def chebyshev_polynomial(self, n, kind='first'):
"""
Generates an endomorphism of this affine line by a Chebyshev polynomial.
Chebyshev polynomials are a sequence of recursively defined orthogonal
polynomials. Chebyshev of the first kind are defined as `T_0(x) = 1`,
`T_1(x) = x`, and `T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x)`. Chebyshev of
the second kind are defined as `U_0(x) = 1`,
`U_1(x) = 2x`, and `U_{n+1}(x) = 2xU_n(x) - U_{n-1}(x)`.
INPUT:
- ``n`` -- a non-negative integer.
- ``kind`` -- ``first`` or ``second`` specifying which kind of chebyshev the user would like
to generate. Defaults to ``first``.
OUTPUT: :class:`DynamicalSystem_affine`
EXAMPLES::
sage: A.<x> = AffineSpace(QQ, 1)
sage: A.chebyshev_polynomial(5, 'first')
Dynamical System of Affine Space of dimension 1 over Rational Field
Defn: Defined on coordinates by sending (x) to
(16*x^5 - 20*x^3 + 5*x)
::
sage: A.<x> = AffineSpace(QQ, 1)
sage: A.chebyshev_polynomial(3, 'second')
Dynamical System of Affine Space of dimension 1 over Rational Field
Defn: Defined on coordinates by sending (x) to
(8*x^3 - 4*x)
::
sage: A.<x> = AffineSpace(QQ, 1)
sage: A.chebyshev_polynomial(3, 2)
Traceback (most recent call last):
...
ValueError: keyword 'kind' must have a value of either 'first' or 'second'
::
sage: A.<x> = AffineSpace(QQ, 1)
sage: A.chebyshev_polynomial(-4, 'second')
Traceback (most recent call last):
...
ValueError: first parameter 'n' must be a non-negative integer
::
sage: A = AffineSpace(QQ, 2, 'x')
sage: A.chebyshev_polynomial(2)
Traceback (most recent call last):
...
TypeError: affine space must be of dimension 1
"""
if self.dimension_relative() != 1:
raise TypeError("affine space must be of dimension 1")
n = ZZ(n)
if (n < 0):
raise ValueError("first parameter 'n' must be a non-negative integer")
from sage.dynamics.arithmetic_dynamics.affine_ds import DynamicalSystem_affine
if kind == 'first':
return DynamicalSystem_affine([chebyshev_T(n, self.gen(0))], domain=self)
elif kind == 'second':
return DynamicalSystem_affine([chebyshev_U(n, self.gen(0))], domain=self)
else:
raise ValueError("keyword 'kind' must have a value of either 'first' or 'second'")
def chebyshev_polynomial(self, n, kind='first'):
"""
Generates an endomorphism of this affine line by a Chebyshev polynomial.
Chebyshev polynomials are a sequence of recursively defined orthogonal
polynomials. Chebyshev of the first kind are defined as `T_0(x) = 1`,
`T_1(x) = x`, and `T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x)`. Chebyshev of
the second kind are defined as `U_0(x) = 1`,
`U_1(x) = 2x`, and `U_{n+1}(x) = 2xU_n(x) - U_{n-1}(x)`.
INPUT:
- ``n`` -- a non-negative integer.
- ``kind`` -- ``first`` or ``second`` specifying which kind of chebyshev the user would like
to generate. Defaults to ``first``.
OUTPUT: :class:`DynamicalSystem_affine`
EXAMPLES::
sage: A.<x> = AffineSpace(QQ, 1)
sage: A.chebyshev_polynomial(5, 'first')
Dynamical System of Affine Space of dimension 1 over Rational Field
Defn: Defined on coordinates by sending (x) to
(16*x^5 - 20*x^3 + 5*x)
::
sage: A.<x> = AffineSpace(QQ, 1)
sage: A.chebyshev_polynomial(3, 'second')
Dynamical System of Affine Space of dimension 1 over Rational Field
Defn: Defined on coordinates by sending (x) to
(8*x^3 - 4*x)
::
sage: A.<x> = AffineSpace(QQ, 1)
sage: A.chebyshev_polynomial(3, 2)
Traceback (most recent call last):
...
ValueError: keyword 'kind' must have a value of either 'first' or 'second'
::
sage: A.<x> = AffineSpace(QQ, 1)
sage: A.chebyshev_polynomial(-4, 'second')
Traceback (most recent call last):
...
ValueError: first parameter 'n' must be a non-negative integer
::
sage: A = AffineSpace(QQ, 2, 'x')
sage: A.chebyshev_polynomial(2)
Traceback (most recent call last):
...
