SRC/sggev3.f(3)            Library Functions Manual            SRC/sggev3.f(3)

NAME
       SRC/sggev3.f

SYNOPSIS
   Functions/Subroutines
       subroutine sggev3 (jobvl, jobvr, n, a, lda, b, ldb, alphar, alphai,
           beta, vl, ldvl, vr, ldvr, work, lwork, info)
            SGGEV3 computes the eigenvalues and, optionally, the left and/or
           right eigenvectors for GE matrices (blocked algorithm)

Function/Subroutine Documentation
   subroutine sggev3 (character jobvl, character jobvr, integer n, real,
       dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b,
       integer ldb, real, dimension( * ) alphar, real, dimension( * ) alphai,
       real, dimension( * ) beta, real, dimension( ldvl, * ) vl, integer ldvl,
       real, dimension( ldvr, * ) vr, integer ldvr, real, dimension( * ) work,
       integer lwork, integer info)
        SGGEV3 computes the eigenvalues and, optionally, the left and/or right
       eigenvectors for GE matrices (blocked algorithm)

       Purpose:


            SGGEV3 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.

            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).

       Parameters
           JOBVL

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

           JOBVR

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

           N

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

           A

                     A is REAL array, dimension (LDA, N)
                     On entry, the matrix A in the pair (A,B).
                     On exit, A has been overwritten.

           LDA

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

           B

                     B is REAL array, dimension (LDB, N)
                     On entry, the matrix B in the pair (A,B).
                     On exit, B has been overwritten.

           LDB

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

           ALPHAR

                     ALPHAR is REAL array, dimension (N)

           ALPHAI

                     ALPHAI is REAL array, dimension (N)

           BETA

                     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

                     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 is scaled so the largest component has
                     abs(real part)+abs(imag. part)=1.
                     Not referenced if JOBVL = 'N'.

           LDVL

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

           VR

                     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 is scaled so the largest component has
                     abs(real part)+abs(imag. part)=1.
                     Not referenced if JOBVR = 'N'.

           LDVR

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

           WORK

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

           LWORK

                     LWORK is INTEGER

                     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.  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 SLAQZ0.
                           =N+2: error return from STGEVC.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Definition at line 223 of file sggev3.f.

Author
       Generated automatically by Doxygen for LAPACK from the source code.

LAPACK                          Version 3.12.0                 SRC/sggev3.f(3)