.Modelica.Math.Matrices.LAPACK.dggev

Information

Lapack documentation
    Purpose
    =======

    DGGEV 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).

    Arguments
    =========

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

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

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

    A       (input/output) DOUBLE PRECISION array, dimension (LDA, N)
            On entry, the matrix A in the pair (A,B).
            On exit, A has been overwritten.

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

    B       (input/output) DOUBLE PRECISION array, dimension (LDB, N)
            On entry, the matrix B in the pair (A,B).
            On exit, B has been overwritten.

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

    ALPHAR  (output) DOUBLE PRECISION array, dimension (N)
    ALPHAI  (output) DOUBLE PRECISION array, dimension (N)
    BETA    (output) DOUBLE PRECISION 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) DOUBLE PRECISION 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    (input) INTEGER
            The leading dimension of the matrix VL. LDVL >= 1, and
            if JOBVL = 'V', LDVL >= N.

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

    WORK    (workspace/output) DOUBLE PRECISION 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.  LWORK >= max(1,8*N).
            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.

    INFO    (output) 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 DHGEQZ.
                  =N+2: error return from DTGEVC.

Interface

pure function dggev
  extends Modelica.Icons.Function;
  input Real A[:, size(A, 1)];
  input Real B[size(A, 1), size(A, 1)];
  input Integer nA = size(A, 1) "The actual dimensions of matrices A and B (the computation is performed for A[1:nA,1:nA], B[1:nA,1:nA])";
  output Real alphaReal[size(A, 1)] "Real part of alpha (eigenvalue=(alphaReal+i*alphaImag)/beta)";
  output Real alphaImag[size(A, 1)] "Imaginary part of alpha";
  output Real beta[size(A, 1)] "Denominator of eigenvalue";
  output Real lEigenVectors[size(A, 1), size(A, 1)] "Left eigenvectors of matrix A";
  output Real rEigenVectors[size(A, 1), size(A, 1)] "Right eigenvectors of matrix A";
  output Integer info;
end dggev;