1876 lines
33 KiB
Python
1876 lines
33 KiB
Python
"""Arithmetics for dense recursive polynomials in ``K[x]`` or ``K[X]``. """
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from sympy.polys.densebasic import (
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dup_slice,
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dup_LC, dmp_LC,
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dup_degree, dmp_degree,
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dup_strip, dmp_strip,
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dmp_zero_p, dmp_zero,
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dmp_one_p, dmp_one,
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dmp_ground, dmp_zeros)
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from sympy.polys.polyerrors import (ExactQuotientFailed, PolynomialDivisionFailed)
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def dup_add_term(f, c, i, K):
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"""
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Add ``c*x**i`` to ``f`` in ``K[x]``.
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Examples
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========
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>>> from sympy.polys import ring, ZZ
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>>> R, x = ring("x", ZZ)
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>>> R.dup_add_term(x**2 - 1, ZZ(2), 4)
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2*x**4 + x**2 - 1
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"""
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if not c:
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return f
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n = len(f)
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m = n - i - 1
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if i == n - 1:
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return dup_strip([f[0] + c] + f[1:])
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else:
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if i >= n:
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return [c] + [K.zero]*(i - n) + f
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else:
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return f[:m] + [f[m] + c] + f[m + 1:]
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def dmp_add_term(f, c, i, u, K):
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"""
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Add ``c(x_2..x_u)*x_0**i`` to ``f`` in ``K[X]``.
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Examples
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========
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>>> from sympy.polys import ring, ZZ
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>>> R, x,y = ring("x,y", ZZ)
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>>> R.dmp_add_term(x*y + 1, 2, 2)
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2*x**2 + x*y + 1
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"""
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if not u:
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return dup_add_term(f, c, i, K)
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v = u - 1
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if dmp_zero_p(c, v):
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return f
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n = len(f)
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m = n - i - 1
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if i == n - 1:
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return dmp_strip([dmp_add(f[0], c, v, K)] + f[1:], u)
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else:
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if i >= n:
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return [c] + dmp_zeros(i - n, v, K) + f
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else:
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return f[:m] + [dmp_add(f[m], c, v, K)] + f[m + 1:]
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def dup_sub_term(f, c, i, K):
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"""
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Subtract ``c*x**i`` from ``f`` in ``K[x]``.
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Examples
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========
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>>> from sympy.polys import ring, ZZ
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>>> R, x = ring("x", ZZ)
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>>> R.dup_sub_term(2*x**4 + x**2 - 1, ZZ(2), 4)
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x**2 - 1
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"""
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if not c:
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return f
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n = len(f)
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m = n - i - 1
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if i == n - 1:
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return dup_strip([f[0] - c] + f[1:])
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else:
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if i >= n:
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return [-c] + [K.zero]*(i - n) + f
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else:
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return f[:m] + [f[m] - c] + f[m + 1:]
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def dmp_sub_term(f, c, i, u, K):
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"""
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Subtract ``c(x_2..x_u)*x_0**i`` from ``f`` in ``K[X]``.
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Examples
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========
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>>> from sympy.polys import ring, ZZ
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>>> R, x,y = ring("x,y", ZZ)
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>>> R.dmp_sub_term(2*x**2 + x*y + 1, 2, 2)
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x*y + 1
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"""
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if not u:
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return dup_add_term(f, -c, i, K)
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v = u - 1
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if dmp_zero_p(c, v):
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return f
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n = len(f)
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m = n - i - 1
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if i == n - 1:
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return dmp_strip([dmp_sub(f[0], c, v, K)] + f[1:], u)
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else:
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if i >= n:
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return [dmp_neg(c, v, K)] + dmp_zeros(i - n, v, K) + f
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else:
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return f[:m] + [dmp_sub(f[m], c, v, K)] + f[m + 1:]
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def dup_mul_term(f, c, i, K):
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"""
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Multiply ``f`` by ``c*x**i`` in ``K[x]``.
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Examples
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========
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>>> from sympy.polys import ring, ZZ
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>>> R, x = ring("x", ZZ)
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>>> R.dup_mul_term(x**2 - 1, ZZ(3), 2)
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3*x**4 - 3*x**2
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"""
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if not c or not f:
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return []
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else:
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return [ cf * c for cf in f ] + [K.zero]*i
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def dmp_mul_term(f, c, i, u, K):
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"""
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Multiply ``f`` by ``c(x_2..x_u)*x_0**i`` in ``K[X]``.
