dgges3.f - Man Page

subroutine dgges3 (jobvsl, jobvsr, sort, selctg, n, a, lda, b, ldb, sdim, alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr, work, lwork, bwork, info)
DGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices (blocked algorithm)

DGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices (blocked algorithm)  

Purpose:

 DGGES3 computes for a pair of N-by-N real nonsymmetric matrices (A,B),
 the generalized eigenvalues, the generalized real Schur form (S,T),
 optionally, the left and/or right matrices of Schur vectors (VSL and
 VSR). This gives the generalized Schur factorization

          (A,B) = ( (VSL)*S*(VSR)**T, (VSL)*T*(VSR)**T )

 Optionally, it also orders the eigenvalues so that a selected cluster
 of eigenvalues appears in the leading diagonal blocks of the upper
 quasi-triangular matrix S and the upper triangular matrix T.The
 leading columns of VSL and VSR then form an orthonormal basis for the
 corresponding left and right eigenspaces (deflating subspaces).

 (If only the generalized eigenvalues are needed, use the driver
 DGGEV instead, which is faster.)

 A generalized eigenvalue for a pair of matrices (A,B) is a scalar w
 or a ratio alpha/beta = w, such that  A - w*B is singular.  It is
 usually represented as the pair (alpha,beta), as there is a
 reasonable interpretation for beta=0 or both being zero.

 A pair of matrices (S,T) is in generalized real Schur form if T is
 upper triangular with non-negative diagonal and S is block upper
 triangular with 1-by-1 and 2-by-2 blocks.  1-by-1 blocks correspond
 to real generalized eigenvalues, while 2-by-2 blocks of S will be
 'standardized' by making the corresponding elements of T have the
 form:
         [  a  0  ]
         [  0  b  ]

 and the pair of corresponding 2-by-2 blocks in S and T will have a
 complex conjugate pair of generalized eigenvalues.

Parameters

JOBVSL

          JOBVSL is CHARACTER*1
          = 'N':  do not compute the left Schur vectors;
          = 'V':  compute the left Schur vectors.

JOBVSR

          JOBVSR is CHARACTER*1
          = 'N':  do not compute the right Schur vectors;
          = 'V':  compute the right Schur vectors.

SORT

          SORT is CHARACTER*1
          Specifies whether or not to order the eigenvalues on the
          diagonal of the generalized Schur form.
          = 'N':  Eigenvalues are not ordered;
          = 'S':  Eigenvalues are ordered (see SELCTG);

SELCTG

          SELCTG is a LOGICAL FUNCTION of three DOUBLE PRECISION arguments
          SELCTG must be declared EXTERNAL in the calling subroutine.
          If SORT = 'N', SELCTG is not referenced.
          If SORT = 'S', SELCTG is used to select eigenvalues to sort
          to the top left of the Schur form.
          An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if
          SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either
          one of a complex conjugate pair of eigenvalues is selected,
          then both complex eigenvalues are selected.

          Note that in the ill-conditioned case, a selected complex
          eigenvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(j),
          BETA(j)) = .TRUE. after ordering. INFO is to be set to N+2
          in this case.

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 first of the pair of matrices.
          On exit, A has been overwritten by its generalized Schur
          form S.

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 second of the pair of matrices.
          On exit, B has been overwritten by its generalized Schur
          form T.

LDB

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

SDIM

          SDIM is INTEGER
          If SORT = 'N', SDIM = 0.
          If SORT = 'S', SDIM = number of eigenvalues (after sorting)
          for which SELCTG is true.  (Complex conjugate pairs for which
          SELCTG is true for either eigenvalue count as 2.)

ALPHAR

          ALPHAR is DOUBLE PRECISION array, dimension (N)

ALPHAI

          ALPHAI is DOUBLE PRECISION array, dimension (N)

BETA

          BETA is DOUBLE PRECISION array, dimension (N)
          On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
          be the generalized eigenvalues.  ALPHAR(j) + ALPHAI(j)*i,
          and  BETA(j),j=1,...,N are the diagonals of the complex Schur
          form (S,T) that would result if the 2-by-2 diagonal blocks of
          the real Schur form of (A,B) were further reduced to
          triangular form using 2-by-2 complex unitary transformations.
          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.
          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).

VSL

          VSL is DOUBLE PRECISION array, dimension (LDVSL,N)
          If JOBVSL = 'V', VSL will contain the left Schur vectors.
          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', VSR will contain the right Schur vectors.
          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.

          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.

BWORK

          BWORK is LOGICAL array, dimension (N)
          Not referenced if SORT = '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.  (A,B) are not in Schur
                form, 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 DLAQZ0.
                =N+2: after reordering, roundoff changed values of
                      some complex eigenvalues so that leading
                      eigenvalues in the Generalized Schur form no
                      longer satisfy SELCTG=.TRUE.  This could also
                      be caused due to scaling.
                =N+3: reordering failed in DTGSEN.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 279 of file dgges3.f.