SGGEVX(3)      LAPACK routine of NEC Numeric Library Collection      SGGEVX(3)



NAME
       SGGEVX

SYNOPSIS
       SUBROUTINE SGGEVX (BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB,
           ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, ILO, IHI, LSCALE, RSCALE,
           ABNRM, BBNRM, RCONDE, RCONDV, WORK, LWORK, IWORK, BWORK, INFO)



PURPOSE
            SGGEVX computes for a pair of N-by-N real nonsymmetric matrices (A,B)
            the generalized eigenvalues, and optionally, the left and/or right
            generalized eigenvectors.

            Optionally also, it computes a balancing transformation to improve
            the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
            LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for
            the eigenvalues (RCONDE), and reciprocal condition numbers for the
            right eigenvectors (RCONDV).

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

            The right eigenvector v(j) corresponding to the eigenvalue lambda(j)
            of (A,B) satisfies

                             A * v(j) = lambda(j) * B * v(j) .

            The left eigenvector u(j) corresponding to the eigenvalue lambda(j)
            of (A,B) satisfies

                             u(j)**H * A  = lambda(j) * u(j)**H * B.

            where u(j)**H is the conjugate-transpose of u(j).




ARGUMENTS
           BALANC    (input)
                     BALANC is CHARACTER*1
                     Specifies the balance option to be performed.
                     = 'N':  do not diagonally scale or permute;
                     = 'P':  permute only;
                     = 'S':  scale only;
                     = 'B':  both permute and scale.
                     Computed reciprocal condition numbers will be for the
                     matrices after permuting and/or balancing. Permuting does
                     not change condition numbers (in exact arithmetic), but
                     balancing does.

           JOBVL     (input)
                     JOBVL is CHARACTER*1
                     = 'N':  do not compute the left generalized eigenvectors;
                     = 'V':  compute the left generalized eigenvectors.

           JOBVR     (input)
                     JOBVR is CHARACTER*1
                     = 'N':  do not compute the right generalized eigenvectors;
                     = 'V':  compute the right generalized eigenvectors.

           SENSE     (input)
                     SENSE is CHARACTER*1
                     Determines which reciprocal condition numbers are computed.
                     = 'N': none are computed;
                     = 'E': computed for eigenvalues only;
                     = 'V': computed for eigenvectors only;
                     = 'B': computed for eigenvalues and eigenvectors.

           N         (input)
                     N is INTEGER
                     The order of the matrices A, B, VL, and VR.  N >= 0.

           A         (input/output)
                     A is REAL array, dimension (LDA, N)
                     On entry, the matrix A in the pair (A,B).
                     On exit, A has been overwritten. If JOBVL='V' or JOBVR='V'
                     or both, then A contains the first part of the real Schur
                     form of the "balanced" versions of the input A and B.

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

           B         (input/output)
                     B is REAL array, dimension (LDB, N)
                     On entry, the matrix B in the pair (A,B).
                     On exit, B has been overwritten. If JOBVL='V' or JOBVR='V'
                     or both, then B contains the second part of the real Schur
                     form of the "balanced" versions of the input A and B.

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

           ALPHAR    (output)
                     ALPHAR is REAL array, dimension (N)

           ALPHAI    (output)
                     ALPHAI is REAL array, dimension (N)

           BETA      (output)
                     BETA is REAL array, dimension (N)
                     On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
                     be the generalized eigenvalues.  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
                     ALPHA/BETA. 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).

           VL        (output)
                     VL is REAL array, dimension (LDVL,N)
                     If JOBVL = 'V', the left eigenvectors u(j) are stored one
                     after another in the columns of VL, in the same order as
                     their eigenvalues. If the j-th eigenvalue is real, then
                     u(j) = VL(:,j), the j-th column of VL. If the j-th and
                     (j+1)-th 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 will be scaled so the largest component have
                     abs(real part) + abs(imag. part) = 1.
                     Not referenced if JOBVL = 'N'.

           LDVL      (input)
                     LDVL is INTEGER
                     The leading dimension of the matrix VL. LDVL >= 1, and
                     if JOBVL = 'V', LDVL >= N.

           VR        (output)
                     VR is REAL array, dimension (LDVR,N)
                     If JOBVR = 'V', the right eigenvectors v(j) are stored one
                     after another in the columns of VR, in the same order as
                     their eigenvalues. If the j-th eigenvalue is real, then
                     v(j) = VR(:,j), the j-th column of VR. If the j-th and
                     (j+1)-th eigenvalues form a complex conjugate pair, then
                     v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1).
                     Each eigenvector will be scaled so the largest component have
                     abs(real part) + abs(imag. part) = 1.
                     Not referenced if JOBVR = 'N'.

           LDVR      (input)
                     LDVR is INTEGER
                     The leading dimension of the matrix VR. LDVR >= 1, and
                     if JOBVR = 'V', LDVR >= N.

