ZGGEV3 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.


 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 generalized eigenvector v(j) corresponding to the


 generalized eigenvalue lambda(j) of (A,B) satisfies


              A * v(j) = lambda(j) * B * v(j).


 The left generalized eigenvector u(j) corresponding to the


 generalized eigenvalues 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 COMPLEX*16 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 COMPLEX*16 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).

ALPHA

          ALPHA is COMPLEX*16 array, dimension (N)

BETA

          BETA is COMPLEX*16 array, dimension (N)


          On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the


          generalized eigenvalues.


          Note: the quotients 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

          VL is COMPLEX*16 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 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 COMPLEX*16 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 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 COMPLEX*16 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.


          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

          RWORK is DOUBLE PRECISION array, dimension (8*N)

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 ALPHA(j) and BETA(j) should be


                correct for j=INFO+1,...,N.


          > N:  =N+1: other then QZ iteration failed in DHGEQZ,


                =N+2: error return from DTGEVC.

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NAG Ltd.

January 2015