sggev3.f(3)

NAME
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 SHGEQZ.
                    =N+2: error return from STGEVC.

Author:
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date:
    January 2015

Definition at line 228 of file sggev3.f.

Author

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