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Examples
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========
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>>> from sympy.polys import ring, ZZ
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>>> R, x,y = ring("x,y", ZZ)
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>>> R.dmp_mul_term(x**2*y + x, 3*y, 2)
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3*x**4*y**2 + 3*x**3*y
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"""
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if not u:
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return dup_mul_term(f, c, i, K)
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v = u - 1
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if dmp_zero_p(f, u):
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return f
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if dmp_zero_p(c, v):
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return dmp_zero(u)
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else:
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return [ dmp_mul(cf, c, v, K) for cf in f ] + dmp_zeros(i, v, K)
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def dup_add_ground(f, c, K):
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"""
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Add an element of the ground domain to ``f``.
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Examples
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========
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>>> from sympy.polys import ring, ZZ
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>>> R, x = ring("x", ZZ)
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>>> R.dup_add_ground(x**3 + 2*x**2 + 3*x + 4, ZZ(4))
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x**3 + 2*x**2 + 3*x + 8
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"""
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return dup_add_term(f, c, 0, K)
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def dmp_add_ground(f, c, u, K):
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"""
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Add an element of the ground domain to ``f``.
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Examples
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========
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>>> from sympy.polys import ring, ZZ
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>>> R, x,y = ring("x,y", ZZ)
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>>> R.dmp_add_ground(x**3 + 2*x**2 + 3*x + 4, ZZ(4))
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x**3 + 2*x**2 + 3*x + 8
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"""
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return dmp_add_term(f, dmp_ground(c, u - 1), 0, u, K)
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def dup_sub_ground(f, c, K):
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"""
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Subtract an element of the ground domain from ``f``.
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Examples
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========
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>>> from sympy.polys import ring, ZZ
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>>> R, x = ring("x", ZZ)
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>>> R.dup_sub_ground(x**3 + 2*x**2 + 3*x + 4, ZZ(4))
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x**3 + 2*x**2 + 3*x
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"""
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return dup_sub_term(f, c, 0, K)
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def dmp_sub_ground(f, c, u, K):
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"""
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Subtract an element of the ground domain from ``f``.
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Examples
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========
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>>> from sympy.polys import ring, ZZ
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>>> R, x,y = ring("x,y", ZZ)
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>>> R.dmp_sub_ground(x**3 + 2*x**2 + 3*x + 4, ZZ(4))
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x**3 + 2*x**2 + 3*x
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"""
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return dmp_sub_term(f, dmp_ground(c, u - 1), 0, u, K)
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def dup_mul_ground(f, c, K):
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"""
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Multiply ``f`` by a constant value in ``K[x]``.
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Examples
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========
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>>> from sympy.polys import ring, ZZ
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>>> R, x = ring("x", ZZ)
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>>> R.dup_mul_ground(x**2 + 2*x - 1, ZZ(3))
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3*x**2 + 6*x - 3
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"""
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if not c or not f:
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return []
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else:
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return [ cf * c for cf in f ]
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def dmp_mul_ground(f, c, u, K):
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"""
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Multiply ``f`` by a constant value in ``K[X]``.
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Examples
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========
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>>> from sympy.polys import ring, ZZ
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>>> R, x,y = ring("x,y", ZZ)
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>>> R.dmp_mul_ground(2*x + 2*y, ZZ(3))
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6*x + 6*y
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"""
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if not u:
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return dup_mul_ground(f, c, K)
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v = u - 1
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return [ dmp_mul_ground(cf, c, v, K) for cf in f ]
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def dup_quo_ground(f, c, K):
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"""
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Quotient by a constant in ``K[x]``.
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Examples
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========
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>>> from sympy.polys import ring, ZZ, QQ
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>>> R, x = ring("x", ZZ)
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>>> R.dup_quo_ground(3*x**2 + 2, ZZ(2))
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x**2 + 1
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>>> R, x = ring("x", QQ)
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>>> R.dup_quo_ground(3*x**2 + 2, QQ(2))
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3/2*x**2 + 1
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"""
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if not c:
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raise ZeroDivisionError('polynomial division')
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if not f:
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return f
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if K.is_Field:
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return [ K.quo(cf, c) for cf in f ]
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else:
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return [ cf // c for cf in f ]
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def dmp_quo_ground(f, c, u, K):
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"""
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Quotient by a constant in ``K[X]``.
