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341 lines
11 KiB
341 lines
11 KiB
"""Functions for generating interesting polynomials, e.g. for benchmarking. """
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from sympy.core import Add, Mul, Symbol, sympify, Dummy, symbols
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from sympy.core.containers import Tuple
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from sympy.core.singleton import S
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from sympy.ntheory import nextprime
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from sympy.polys.densearith import (
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dmp_add_term, dmp_neg, dmp_mul, dmp_sqr
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)
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from sympy.polys.densebasic import (
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dmp_zero, dmp_one, dmp_ground,
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dup_from_raw_dict, dmp_raise, dup_random
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)
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from sympy.polys.domains import ZZ
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from sympy.polys.factortools import dup_zz_cyclotomic_poly
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from sympy.polys.polyclasses import DMP
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from sympy.polys.polytools import Poly, PurePoly
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from sympy.polys.polyutils import _analyze_gens
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from sympy.utilities import subsets, public, filldedent
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@public
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def swinnerton_dyer_poly(n, x=None, polys=False):
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"""Generates n-th Swinnerton-Dyer polynomial in `x`.
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Parameters
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----------
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n : int
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`n` decides the order of polynomial
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x : optional
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polys : bool, optional
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``polys=True`` returns an expression, otherwise
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(default) returns an expression.
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"""
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if n <= 0:
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raise ValueError(
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"Cannot generate Swinnerton-Dyer polynomial of order %s" % n)
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if x is not None:
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sympify(x)
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else:
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x = Dummy('x')
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if n > 3:
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from sympy.functions.elementary.miscellaneous import sqrt
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from .numberfields import minimal_polynomial
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p = 2
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a = [sqrt(2)]
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for i in range(2, n + 1):
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p = nextprime(p)
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a.append(sqrt(p))
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return minimal_polynomial(Add(*a), x, polys=polys)
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if n == 1:
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ex = x**2 - 2
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elif n == 2:
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ex = x**4 - 10*x**2 + 1
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elif n == 3:
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ex = x**8 - 40*x**6 + 352*x**4 - 960*x**2 + 576
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return PurePoly(ex, x) if polys else ex
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@public
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def cyclotomic_poly(n, x=None, polys=False):
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"""Generates cyclotomic polynomial of order `n` in `x`.
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Parameters
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----------
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n : int
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`n` decides the order of polynomial
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x : optional
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polys : bool, optional
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``polys=True`` returns an expression, otherwise
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(default) returns an expression.
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"""
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if n <= 0:
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raise ValueError(
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"Cannot generate cyclotomic polynomial of order %s" % n)
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poly = DMP(dup_zz_cyclotomic_poly(int(n), ZZ), ZZ)
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if x is not None:
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poly = Poly.new(poly, x)
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else:
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poly = PurePoly.new(poly, Dummy('x'))
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return poly if polys else poly.as_expr()
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@public
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def symmetric_poly(n, *gens, polys=False):
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"""
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Generates symmetric polynomial of order `n`.
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Parameters
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==========
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polys: bool, optional (default: False)
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Returns a Poly object when ``polys=True``, otherwise
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(default) returns an expression.
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"""
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gens = _analyze_gens(gens)
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if n < 0 or n > len(gens) or not gens:
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raise ValueError("Cannot generate symmetric polynomial of order %s for %s" % (n, gens))
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elif not n:
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poly = S.One
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else:
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poly = Add(*[Mul(*s) for s in subsets(gens, int(n))])
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return Poly(poly, *gens) if polys else poly
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@public
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def random_poly(x, n, inf, sup, domain=ZZ, polys=False):
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"""Generates a polynomial of degree ``n`` with coefficients in
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``[inf, sup]``.
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Parameters
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----------
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x
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`x` is the independent term of polynomial
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n : int
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`n` decides the order of polynomial
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inf
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Lower limit of range in which coefficients lie
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sup
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Upper limit of range in which coefficients lie
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domain : optional
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Decides what ring the coefficients are supposed
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to belong. Default is set to Integers.
