Provided by: liblapack-doc_3.3.1-1_all bug

NAME

       LAPACK-3 - computes for a pair of N-by-N real nonsymmetric matrices (A,B), the generalized
       eigenvalues, the real Schur form (S,T), and,

SYNOPSIS

       SUBROUTINE SGGESX( JOBVSL, JOBVSR, SORT, SELCTG, SENSE, N, A, LDA, B, LDB,  SDIM,  ALPHAR,
                          ALPHAI,  BETA,  VSL,  LDVSL,  VSR,  LDVSR, RCONDE, RCONDV, WORK, LWORK,
                          IWORK, LIWORK, BWORK, INFO )

           CHARACTER      JOBVSL, JOBVSR, SENSE, SORT

           INTEGER        INFO, LDA, LDB, LDVSL, LDVSR, LIWORK, LWORK, N, SDIM

           LOGICAL        BWORK( * )

           INTEGER        IWORK( * )

           REAL           A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B( LDB, * ), BETA( * ),  RCONDE(
                          2 ), RCONDV( 2 ), VSL( LDVSL, * ), VSR( LDVSR, * ), WORK( * )

           LOGICAL        SELCTG

           EXTERNAL       SELCTG

PURPOSE

       SGGESX  computes  for  a  pair of N-by-N real nonsymmetric matrices (A,B), the generalized
       eigenvalues, the real Schur form (S,T), and,
        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; computes
        a reciprocal condition number for the average of the selected
        eigenvalues (RCONDE); and computes a reciprocal condition number for
        the right and left deflating subspaces corresponding to the selected
        eigenvalues (RCONDV). The leading columns of VSL and VSR then form
        an orthonormal basis for the corresponding left and right eigenspaces
        (deflating subspaces).
        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 for 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.

ARGUMENTS

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

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

        SORT    (input) 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  (external procedure) LOGICAL FUNCTION of three REAL 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 a selected complex eigenvalue may no longer satisfy
                SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) = .TRUE. after ordering,
                since ordering may change the value of complex eigenvalues
                (especially if the eigenvalue is ill-conditioned), in this
                case INFO is set to N+3.

        SENSE   (input) CHARACTER*1
                Determines which reciprocal condition numbers are computed.
                = 'N' : None are computed;
                = 'E' : Computed for average of selected eigenvalues only;
                = 'V' : Computed for selected deflating subspaces only;
                = 'B' : Computed for both.
                If SENSE = 'E', 'V', or 'B', SORT must equal 'S'.

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

        A       (input/output) REAL 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     (input) INTEGER
                The leading dimension of A.  LDA >= max(1,N).

        B       (input/output) REAL 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     (input) INTEGER
                The leading dimension of B.  LDB >= max(1,N).

        SDIM    (output) 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  (output) REAL array, dimension (N)
                ALPHAI  (output) REAL array, dimension (N)
                BETA    (output) REAL 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     (output) REAL array, dimension (LDVSL,N)
                If JOBVSL = 'V', VSL will contain the left Schur vectors.
                Not referenced if JOBVSL = 'N'.

        LDVSL   (input) INTEGER
                The leading dimension of the matrix VSL. LDVSL >=1, and
                if JOBVSL = 'V', LDVSL >= N.

        VSR     (output) REAL array, dimension (LDVSR,N)
                If JOBVSR = 'V', VSR will contain the right Schur vectors.
                Not referenced if JOBVSR = 'N'.

        LDVSR   (input) INTEGER
                The leading dimension of the matrix VSR. LDVSR >= 1, and
                if JOBVSR = 'V', LDVSR >= N.

        RCONDE  (output) REAL array, dimension ( 2 )
                If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the
                reciprocal condition numbers for the average of the selected
                eigenvalues.
                Not referenced if SENSE = 'N' or 'V'.

        RCONDV  (output) REAL array, dimension ( 2 )
                If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the
                reciprocal condition numbers for the selected deflating
                subspaces.
                Not referenced if SENSE = 'N' or 'E'.

        WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
                On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

        LWORK   (input) INTEGER
                The dimension of the array WORK.
                If N = 0, LWORK >= 1, else if SENSE = 'E', 'V', or 'B',
                LWORK >= max( 8*N, 6*N+16, 2*SDIM*(N-SDIM) ), else
                LWORK >= max( 8*N, 6*N+16 ).
                Note that 2*SDIM*(N-SDIM) <= N*N/2.
                Note also that an error is only returned if
                LWORK < max( 8*N, 6*N+16), but if SENSE = 'E' or 'V' or 'B'
                this may not be large enough.
                If LWORK = -1, then a workspace query is assumed; the routine
                only calculates the bound on the optimal size of the WORK
                array and the minimum size of the IWORK array, returns these
                values as the first entries of the WORK and IWORK arrays, and
                no error message related to LWORK or LIWORK is issued by
                XERBLA.

        IWORK   (workspace) INTEGER array, dimension (MAX(1,LIWORK))
                On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.

        LIWORK  (input) INTEGER
                The dimension of the array IWORK.
                If SENSE = 'N' or N = 0, LIWORK >= 1, otherwise
                LIWORK >= N+6.
                If LIWORK = -1, then a workspace query is assumed; the
                routine only calculates the bound on the optimal size of the
                WORK array and the minimum size of the IWORK array, returns
                these values as the first entries of the WORK and IWORK
                arrays, and no error message related to LWORK or LIWORK is
                issued by XERBLA.

        BWORK   (workspace) LOGICAL array, dimension (N)
                Not referenced if SORT = 'N'.

        INFO    (output) 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 SHGEQZ
                =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 STGSEN.

FURTHER DETAILS

        An approximate (asymptotic) bound on the average absolute error of
        the selected eigenvalues is
             EPS * norm((A, B)) / RCONDE( 1 ).
        An approximate (asymptotic) bound on the maximum angular error in
        the computed deflating subspaces is
             EPS * norm((A, B)) / RCONDV( 2 ).
        See LAPACK User's Guide, section 4.11 for more information.

 LAPACK driver routine (version 3.2.1)      April 2011                            SGGESX(3lapack)