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Examples
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========
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>>> from sympy.polys import ring, ZZ, QQ
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>>> R, x,y = ring("x,y", ZZ)
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>>> R.dmp_quo_ground(2*x**2*y + 3*x, ZZ(2))
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x**2*y + x
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>>> R, x,y = ring("x,y", QQ)
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>>> R.dmp_quo_ground(2*x**2*y + 3*x, QQ(2))
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x**2*y + 3/2*x
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"""
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if not u:
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return dup_quo_ground(f, c, K)
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v = u - 1
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return [ dmp_quo_ground(cf, c, v, K) for cf in f ]
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def dup_exquo_ground(f, c, K):
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"""
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Exact quotient by a constant in ``K[x]``.
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Examples
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========
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>>> from sympy.polys import ring, QQ
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>>> R, x = ring("x", QQ)
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>>> R.dup_exquo_ground(x**2 + 2, QQ(2))
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1/2*x**2 + 1
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"""
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if not c:
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raise ZeroDivisionError('polynomial division')
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if not f:
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return f
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return [ K.exquo(cf, c) for cf in f ]
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def dmp_exquo_ground(f, c, u, K):
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"""
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Exact quotient by a constant in ``K[X]``.
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Examples
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========
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>>> from sympy.polys import ring, QQ
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>>> R, x,y = ring("x,y", QQ)
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>>> R.dmp_exquo_ground(x**2*y + 2*x, QQ(2))
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1/2*x**2*y + x
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"""
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if not u:
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return dup_exquo_ground(f, c, K)
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v = u - 1
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return [ dmp_exquo_ground(cf, c, v, K) for cf in f ]
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def dup_lshift(f, n, K):
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"""
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Efficiently multiply ``f`` by ``x**n`` in ``K[x]``.
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Examples
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========
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>>> from sympy.polys import ring, ZZ
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>>> R, x = ring("x", ZZ)
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>>> R.dup_lshift(x**2 + 1, 2)
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x**4 + x**2
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"""
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if not f:
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return f
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else:
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return f + [K.zero]*n
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def dup_rshift(f, n, K):
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"""
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Efficiently divide ``f`` by ``x**n`` in ``K[x]``.
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Examples
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========
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>>> from sympy.polys import ring, ZZ
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>>> R, x = ring("x", ZZ)
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>>> R.dup_rshift(x**4 + x**2, 2)
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x**2 + 1
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>>> R.dup_rshift(x**4 + x**2 + 2, 2)
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x**2 + 1
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"""
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return f[:-n]
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def dup_abs(f, K):
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"""
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Make all coefficients positive in ``K[x]``.
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Examples
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========
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>>> from sympy.polys import ring, ZZ
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>>> R, x = ring("x", ZZ)
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>>> R.dup_abs(x**2 - 1)
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x**2 + 1
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"""
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return [ K.abs(coeff) for coeff in f ]
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def dmp_abs(f, u, K):
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"""
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Make all coefficients positive in ``K[X]``.
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Examples
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========
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>>> from sympy.polys import ring, ZZ
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>>> R, x,y = ring("x,y", ZZ)
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>>> R.dmp_abs(x**2*y - x)
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x**2*y + x
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"""
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if not u:
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return dup_abs(f, K)
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v = u - 1
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return [ dmp_abs(cf, v, K) for cf in f ]
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def dup_neg(f, K):
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"""
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Negate a polynomial in ``K[x]``.
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Examples
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========
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>>> from sympy.polys import ring, ZZ
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>>> R, x = ring("x", ZZ)
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>>> R.dup_neg(x**2 - 1)
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-x**2 + 1
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"""
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return [ -coeff for coeff in f ]
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def dmp_neg(f, u, K):
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"""
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Negate a polynomial in ``K[X]``.
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Examples
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========
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>>> from sympy.polys import ring, ZZ
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>>> R, x,y = ring("x,y", ZZ)
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>>> R.dmp_neg(x**2*y - x)
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-x**2*y + x
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"""
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if not u:
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return dup_neg(f, K)
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v = u - 1
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return [ dmp_neg(cf, v, K) for cf in f ]
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def dup_add(f, g, K):
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"""
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Add dense polynomials in ``K[x]``.
