This routine is deprecated and has been replaced by routine SGGEV.


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


 eigenvectors of a real 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 REAL array, dimension (LDA, N)


          On entry, the matrix A.


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


          contains the real 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


          blocks from the Schur form will be correct.  See SGGHRD and


          SHGEQZ for details.

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.


          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 those elements of


          B corresponding to the diagonal blocks from the Schur form of


          A will be correct.  See SGGHRD and SHGEQZ for details.

LDB

          LDB is INTEGER


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

ALPHAR

          ALPHAR is REAL array, dimension (N)


          The real parts of each scalar alpha defining an eigenvalue of


          GNEP.

ALPHAI

          ALPHAI is REAL array, dimension (N)


          The imaginary parts of each scalar alpha defining an


          eigenvalue of GNEP.  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) = -ALPHAI(j).

BETA

          BETA is REAL array, dimension (N)


          The scalars beta that define the eigenvalues of GNEP.


          


          Together, the quantities alpha = (ALPHAR(j),ALPHAI(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 REAL 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.


          If the j-th eigenvalue is real, then u(j) = VL(:,j).


          If the j-th and (j+1)-st 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 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 REAL 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.


          If the j-th eigenvalue is real, then x(j) = VR(:,j).


          If the j-th and (j+1)-st eigenvalues form a complex conjugate


          pair, then


            x(j) = VR(:,j) + i*VR(:,j+1)


          and


            x(j+1) = VR(:,j) - i*VR(:,j+1).


          Each eigenvector is scaled so that its largest component has


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


          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 REAL 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 must generally be larger.


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


          blocksizes (for SGEQRF, SORMQR, and SORGQR.)  Then compute:


          NB  -- MAX of the blocksizes for SGEQRF, SORMQR, and SORGQR;


          The optimal LWORK is:


              2*N + MAX( 6*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.

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:  errors that usually indicate LAPACK problems:


                =N+1: error return from SGGBAL


                =N+2: error return from SGEQRF


                =N+3: error return from SORMQR


                =N+4: error return from SORGQR


                =N+5: error return from SGGHRD


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


                                                iteration)


                =N+7: error return from STGEVC


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


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


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

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

November 2011

  Balancing


  ---------


  This driver calls SGGBAL 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, SGGBAK 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 real Schur


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


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


  [*] See SHGEQZ, SGEGS, or read the book "Matrix Computations",


      by Golub & van Loan, pub. by Johns Hopkins U. Press.