SGGEV3 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 REAL 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 REAL 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 REAL array, dimension (N)

ALPHAI

          ALPHAI is REAL array, dimension (N)

BETA

          BETA is REAL 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 REAL 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 REAL 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 REAL array, dimension (MAX(1,LWORK))


          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER


          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 SHGEQZ.


                =N+2: error return from STGEVC.

Univ. of Tennessee

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NAG Ltd.

January 2015