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Examples
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========
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>>> from sympy.polys import ring, ZZ
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>>> R, x = ring("x", ZZ)
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>>> R.dup_add(x**2 - 1, x - 2)
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x**2 + x - 3
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"""
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if not f:
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return g
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if not g:
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return f
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df = dup_degree(f)
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dg = dup_degree(g)
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if df == dg:
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return dup_strip([ a + b for a, b in zip(f, g) ])
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else:
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k = abs(df - dg)
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if df > dg:
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h, f = f[:k], f[k:]
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else:
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h, g = g[:k], g[k:]
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return h + [ a + b for a, b in zip(f, g) ]
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def dmp_add(f, g, u, K):
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"""
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Add dense polynomials in ``K[X]``.
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Examples
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========
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>>> from sympy.polys import ring, ZZ
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>>> R, x,y = ring("x,y", ZZ)
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>>> R.dmp_add(x**2 + y, x**2*y + x)
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x**2*y + x**2 + x + y
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"""
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if not u:
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return dup_add(f, g, K)
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df = dmp_degree(f, u)
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if df < 0:
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return g
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dg = dmp_degree(g, u)
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if dg < 0:
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return f
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v = u - 1
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if df == dg:
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return dmp_strip([ dmp_add(a, b, v, K) for a, b in zip(f, g) ], u)
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else:
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k = abs(df - dg)
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if df > dg:
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h, f = f[:k], f[k:]
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else:
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h, g = g[:k], g[k:]
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return h + [ dmp_add(a, b, v, K) for a, b in zip(f, g) ]
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def dup_sub(f, g, K):
