This routine is deprecated and has been replaced by routine ZGGEV.


 ZGEGV computes the eigenvalues and, optionally, the left and/or right


 eigenvectors of a complex matrix pair (A,B).


 Given two square matrices A and B,


 the generalized nonsymmetric eigenvalue problem (GNEP) is to find the


 eigenvalues lambda and corresponding (non-zero) eigenvectors x such


 that


    A*x = lambda*B*x.


 An alternate form is to find the eigenvalues mu and corresponding


 eigenvectors y such that


    mu*A*y = B*y.


 These two forms are equivalent with mu = 1/lambda and x = y if


 neither lambda nor mu is zero.  In order to deal with the case that


 lambda or mu is zero or small, two values alpha and beta are returned


 for each eigenvalue, such that lambda = alpha/beta and


 mu = beta/alpha.


 The vectors x and y in the above equations are right eigenvectors of


 the matrix pair (A,B).  Vectors u and v satisfying


    u**H*A = lambda*u**H*B  or  mu*v**H*A = v**H*B


 are left eigenvectors of (A,B).


 Note: this routine performs "full balancing" on A and B

JOBVL

          JOBVL is CHARACTER*1


          = 'N':  do not compute the left generalized eigenvectors;


          = 'V':  compute the left generalized eigenvectors (returned


                  in VL).

JOBVR

          JOBVR is CHARACTER*1


          = 'N':  do not compute the right generalized eigenvectors;


          = 'V':  compute the right generalized eigenvectors (returned


                  in VR).

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.


          If JOBVL = 'V' or JOBVR = 'V', then on exit A


          contains the Schur form of A from the generalized Schur


          factorization of the pair (A,B) after balancing.  If no


          eigenvectors were computed, then only the diagonal elements


          of the Schur form will be correct.  See ZGGHRD and ZHGEQZ


          for details.

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.


          If JOBVL = 'V' or JOBVR = 'V', then on exit B contains the


          upper triangular matrix obtained from B in the generalized


          Schur factorization of the pair (A,B) after balancing.


          If no eigenvectors were computed, then only the diagonal


          elements of B will be correct.  See ZGGHRD and ZHGEQZ for


          details.

LDB

          LDB is INTEGER


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

ALPHA

          ALPHA is COMPLEX*16 array, dimension (N)


          The complex scalars alpha that define the eigenvalues of


          GNEP.

BETA

          BETA is COMPLEX*16 array, dimension (N)


          The complex scalars beta that define the eigenvalues of GNEP.


          


          Together, the quantities alpha = ALPHA(j) and beta = BETA(j)


          represent the j-th eigenvalue of the matrix pair (A,B), in


          one of the forms lambda = alpha/beta or mu = beta/alpha.


          Since either lambda or mu may overflow, they should not,


          in general, be computed.

VL

          VL is COMPLEX*16 array, dimension (LDVL,N)


          If JOBVL = 'V', the left eigenvectors u(j) are stored


          in the columns of VL, in the same order as their eigenvalues.


          Each eigenvector is scaled so that its largest component has


          abs(real part) + abs(imag. part) = 1, except for eigenvectors


          corresponding to an eigenvalue with alpha = beta = 0, which


          are set to zero.


          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 eigenvectors x(j) are stored


          in the columns of VR, in the same order as their eigenvalues.


          Each eigenvector is scaled so that its largest component has


          abs(real part) + abs(imag. part) = 1, except for eigenvectors


          corresponding to an eigenvalue with alpha = beta = 0, which


          are set to zero.


          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.  LWORK >= max(1,2*N).


          For good performance, LWORK must generally be larger.


          To compute the optimal value of LWORK, call ILAENV to get


          blocksizes (for ZGEQRF, ZUNMQR, and ZUNGQR.)  Then compute:


          NB  -- MAX of the blocksizes for ZGEQRF, ZUNMQR, and ZUNGQR;


          The optimal LWORK is  MAX( 2*N, N*(NB+1) ).


          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:  errors that usually indicate LAPACK problems:


                =N+1: error return from ZGGBAL


                =N+2: error return from ZGEQRF


                =N+3: error return from ZUNMQR


                =N+4: error return from ZUNGQR


                =N+5: error return from ZGGHRD


                =N+6: error return from ZHGEQZ (other than failed


                                               iteration)


                =N+7: error return from ZTGEVC


                =N+8: error return from ZGGBAK (computing VL)


                =N+9: error return from ZGGBAK (computing VR)


                =N+10: error return from ZLASCL (various calls)

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

November 2011

  Balancing


  ---------


  This driver calls ZGGBAL to both permute and scale rows and columns


  of A and B.  The permutations PL and PR are chosen so that PL*A*PR


  and PL*B*R will be upper triangular except for the diagonal blocks


  A(i:j,i:j) and B(i:j,i:j), with i and j as close together as


  possible.  The diagonal scaling matrices DL and DR are chosen so


  that the pair  DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to


  one (except for the elements that start out zero.)


  After the eigenvalues and eigenvectors of the balanced matrices


  have been computed, ZGGBAK transforms the eigenvectors back to what


  they would have been (in perfect arithmetic) if they had not been


  balanced.


  Contents of A and B on Exit


  -------- -- - --- - -- ----


  If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or


  both), then on exit the arrays A and B will contain the complex Schur


  form[*] of the "balanced" versions of A and B.  If no eigenvectors


  are computed, then only the diagonal blocks will be correct.


  [*] In other words, upper triangular form.