.Modelica.Math.Matrices.LAPACK.dtrevc

Information

Lapack documentation
    Purpose
    =======

    DTREVC computes some or all of the right and/or left eigenvectors of
    a real upper quasi-triangular matrix T.
    Matrices of this type are produced by the Schur factorization of
    a real general matrix:  A = Q*T*Q**T, as computed by DHSEQR.

    The right eigenvector x and the left eigenvector y of T corresponding
    to an eigenvalue w are defined by:

       T*x = w*x,     (y**H)*T = w*(y**H)

    where y**H denotes the conjugate transpose of y.
    The eigenvalues are not input to this routine, but are read directly
    from the diagonal blocks of T.

    This routine returns the matrices X and/or Y of right and left
    eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an
    input matrix.  If Q is the orthogonal factor that reduces a matrix
    A to Schur form T, then Q*X and Q*Y are the matrices of right and
    left eigenvectors of A.

    Arguments
    =========

    SIDE    (input) CHARACTER*1
            = 'R':  compute right eigenvectors only;
            = 'L':  compute left eigenvectors only;
            = 'B':  compute both right and left eigenvectors.

    HOWMNY  (input) CHARACTER*1
            = 'A':  compute all right and/or left eigenvectors;
            = 'B':  compute all right and/or left eigenvectors,
                    backtransformed by the matrices in VR and/or VL;
            = 'S':  compute selected right and/or left eigenvectors,
                    as indicated by the logical array SELECT.

    SELECT  (input/output) LOGICAL array, dimension (N)
            If HOWMNY = 'S', SELECT specifies the eigenvectors to be
            computed.
            If w(j) is a real eigenvalue, the corresponding real
            eigenvector is computed if SELECT(j) is .TRUE..
            If w(j) and w(j+1) are the real and imaginary parts of a
            complex eigenvalue, the corresponding complex eigenvector is
            computed if either SELECT(j) or SELECT(j+1) is .TRUE., and
            on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is set to
            .FALSE..
            Not referenced if HOWMNY = 'A' or 'B'.

    N       (input) INTEGER
            The order of the matrix T. N >= 0.

    T       (input) DOUBLE PRECISION array, dimension (LDT,N)
            The upper quasi-triangular matrix T in Schur canonical form.

    LDT     (input) INTEGER
            The leading dimension of the array T. LDT >= max(1,N).

    VL      (input/output) DOUBLE PRECISION array, dimension (LDVL,MM)
            On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
            contain an N-by-N matrix Q (usually the orthogonal matrix Q
            of Schur vectors returned by DHSEQR).
            On exit, if SIDE = 'L' or 'B', VL contains:
            if HOWMNY = 'A', the matrix Y of left eigenvectors of T;
            if HOWMNY = 'B', the matrix Q*Y;
            if HOWMNY = 'S', the left eigenvectors of T specified by
                             SELECT, stored consecutively in the columns
                             of VL, in the same order as their
                             eigenvalues.
            A complex eigenvector corresponding to a complex eigenvalue
            is stored in two consecutive columns, the first holding the
            real part, and the second the imaginary part.
            Not referenced if SIDE = 'R'.

    LDVL    (input) INTEGER
            The leading dimension of the array VL.  LDVL >= 1, and if
            SIDE = 'L' or 'B', LDVL >= N.

    VR      (input/output) DOUBLE PRECISION array, dimension (LDVR,MM)
            On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
            contain an N-by-N matrix Q (usually the orthogonal matrix Q
            of Schur vectors returned by DHSEQR).
            On exit, if SIDE = 'R' or 'B', VR contains:
            if HOWMNY = 'A', the matrix X of right eigenvectors of T;
            if HOWMNY = 'B', the matrix Q*X;
            if HOWMNY = 'S', the right eigenvectors of T specified by
                             SELECT, stored consecutively in the columns
                             of VR, in the same order as their
                             eigenvalues.
            A complex eigenvector corresponding to a complex eigenvalue
            is stored in two consecutive columns, the first holding the
            real part and the second the imaginary part.
            Not referenced if SIDE = 'L'.

    LDVR    (input) INTEGER
            The leading dimension of the array VR.  LDVR >= 1, and if
            SIDE = 'R' or 'B', LDVR >= N.

    MM      (input) INTEGER
            The number of columns in the arrays VL and/or VR. MM >= M.

    M       (output) INTEGER
            The number of columns in the arrays VL and/or VR actually
            used to store the eigenvectors.
            If HOWMNY = 'A' or 'B', M is set to N.
            Each selected real eigenvector occupies one column and each
            selected complex eigenvector occupies two columns.

    WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)

    INFO    (output) INTEGER
            = 0:  successful exit
            < 0:  if INFO = -i, the i-th argument had an illegal value

    Further Details
    ===============

    The algorithm used in this program is basically backward (forward)
    substitution, with scaling to make the code robust against
    possible overflow.

    Each eigenvector is normalized so that the element of largest
    magnitude has magnitude 1; here the magnitude of a complex number
    (x,y) is taken to be |x| + |y|.

Interface

pure function dtrevc
  extends Modelica.Icons.Function;
  input Real T[:, size(T, 1)] "Upper quasi triangular matrix";
  input String side = "R" "Specify which eigenvectors";
  input String howmny = "B" "Specify how many eigenvectors";
  input Real Q[size(T, 1), size(T, 1)] "Orthogonal matrix Q of Schur vectors returned by DHSEQR";
  output Real lEigenVectors[size(T, 1), size(T, 1)] = Q "Left eigenvectors of matrix T";
  output Real rEigenVectors[size(T, 1), size(T, 1)] = Q "Right eigenvectors of matrix T";
  output Integer info;
end dtrevc;