           ILO       (output)
                     ILO is INTEGER

           IHI       (output)
                     IHI is INTEGER
                     ILO and IHI are integer values such that on exit
                     A(i,j) = 0 and B(i,j) = 0 if i > j and
                     j = 1,...,ILO-1 or i = IHI+1,...,N.
                     If BALANC = 'N' or 'S', ILO = 1 and IHI = N.

           LSCALE    (output)
                     LSCALE is REAL array, dimension (N)
                     Details of the permutations and scaling factors applied
                     to the left side of A and B.  If PL(j) is the index of the
                     row interchanged with row j, and DL(j) is the scaling
                     factor applied to row j, then
                       LSCALE(j) = PL(j)  for j = 1,...,ILO-1
                                 = DL(j)  for j = ILO,...,IHI
                                 = PL(j)  for j = IHI+1,...,N.
                     The order in which the interchanges are made is N to IHI+1,
                     then 1 to ILO-1.

           RSCALE    (output)
                     RSCALE is REAL array, dimension (N)
                     Details of the permutations and scaling factors applied
                     to the right side of A and B.  If PR(j) is the index of the
                     column interchanged with column j, and DR(j) is the scaling
                     factor applied to column j, then
                       RSCALE(j) = PR(j)  for j = 1,...,ILO-1
                                 = DR(j)  for j = ILO,...,IHI
                                 = PR(j)  for j = IHI+1,...,N
                     The order in which the interchanges are made is N to IHI+1,
                     then 1 to ILO-1.

           ABNRM     (output)
                     ABNRM is REAL
                     The one-norm of the balanced matrix A.

           BBNRM     (output)
                     BBNRM is REAL
                     The one-norm of the balanced matrix B.

           RCONDE    (output)
                     RCONDE is REAL array, dimension (N)
                     If SENSE = 'E' or 'B', the reciprocal condition numbers of
                     the eigenvalues, stored in consecutive elements of the array.
                     For a complex conjugate pair of eigenvalues two consecutive
                     elements of RCONDE are set to the same value. Thus RCONDE(j),
                     RCONDV(j), and the j-th columns of VL and VR all correspond
                     to the j-th eigenpair.
                     If SENSE = 'N' or 'V', RCONDE is not referenced.

           RCONDV    (output)
                     RCONDV is REAL array, dimension (N)
                     If SENSE = 'V' or 'B', the estimated reciprocal condition
                     numbers of the eigenvectors, stored in consecutive elements
                     of the array. For a complex eigenvector two consecutive
                     elements of RCONDV are set to the same value. If the
                     eigenvalues cannot be reordered to compute RCONDV(j),
                     RCONDV(j) is set to 0; this can only occur when the true
                     value would be very small anyway.
                     If SENSE = 'N' or 'E', RCONDV is not referenced.

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

           LWORK     (input)
                     LWORK is INTEGER
                     The dimension of the array WORK. LWORK >= max(1,2*N).
                     If BALANC = 'S' or 'B', or JOBVL = 'V', or JOBVR = 'V',
                     LWORK >= max(1,6*N).
                     If SENSE = 'E', LWORK >= max(1,10*N).
                     If SENSE = 'V' or 'B', LWORK >= 2*N*N+8*N+16.

                     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.

           IWORK     (output)
                     IWORK is INTEGER array, dimension (N+6)
                     If SENSE = 'E', IWORK is not referenced.

           BWORK     (output)
                     BWORK is LOGICAL array, dimension (N)
                     If SENSE = 'N', BWORK is not referenced.

           INFO      (output)
                     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:  =N+1: other than QZ iteration failed in SHGEQZ.
                           =N+2: error return from STGEVC.






FURTHER DETAILS
             Balancing a matrix pair (A,B) includes, first, permuting rows and
             columns to isolate eigenvalues, second, applying diagonal similarity
             transformation to the rows and columns to make the rows and columns
             as close in norm as possible. The computed reciprocal condition
             numbers correspond to the balanced matrix. Permuting rows and columns
             will not change the condition numbers (in exact arithmetic) but
             diagonal scaling will.  For further explanation of balancing, see
             section 4.11.1.2 of LAPACK Users' Guide.

             An approximate error bound on the chordal distance between the i-th
             computed generalized eigenvalue w and the corresponding exact
             eigenvalue lambda is

                  chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I)

             An approximate error bound for the angle between the i-th computed
             eigenvector VL(i) or VR(i) is given by

                  EPS * norm(ABNRM, BBNRM) / DIF(i).

             For further explanation of the reciprocal condition numbers RCONDE
             and RCONDV, see section 4.11 of LAPACK User's Guide.



LAPACK routine                  31 October 2017                      SGGEVX(3)