TypeError: affine space must be of dimension 1
"""
if self.dimension_relative() != 1:
raise TypeError("affine space must be of dimension 1")
n = ZZ(n)
if (n < 0):
raise ValueError("first parameter 'n' must be a non-negative integer")
from sage.dynamics.arithmetic_dynamics.affine_ds import DynamicalSystem_affine
if kind == 'first':
return DynamicalSystem_affine([chebyshev_T(n, self.gen(0))], domain=self)
elif kind == 'second':
return DynamicalSystem_affine([chebyshev_U(n, self.gen(0))], domain=self)
else:
raise ValueError("keyword 'kind' must have a value of either 'first' or 'second'")
def chebyshev_polynomial(self, n, kind='first', monic=False):
"""
Generates an endomorphism of this affine line by a Chebyshev polynomial.
Chebyshev polynomials are a sequence of recursively defined orthogonal
polynomials. Chebyshev of the first kind are defined as `T_0(x) = 1`,
`T_1(x) = x`, and `T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x)`. Chebyshev of
the second kind are defined as `U_0(x) = 1`,
`U_1(x) = 2x`, and `U_{n+1}(x) = 2xU_n(x) - U_{n-1}(x)`.
INPUT:
- ``n`` -- a non-negative integer.
- ``kind`` -- ``first`` or ``second`` specifying which kind of chebyshev the user would like
to generate. Defaults to ``first``.
- ``monic`` -- ``True`` or ``False`` specifying if the polynomial defining the system
should be monic or not. Defaults to ``False``.
OUTPUT: :class:`DynamicalSystem_affine`
EXAMPLES::
sage: A.<x> = AffineSpace(QQ, 1)
sage: A.chebyshev_polynomial(5, 'first')
Dynamical System of Affine Space of dimension 1 over Rational Field
Defn: Defined on coordinates by sending (x) to
(16*x^5 - 20*x^3 + 5*x)
::
sage: A.<x> = AffineSpace(QQ, 1)
sage: A.chebyshev_polynomial(3, 'second')
Dynamical System of Affine Space of dimension 1 over Rational Field
Defn: Defined on coordinates by sending (x) to
(8*x^3 - 4*x)
::
sage: A.<x> = AffineSpace(QQ, 1)
sage: A.chebyshev_polynomial(3, 2)
Traceback (most recent call last):
...
ValueError: keyword 'kind' must have a value of either 'first' or 'second'
::
sage: A.<x> = AffineSpace(QQ, 1)
sage: A.chebyshev_polynomial(-4, 'second')
Traceback (most recent call last):
...
ValueError: first parameter 'n' must be a non-negative integer
::
sage: A = AffineSpace(QQ, 2, 'x')
sage: A.chebyshev_polynomial(2)
Traceback (most recent call last):
...
TypeError: affine space must be of dimension 1
::
sage: A.<x> = AffineSpace(QQ, 1)
sage: A.chebyshev_polynomial(7, monic=True)
Dynamical System of Affine Space of dimension 1 over Rational Field
Defn: Defined on coordinates by sending (x) to
(x^7 - 7*x^5 + 14*x^3 - 7*x)
::
sage: F.<t> = FunctionField(QQ)
sage: A.<x> = AffineSpace(F,1)
sage: A.chebyshev_polynomial(4, monic=True)
Dynamical System of Affine Space of dimension 1 over Rational function field in t over Rational Field
Defn: Defined on coordinates by sending (x) to
(x^4 + (-4)*x^2 + 2)
"""
if self.dimension_relative() != 1:
raise TypeError("affine space must be of dimension 1")
n = ZZ(n)
if (n < 0):
raise ValueError("first parameter 'n' must be a non-negative integer")
from sage.dynamics.arithmetic_dynamics.affine_ds import DynamicalSystem_affine
if kind == 'first':
if monic and self.base().characteristic() != 2:
f = DynamicalSystem_affine([chebyshev_T(n, self.gen(0))], domain=self)
f = f.homogenize(1)
f = f.conjugate(matrix([[1/ZZ(2), 0],[0, 1]]))
f = f.dehomogenize(1)
return f
return DynamicalSystem_affine([chebyshev_T(n, self.gen(0))], domain=self)
elif kind == 'second':
if monic and self.base().characteristic() != 2:
f = DynamicalSystem_affine([chebyshev_T(n, self.gen(0))], domain=self)
f = f.homogenize(1)
f = f.conjugate(matrix([[1/ZZ(2), 0],[0, 1]]))
f = f.dehomogenize(1)
return f
return DynamicalSystem_affine([chebyshev_U(n, self.gen(0))], domain=self)
else:
raise ValueError("keyword 'kind' must have a value of either 'first' or 'second'")