dggev3.f3

!>
!> 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).
!>
!> 

JOBVL

!>          JOBVL is CHARACTER*1
!>          = 'N':  do not compute the left generalized eigenvectors;
!>          = 'V':  compute the left generalized eigenvectors.
!> 

JOBVR

!>          JOBVR is CHARACTER*1
!>          = 'N':  do not compute the right generalized eigenvectors;
!>          = 'V':  compute the right generalized eigenvectors.
!> 

N

!>          N is INTEGER
!>          The order of the matrices A, B, VL, and VR.  N >= 0.
!> 

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.
!> 

LDA

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

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.
!> 

LDB

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

ALPHAR

!>          ALPHAR is DOUBLE PRECISION array, dimension (N)
!> 

ALPHAI

!>          ALPHAI is DOUBLE PRECISION array, dimension (N)
!> 

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).
!> 

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'.
!> 

LDVL

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

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'.
!> 

LDVR

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

WORK

!>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
!>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
!> 

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.
!> 

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.
!> 

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