		double precision, dimension( ldb, * )	b,
		integer	ldb,
		double precision, dimension( * )	alphar,
		double precision, dimension( * )	alphai,
		double precision, dimension( * )	beta,
		double precision, dimension( ldvl, * )	vl,
		integer	ldvl,
		double precision, dimension( ldvr, * )	vr,
		integer	ldvr,
		double precision, dimension( * )	work,
		integer	lwork,
		integer	info )

DGGEV3 computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices (blocked algorithm)

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Purpose:

!>
!> DGGEV3 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.
!>
!> 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.
!>
!> 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).
!>
!> 

Parameters

[in]	JOBVL	!> JOBVL is CHARACTER*1 !> = 'N': do not compute the left generalized eigenvectors; !> = 'V': compute the left generalized eigenvectors. !>
[in]	JOBVR	!> JOBVR is CHARACTER*1 !> = 'N': do not compute the right generalized eigenvectors; !> = 'V': compute the right generalized eigenvectors. !>
[in]	N	!> N is INTEGER !> The order of the matrices A, B, VL, and VR. N >= 0. !>
[in,out]	A	!> A is DOUBLE PRECISION array, dimension (LDA, N) !> On entry, the matrix A in the pair (A,B). !> On exit, A has been overwritten. !>
[in]	LDA	!> LDA is INTEGER !> The leading dimension of A. LDA >= max(1,N). !>
[in,out]	B	!> B is DOUBLE PRECISION array, dimension (LDB, N) !> On entry, the matrix B in the pair (A,B). !> On exit, B has been overwritten. !>
[in]	LDB	!> LDB is INTEGER !> The leading dimension of B. LDB >= max(1,N). !>
[out]	ALPHAR	!> ALPHAR is DOUBLE PRECISION array, dimension (N) !>
[out]	ALPHAI	!> ALPHAI is DOUBLE PRECISION array, dimension (N) !>
[out]	BETA	!> BETA is 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). !>
[out]	VL	!> VL is 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 is scaled so the largest component has !> abs(real part)+abs(imag. part)=1. !> Not referenced if JOBVL = 'N'. !>
[in]	LDVL	!> LDVL is INTEGER !> The leading dimension of the matrix VL. LDVL >= 1, and !> if JOBVL = 'V', LDVL >= N. !>
[out]	VR	!> VR is 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 is scaled so the largest component has !> abs(real part)+abs(imag. part)=1. !> Not referenced if JOBVR = 'N'. !>
[in]	LDVR	!> LDVR is INTEGER !> The leading dimension of the matrix VR. LDVR >= 1, and !> if JOBVR = 'V', LDVR >= N. !>
[out]	WORK	!> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) !> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. !>
[in]	LWORK	!> LWORK is INTEGER. !> The dimension of the array WORK. LWORK >= MAX(1,8*N). !> For good performance, LWORK should generally be larger. !> !> 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. !>
[out]	INFO	!> INFO is 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 DLAQZ0. !> =N+2: error return from DTGEVC. !>

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.