LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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◆ dgges()

subroutine dgges ( character jobvsl,
character jobvsr,
character sort,
external selctg,
integer n,
double precision, dimension( lda, * ) a,
integer lda,
double precision, dimension( ldb, * ) b,
integer ldb,
integer sdim,
double precision, dimension( * ) alphar,
double precision, dimension( * ) alphai,
double precision, dimension( * ) beta,
double precision, dimension( ldvsl, * ) vsl,
integer ldvsl,
double precision, dimension( ldvsr, * ) vsr,
integer ldvsr,
double precision, dimension( * ) work,
integer lwork,
logical, dimension( * ) bwork,
integer info )

DGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices

Download DGGES + dependencies [TGZ] [ZIP] [TXT]

Purpose:
!>
!> DGGES 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
!>  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
[in]JOBVSL
!>          JOBVSL is CHARACTER*1
!>          = 'N':  do not compute the left Schur vectors;
!>          = 'V':  compute the left Schur vectors.
!>
[in]JOBVSR
!>          JOBVSR is CHARACTER*1
!>          = 'N':  do not compute the right Schur vectors;
!>          = 'V':  compute the right Schur vectors.
!>
[in]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);
!>
[in]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.
!>
[in]N
!>          N is INTEGER
!>          The order of the matrices A, B, VSL, and VSR.  N >= 0.
!>
[in,out]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.
!>
[in]LDA
!>          LDA is INTEGER
!>          The leading dimension of A.  LDA >= max(1,N).
!>
[in,out]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.
!>
[in]LDB
!>          LDB is INTEGER
!>          The leading dimension of B.  LDB >= max(1,N).
!>
[out]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.)
!>
[out]ALPHAR
!>          ALPHAR is DOUBLE PRECISION array, dimension (N)
!>
[out]ALPHAI
!>          ALPHAI is DOUBLE PRECISION array, dimension (N)
!>
[out]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).
!>
[out]VSL
!>          VSL is DOUBLE PRECISION array, dimension (LDVSL,N)
!>          If JOBVSL = 'V', VSL will contain the left Schur vectors.
!>          Not referenced if JOBVSL = 'N'.
!>
[in]LDVSL
!>          LDVSL is INTEGER
!>          The leading dimension of the matrix VSL. LDVSL >=1, and
!>          if JOBVSL = 'V', LDVSL >= N.
!>
[out]VSR
!>          VSR is DOUBLE PRECISION array, dimension (LDVSR,N)
!>          If JOBVSR = 'V', VSR will contain the right Schur vectors.
!>          Not referenced if JOBVSR = 'N'.
!>
[in]LDVSR
!>          LDVSR is INTEGER
!>          The leading dimension of the matrix VSR. LDVSR >= 1, and
!>          if JOBVSR = 'V', LDVSR >= N.
!>
[out]WORK
!>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
!>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
!>
[in]LWORK
!>          LWORK is INTEGER
!>          The dimension of the array WORK.
!>          If N = 0, LWORK >= 1, else LWORK >= MAX(8*N,6*N+16).
!>          For good performance, LWORK must generally be larger.
!>
!>          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.
!>
[out]BWORK
!>          BWORK is LOGICAL array, dimension (N)
!>          Not referenced if SORT = 'N'.
!>
[out]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 DHGEQZ.
!>                =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.