from sympy.core.singleton import S from sympy.core.sympify import _sympify from sympy.polys.polytools import Poly from .matexpr import MatrixExpr class CompanionMatrix(MatrixExpr): """A symbolic companion matrix of a polynomial. Examples ======== >>> from sympy import Poly, Symbol, symbols >>> from sympy.matrices.expressions import CompanionMatrix >>> x = Symbol('x') >>> c0, c1, c2, c3, c4 = symbols('c0:5') >>> p = Poly(c0 + c1*x + c2*x**2 + c3*x**3 + c4*x**4 + x**5, x) >>> CompanionMatrix(p) CompanionMatrix(Poly(x**5 + c4*x**4 + c3*x**3 + c2*x**2 + c1*x + c0, x, domain='ZZ[c0,c1,c2,c3,c4]')) """ def __new__(cls, poly): poly = _sympify(poly) if not isinstance(poly, Poly): raise ValueError("{} must be a Poly instance.".format(poly)) if not poly.is_monic: raise ValueError("{} must be a monic polynomial.".format(poly)) if not poly.is_univariate: raise ValueError( "{} must be a univariate polynomial.".format(poly)) if not poly.degree() >= 1: raise ValueError( "{} must have degree not less than 1.".format(poly)) return super().__new__(cls, poly) @property def shape(self): poly = self.args[0] size = poly.degree() return size, size def _entry(self, i, j): if j == self.cols - 1: return -self.args[0].all_coeffs()[-1 - i] elif i == j + 1: return S.One return S.Zero def as_explicit(self): from sympy.matrices.immutable import ImmutableDenseMatrix return ImmutableDenseMatrix.companion(self.args[0])