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polys : bool, optional
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``polys=True`` returns an expression, otherwise
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(default) returns an expression.
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"""
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poly = Poly(dup_random(n, inf, sup, domain), x, domain=domain)
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return poly if polys else poly.as_expr()
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@public
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def interpolating_poly(n, x, X='x', Y='y'):
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"""Construct Lagrange interpolating polynomial for ``n``
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data points. If a sequence of values are given for ``X`` and ``Y``
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then the first ``n`` values will be used.
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"""
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ok = getattr(x, 'free_symbols', None)
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if isinstance(X, str):
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X = symbols("%s:%s" % (X, n))
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elif ok and ok & Tuple(*X).free_symbols:
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ok = False
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if isinstance(Y, str):
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Y = symbols("%s:%s" % (Y, n))
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elif ok and ok & Tuple(*Y).free_symbols:
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ok = False
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if not ok:
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raise ValueError(filldedent('''
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Expecting symbol for x that does not appear in X or Y.
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Use `interpolate(list(zip(X, Y)), x)` instead.'''))
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coeffs = []
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numert = Mul(*[x - X[i] for i in range(n)])
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for i in range(n):
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numer = numert/(x - X[i])
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denom = Mul(*[(X[i] - X[j]) for j in range(n) if i != j])
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coeffs.append(numer/denom)
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return Add(*[coeff*y for coeff, y in zip(coeffs, Y)])
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def fateman_poly_F_1(n):
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"""Fateman's GCD benchmark: trivial GCD """
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Y = [Symbol('y_' + str(i)) for i in range(n + 1)]
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y_0, y_1 = Y[0], Y[1]
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u = y_0 + Add(*Y[1:])
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v = y_0**2 + Add(*[y**2 for y in Y[1:]])
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F = ((u + 1)*(u + 2)).as_poly(*Y)
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G = ((v + 1)*(-3*y_1*y_0**2 + y_1**2 - 1)).as_poly(*Y)
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H = Poly(1, *Y)
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return F, G, H
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def dmp_fateman_poly_F_1(n, K):
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"""Fateman's GCD benchmark: trivial GCD """
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u = [K(1), K(0)]
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for i in range(n):
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u = [dmp_one(i, K), u]
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v = [K(1), K(0), K(0)]
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for i in range(0, n):
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v = [dmp_one(i, K), dmp_zero(i), v]
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m = n - 1
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U = dmp_add_term(u, dmp_ground(K(1), m), 0, n, K)
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V = dmp_add_term(u, dmp_ground(K(2), m), 0, n, K)
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f = [[-K(3), K(0)], [], [K(1), K(0), -K(1)]]
