Using the Hankel matrix of a noncommutative polynomial one can determine the rank (Matrices de Hankel by Fliess). The extended Ho-Algorithmus (Fornasini+Marchesini) is used to create a minimal linear representation. Another approach is implemented in NonCommutativeRationalFunctions.
- display: Record(rows: List FreeMonoid VAR, cols: List FreeMonoid VAR, H: Matrix F) -> OutputForm
display(sys)prints the Hankel matrix in an extended form with row-indices as column 0 and column-indices as row 0.
- findNonZeroEntry: (Matrix F, NonNegativeInteger, NonNegativeInteger) -> List NonNegativeInteger
findNonZeroEntry(A,i,j)returns the indices [
l] with i<=k<=m and j<=l<=n such that A(
l) is non-zero and [0,0] otherwise. A is of size
- hankelIndices: XDistributedPolynomial(VAR, F) -> List FreeMonoid VAR
hankelIndices(p)returns a list of all left and right factors of the monomials of a given multivariate noncommutative polynomial. Factorization:
- hankelMatrix: (XDistributedPolynomial(VAR, F), VAR) -> Matrix F
hankelMatrix(p,x)returns a matrix with the entries of the coefficients of
v) where the monomials factorizes through
w= u*x*v. This matrix is indexed by all words of the Hankel matrix.
- hankelMatrix: XDistributedPolynomial(VAR, F) -> Matrix F
hankelMatrix(p)returns the Hankel matrix
p) of given polynomial
p, i.e. the entries at (
v) are the coefficients of the monomials
w= u*v. Rows and columns are indexed by words.
- hankelSystem: XDistributedPolynomial(VAR, F) -> Record(rows: List FreeMonoid VAR, cols: List FreeMonoid VAR, H: Matrix F)
hankelSystem(p)creates a Hankel-matrix for the polynomial
pwith respect to the basis of all factors in
p. Row- and column-indices can be different.
- minimalMatrix: (Matrix F, NonNegativeInteger) -> Matrix F
minimalMatrix(A,r)returns the minimal (upper-left) submatrix of A such that the rank corresponds to the given.
- minimalRepresentation: XDistributedPolynomial(VAR, F) -> Record(alpha: Matrix F, mu: List Matrix F, var: List VAR, beta: Matrix F)
minimalRepresentation(p)returns a minimal representation using the generalized Ho algorithm [Fornasini, 1978]
p= sum_w alpha*mu(
w)*beta*w [Theorem 3.3, Salomaa–Soittola 1978]