This routine is deprecated and has been replaced by routine DGGES.


 DGEGS computes the eigenvalues, real Schur form, and, optionally,


 left and or/right Schur vectors of a real matrix pair (A,B).


 Given two square matrices A and B, the generalized real Schur


 factorization has the form


   A = Q*S*Z**T,  B = Q*T*Z**T


 where Q and Z are orthogonal matrices, T is upper triangular, and S


 is an upper quasi-triangular matrix with 1-by-1 and 2-by-2 diagonal


 blocks, the 2-by-2 blocks corresponding to complex conjugate pairs


 of eigenvalues of (A,B).  The columns of Q are the left Schur vectors


 and the columns of Z are the right Schur vectors.


 If only the eigenvalues of (A,B) are needed, the driver routine


 DGEGV should be used instead.  See DGEGV for a description of the


 eigenvalues of the generalized nonsymmetric eigenvalue problem


 (GNEP).

JOBVSL

          JOBVSL is CHARACTER*1


          = 'N':  do not compute the left Schur vectors;


          = 'V':  compute the left Schur vectors (returned in VSL).

JOBVSR

          JOBVSR is CHARACTER*1


          = 'N':  do not compute the right Schur vectors;


          = 'V':  compute the right Schur vectors (returned in VSR).

N

          N is INTEGER


          The order of the matrices A, B, VSL, and VSR.  N >= 0.

A

          A is DOUBLE PRECISION array, dimension (LDA, N)


          On entry, the matrix A.


          On exit, the upper quasi-triangular matrix S from the


          generalized real Schur factorization.

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.


          On exit, the upper triangular matrix T from the generalized


          real Schur factorization.

LDB

          LDB is INTEGER


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

ALPHAR

          ALPHAR is DOUBLE PRECISION array, dimension (N)


          The real parts of each scalar alpha defining an eigenvalue


          of GNEP.

ALPHAI

          ALPHAI is DOUBLE PRECISION 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 DOUBLE PRECISION 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.

VSL

          VSL is DOUBLE PRECISION array, dimension (LDVSL,N)


          If JOBVSL = 'V', the matrix of left Schur vectors Q.


          Not referenced if JOBVSL = 'N'.

LDVSL

          LDVSL is INTEGER


          The leading dimension of the matrix VSL. LDVSL >=1, and


          if JOBVSL = 'V', LDVSL >= N.

VSR

          VSR is DOUBLE PRECISION array, dimension (LDVSR,N)


          If JOBVSR = 'V', the matrix of right Schur vectors Z.


          Not referenced if JOBVSR = 'N'.

LDVSR

          LDVSR is INTEGER


          The leading dimension of the matrix VSR. LDVSR >= 1, and


          if JOBVSR = 'V', LDVSR >= 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(1,4*N).


          For good performance, LWORK must generally be larger.


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


          blocksizes (for DGEQRF, DORMQR, and DORGQR.)  Then compute:


          NB  -- MAX of the blocksizes for DGEQRF, DORMQR, and DORGQR


          The optimal LWORK is  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.

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.  (A,B) are not in Schur


                form, 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 DGGBAL


                =N+2: error return from DGEQRF


                =N+3: error return from DORMQR


                =N+4: error return from DORGQR


                =N+5: error return from DGGHRD


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


                                                iteration)


                =N+7: error return from DGGBAK (computing VSL)


                =N+8: error return from DGGBAK (computing VSR)


                =N+9: error return from DLASCL (various places)

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

November 2011