|
|
"""
|
|
Subtract dense polynomials in ``K[x]``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.polys import ring, ZZ
|
|
>>> R, x = ring("x", ZZ)
|
|
|
|
>>> R.dup_sub(x**2 - 1, x - 2)
|
|
x**2 - x + 1
|
|
|
|
"""
|
|
if not f:
|
|
return dup_neg(g, K)
|
|
if not g:
|
|
return f
|
|
|
|
df = dup_degree(f)
|
|
dg = dup_degree(g)
|
|
|
|
if df == dg:
|
|
return dup_strip([ a - b for a, b in zip(f, g) ])
|
|
else:
|
|
k = abs(df - dg)
|
|
|
|
if df > dg:
|
|
h, f = f[:k], f[k:]
|
|
else:
|
|
h, g = dup_neg(g[:k], K), g[k:]
|
|
|
|
return h + [ a - b for a, b in zip(f, g) ]
|
|
|
|
|
|
def dmp_sub(f, g, u, K):
|
|
"""
|
|
Subtract dense polynomials in ``K[X]``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.polys import ring, ZZ
|
|
>>> R, x,y = ring("x,y", ZZ)
|
|
|
|
>>> R.dmp_sub(x**2 + y, x**2*y + x)
|
|
-x**2*y + x**2 - x + y
|
|
|
|
"""
|
|
if not u:
|
|
return dup_sub(f, g, K)
|
|
|
|
df = dmp_degree(f, u)
|
|
|
|
if df < 0:
|
|
return dmp_neg(g, u, K)
|
|
|
|
dg = dmp_degree(g, u)
|
|
|
|
if dg < 0:
|
|
return f
|
|
|
|
v = u - 1
|
|
|
|
if df == dg:
|
|
return dmp_strip([ dmp_sub(a, b, v, K) for a, b in zip(f, g) ], u)
|
|
else:
|
|
k = abs(df - dg)
|
|
|
|
if df > dg:
|
|
h, f = f[:k], f[k:]
|
|
else:
|
|
h, g = dmp_neg(g[:k], u, K), g[k:]
|
|
|
|
return h + [ dmp_sub(a, b, v, K) for a, b in zip(f, g) ]
|
|
|
|
|
|
def dup_add_mul(f, g, h, K):
|
|
"""
|
|
Returns ``f + g*h`` where ``f, g, h`` are in ``K[x]``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.polys import ring, ZZ
|
|
>>> R, x = ring("x", ZZ)
|
|
|
|
>>> R.dup_add_mul(x**2 - 1, x - 2, x + 2)
|
|
2*x**2 - 5
|
|
|
|
"""
|
|
return dup_add(f, dup_mul(g, h, K), K)
|
|
|
|
|
|
def dmp_add_mul(f, g, h, u, K):
|
|
"""
|
|
Returns ``f + g*h`` where ``f, g, h`` are in ``K[X]``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.polys import ring, ZZ
|
|
>>> R, x,y = ring("x,y", ZZ)
|
|
|
|
>>> R.dmp_add_mul(x**2 + y, x, x + 2)
|
|
2*x**2 + 2*x + y
|
|
|
|
"""
|
|
return dmp_add(f, dmp_mul(g, h, u, K), u, K)
|
|
|
|
|
|
def dup_sub_mul(f, g, h, K):
|
|
"""
|
|
Returns ``f - g*h`` where ``f, g, h`` are in ``K[x]``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.polys import ring, ZZ
|
|
>>> R, x = ring("x", ZZ)
|
|
|
|
>>> R.dup_sub_mul(x**2 - 1, x - 2, x + 2)
|
|
3
|
|
|
|
"""
|
|
return dup_sub(f, dup_mul(g, h, K), K)
|
|
|
|
|
|
def dmp_sub_mul(f, g, h, u, K):
|
|
"""
|
|
Returns ``f - g*h`` where ``f, g, h`` are in ``K[X]``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.polys import ring, ZZ
|
|
>>> R, x,y = ring("x,y", ZZ)
|
|
|
|
>>> R.dmp_sub_mul(x**2 + y, x, x + 2)
|
|
-2*x + y
|
|
|
|
"""
|
|
return dmp_sub(f, dmp_mul(g, h, u, K), u, K)
|
|
|
|
|
|
def dup_mul(f, g, K):
|
|
"""
|
|
Multiply dense polynomials in ``K[x]``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.polys import ring, ZZ
|
|
>>> R, x = ring("x", ZZ)
|
|
|
|
>>> R.dup_mul(x - 2, x + 2)
|
|
x**2 - 4
|
|
|
|
"""
|
|
if f == g:
|
|
return dup_sqr(f, K)
|
|
|
|
if not (f and g):
|
|
return []
|
|
|
|
df = dup_degree(f)
|
|
dg = dup_degree(g)
|
|
|
|
n = max(df, dg) + 1
|
|
|
|
if n < 100:
|
|
h = []
|
|
|
|
for i in range(0, df + dg + 1):
|
|
coeff = K.zero
|
|
|
|
for j in range(max(0, i - dg), min(df, i) + 1):
|
|
coeff += f[j]*g[i - j]
|
|
|
|
h.append(coeff)
|
|
|
|
return dup_strip(h)
|
|
else:
|
|
# Use Karatsuba's algorithm (divide and conquer), see e.g.:
|
|
# Joris van der Hoeven, Relax But Don't Be Too Lazy,
|
|
# J. Symbolic Computation, 11 (2002), section 3.1.1.
|
|
n2 = n//2
|
|
|
|
fl, gl = dup_slice(f, 0, n2, K), dup_slice(g, 0, n2, K)
|
|
|
|