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W = dmp_add_term(v, dmp_ground(K(1), m), 0, n, K)
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Y = dmp_raise(f, m, 1, K)
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F = dmp_mul(U, V, n, K)
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G = dmp_mul(W, Y, n, K)
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H = dmp_one(n, K)
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return F, G, H
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def fateman_poly_F_2(n):
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"""Fateman's GCD benchmark: linearly dense quartic inputs """
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Y = [Symbol('y_' + str(i)) for i in range(n + 1)]
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y_0 = Y[0]
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u = Add(*Y[1:])
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H = Poly((y_0 + u + 1)**2, *Y)
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F = Poly((y_0 - u - 2)**2, *Y)
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G = Poly((y_0 + u + 2)**2, *Y)
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return H*F, H*G, H
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def dmp_fateman_poly_F_2(n, K):
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"""Fateman's GCD benchmark: linearly dense quartic inputs """
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u = [K(1), K(0)]
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for i in range(n - 1):
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u = [dmp_one(i, K), u]
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m = n - 1
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v = dmp_add_term(u, dmp_ground(K(2), m - 1), 0, n, K)
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f = dmp_sqr([dmp_one(m, K), dmp_neg(v, m, K)], n, K)
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g = dmp_sqr([dmp_one(m, K), v], n, K)
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v = dmp_add_term(u, dmp_one(m - 1, K), 0, n, K)
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h = dmp_sqr([dmp_one(m, K), v], n, K)
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return dmp_mul(f, h, n, K), dmp_mul(g, h, n, K), h
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def fateman_poly_F_3(n):
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"""Fateman's GCD benchmark: sparse inputs (deg f ~ vars f) """
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Y = [Symbol('y_' + str(i)) for i in range(n + 1)]
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y_0 = Y[0]
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u = Add(*[y**(n + 1) for y in Y[1:]])
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H = Poly((y_0**(n + 1) + u + 1)**2, *Y)
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F = Poly((y_0**(n + 1) - u - 2)**2, *Y)
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G = Poly((y_0**(n + 1) + u + 2)**2, *Y)
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return H*F, H*G, H
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def dmp_fateman_poly_F_3(n, K):
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"""Fateman's GCD benchmark: sparse inputs (deg f ~ vars f) """
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u = dup_from_raw_dict({n + 1: K.one}, K)
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for i in range(0, n - 1):
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u = dmp_add_term([u], dmp_one(i, K), n + 1, i + 1, K)
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v = dmp_add_term(u, dmp_ground(K(2), n - 2), 0, n, K)
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f = dmp_sqr(
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dmp_add_term([dmp_neg(v, n - 1, K)], dmp_one(n - 1, K), n + 1, n, K), n, K)
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g = dmp_sqr(dmp_add_term([v], dmp_one(n - 1, K), n + 1, n, K), n, K)
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v = dmp_add_term(u, dmp_one(n - 2, K), 0, n - 1, K)
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h = dmp_sqr(dmp_add_term([v], dmp_one(n - 1, K), n + 1, n, K), n, K)
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return dmp_mul(f, h, n, K), dmp_mul(g, h, n, K), h
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# A few useful polynomials from Wang's paper ('78).
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from sympy.polys.rings import ring
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def _f_0():
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R, x, y, z = ring("x,y,z", ZZ)
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return x**2*y*z**2 + 2*x**2*y*z + 3*x**2*y + 2*x**2 + 3*x + 4*y**2*z**2 + 5*y**2*z + 6*y**2 + y*z**2 + 2*y*z + y + 1
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def _f_1():
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R, x, y, z = ring("x,y,z", ZZ)
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return x**3*y*z + x**2*y**2*z**2 + x**2*y**2 + 20*x**2*y*z + 30*x**2*y + x**2*z**2 + 10*x**2*z + x*y**3*z + 30*x*y**2*z + 20*x*y**2 + x*y*z**3 + 10*x*y*z**2 + x*y*z + 610*x*y + 20*x*z**2 + 230*x*z + 300*x + y**2*z**2 + 10*y**2*z + 30*y*z**2 + 320*y*z + 200*y + 600*z + 6000
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def _f_2():
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R, x, y, z = ring("x,y,z", ZZ)
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return x**5*y**3 + x**5*y**2*z + x**5*y*z**2 + x**5*z**3 + x**3*y**2 + x**3*y*z + 90*x**3*y + 90*x**3*z + x**2*y**2*z - 11*x**2*y**2 + x**2*z**3 - 11*x**2*z**2 + y*z - 11*y + 90*z - 990
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def _f_3():
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R, x, y, z = ring("x,y,z", ZZ)
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return x**5*y**2 + x**4*z**4 + x**4 + x**3*y**3*z + x**3*z + x**2*y**4 + x**2*y**3*z**3 + x**2*y*z**5 + x**2*y*z + x*y**2*z**4 + x*y**2 + x*y*z**7 + x*y*z**3 + x*y*z**2 + y**2*z + y*z**4
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def _f_4():
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R, x, y, z = ring("x,y,z", ZZ)
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return -x**9*y**8*z - x**8*y**5*z**3 - x**7*y**12*z**2 - 5*x**7*y**8 - x**6*y**9*z**4 + x**6*y**7*z**3 + 3*x**6*y**7*z - 5*x**6*y**5*z**2 - x**6*y**4*z**3 + x**5*y**4*z**5 + 3*x**5*y**4*z**3 - x**5*y*z**5 + x**4*y**11*z**4 + 3*x**4*y**11*z**2 - x**4*y**8*z**4 + 5*x**4*y**7*z**2 + 15*x**4*y**7 - 5*x**4*y**4*z**2 + x**3*y**8*z**6 + 3*x**3*y**8*z**4 - x**3*y**5*z**6 + 5*x**3*y**4*z**4 + 15*x**3*y**4*z**2 + x**3*y**3*z**5 + 3*x**3*y**3*z**3 - 5*x**3*y*z**4 + x**2*z**7 + 3*x**2*z**5 + x*y**7*z**6 + 3*x*y**7*z**4 + 5*x*y**3*z**4 + 15*x*y**3*z**2 + y**4*z**8 + 3*y**4*z**6 + 5*z**6 + 15*z**4
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def _f_5():
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R, x, y, z = ring("x,y,z", ZZ)
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return -x**3 - 3*x**2*y + 3*x**2*z - 3*x*y**2 + 6*x*y*z - 3*x*z**2 - y**3 + 3*y**2*z - 3*y*z**2 + z**3
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def _f_6():
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R, x, y, z, t = ring("x,y,z,t", ZZ)
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return 2115*x**4*y + 45*x**3*z**3*t**2 - 45*x**3*t**2 - 423*x*y**4 - 47*x*y**3 + 141*x*y*z**3 + 94*x*y*z*t - 9*y**3*z**3*t**2 + 9*y**3*t**2 - y**2*z**3*t**2 + y**2*t**2 + 3*z**6*t**2 + 2*z**4*t**3 - 3*z**3*t**2 - 2*z*t**3
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def _w_1():
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R, x, y, z = ring("x,y,z", ZZ)
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return 4*x**6*y**4*z**2 + 4*x**6*y**3*z**3 - 4*x**6*y**2*z**4 - 4*x**6*y*z**5 + x**5*y**4*z**3 + 12*x**5*y**3*z - x**5*y**2*z**5 + 12*x**5*y**2*z**2 - 12*x**5*y*z**3 - 12*x**5*z**4 + 8*x**4*y**4 + 6*x**4*y**3*z**2 + 8*x**4*y**3*z - 4*x**4*y**2*z**4 + 4*x**4*y**2*z**3 - 8*x**4*y**2*z**2 - 4*x**4*y*z**5 - 2*x**4*y*z**4 - 8*x**4*y*z**3 + 2*x**3*y**4*z + x**3*y**3*z**3 - x**3*y**2*z**5 - 2*x**3*y**2*z**3 + 9*x**3*y**2*z - 12*x**3*y*z**3 + 12*x**3*y*z**2 - 12*x**3*z**4 + 3*x**3*z**3 + 6*x**2*y**3 - 6*x**2*y**2*z**2 + 8*x**2*y**2*z - 2*x**2*y*z**4 - 8*x**2*y*z**3 + 2*x**2*y*z**2 + 2*x*y**3*z - 2*x*y**2*z**3 - 3*x*y*z + 3*x*z**3 - 2*y**2 + 2*y*z**2
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def _w_2():
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R, x, y = ring("x,y", ZZ)
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return 24*x**8*y**3 + 48*x**8*y**2 + 24*x**7*y**5 - 72*x**7*y**2 + 25*x**6*y**4 + 2*x**6*y**3 + 4*x**6*y + 8*x**6 + x**5*y**6 + x**5*y**3 - 12*x**5 + x**4*y**5 - x**4*y**4 - 2*x**4*y**3 + 292*x**4*y**2 - x**3*y**6 + 3*x**3*y**3 - x**2*y**5 + 12*x**2*y**3 + 48*x**2 - 12*y**3
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def f_polys():
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return _f_0(), _f_1(), _f_2(), _f_3(), _f_4(), _f_5(), _f_6()
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def w_polys():
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return _w_1(), _w_2()
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