fh = dup_rshift(dup_slice(f, n2, n, K), n2, K)
|
|
gh = dup_rshift(dup_slice(g, n2, n, K), n2, K)
|
|
|
|
lo, hi = dup_mul(fl, gl, K), dup_mul(fh, gh, K)
|
|
|
|
mid = dup_mul(dup_add(fl, fh, K), dup_add(gl, gh, K), K)
|
|
mid = dup_sub(mid, dup_add(lo, hi, K), K)
|
|
|
|
return dup_add(dup_add(lo, dup_lshift(mid, n2, K), K),
|
|
dup_lshift(hi, 2*n2, K), K)
|
|
|
|
|
|
def dmp_mul(f, g, u, K):
|
|
"""
|
|
Multiply dense polynomials in ``K[X]``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.polys import ring, ZZ
|
|
>>> R, x,y = ring("x,y", ZZ)
|
|
|
|
>>> R.dmp_mul(x*y + 1, x)
|
|
x**2*y + x
|
|
|
|
"""
|
|
if not u:
|
|
return dup_mul(f, g, K)
|
|
|
|
if f == g:
|
|
return dmp_sqr(f, u, K)
|
|
|
|
df = dmp_degree(f, u)
|
|
|
|
if df < 0:
|
|
return f
|
|
|
|
dg = dmp_degree(g, u)
|
|
|
|
if dg < 0:
|
|
return g
|
|
|
|
h, v = [], u - 1
|
|
|
|
for i in range(0, df + dg + 1):
|
|
coeff = dmp_zero(v)
|
|
|
|
for j in range(max(0, i - dg), min(df, i) + 1):
|
|
coeff = dmp_add(coeff, dmp_mul(f[j], g[i - j], v, K), v, K)
|
|
|
|
h.append(coeff)
|
|
|
|
return dmp_strip(h, u)
|
|
|
|
|
|
def dup_sqr(f, K):
|
|
"""
|
|
Square dense polynomials in ``K[x]``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.polys import ring, ZZ
|
|
>>> R, x = ring("x", ZZ)
|
|
|
|
>>> R.dup_sqr(x**2 + 1)
|
|
x**4 + 2*x**2 + 1
|
|
|
|
"""
|
|
df, h = len(f) - 1, []
|
|
|
|
for i in range(0, 2*df + 1):
|
|
c = K.zero
|
|
|
|
jmin = max(0, i - df)
|
|
jmax = min(i, df)
|
|
|
|
n = jmax - jmin + 1
|
|
|
|
jmax = jmin + n // 2 - 1
|
|
|
|
for j in range(jmin, jmax + 1):
|
|
c += f[j]*f[i - j]
|
|
|
|
c += c
|
|
|
|
if n & 1:
|
|
elem = f[jmax + 1]
|
|
c += elem**2
|
|
|
|
h.append(c)
|
|
|
|
return dup_strip(h)
|
|
|
|
|
|
def dmp_sqr(f, u, K):
|
|
"""
|
|
Square dense polynomials in ``K[X]``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.polys import ring, ZZ
|
|
>>> R, x,y = ring("x,y", ZZ)
|
|
|
|
>>> R.dmp_sqr(x**2 + x*y + y**2)
|
|
x**4 + 2*x**3*y + 3*x**2*y**2 + 2*x*y**3 + y**4
|
|
|
|
"""
|
|
if not u:
|
|
return dup_sqr(f, K)
|
|
|
|
df = dmp_degree(f, u)
|
|
|
|
if df < 0:
|
|
return f
|
|
|
|
h, v = [], u - 1
|
|
|
|
for i in range(0, 2*df + 1):
|
|
c = dmp_zero(v)
|
|
|
|
jmin = max(0, i - df)
|
|
jmax = min(i, df)
|
|
|
|
n = jmax - jmin + 1
|
|
|
|
jmax = jmin + n // 2 - 1
|
|
|
|
for j in range(jmin, jmax + 1):
|
|
c = dmp_add(c, dmp_mul(f[j], f[i - j], v, K), v, K)
|
|
|
|
c = dmp_mul_ground(c, K(2), v, K)
|
|
|
|
if n & 1:
|
|
elem = dmp_sqr(f[jmax + 1], v, K)
|
|
c = dmp_add(c, elem, v, K)
|
|
|
|
h.append(c)
|
|
|
|
return dmp_strip(h, u)
|
|
|
|
|
|
def dup_pow(f, n, K):
|
|
"""
|
|
Raise ``f`` to the ``n``-th power in ``K[x]``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.polys import ring, ZZ
|
|
>>> R, x = ring("x", ZZ)
|
|
|
|
>>> R.dup_pow(x - 2, 3)
|
|
x**3 - 6*x**2 + 12*x - 8
|
|
|
|
"""
|
|
if not n:
|
|
return [K.one]
|
|
if n < 0:
|
|
raise ValueError("Cannot raise polynomial to a negative power")
|
|
if n == 1 or not f or f == [K.one]:
|
|
return f
|
|
|
|
g = [K.one]
|
|
|
|
while True:
|
|
n, m = n//2, n
|
|
|
|
if m % 2:
|
|
g = dup_mul(g, f, K)
|
|
|
|
if not n:
|
|
break
|
|
|
|
f = dup_sqr(f, K)
|
|
|
|
return g
|
|
|
|
|
|
def dmp_pow(f, n, u, K):
|
|
"""
|
|
Raise ``f`` to the ``n``-th power in ``K[X]``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.polys import ring, ZZ
|
|
>>> R, x,y = ring("x,y", ZZ)
|
|
|
|
>>> R.dmp_pow(x*y + 1, 3)
|
|
x**3*y**3 + 3*x**2*y**2 + 3*x*y + 1
|
|
|
|
"""
|
|
if not u:
|
|
return dup_pow(f, n, K)
|
|
|
|
if not n:
|
|
return dmp_one(u, K)
|
|
if n < 0:
|
|
raise ValueError("Cannot raise polynomial to a negative power")
|
|
if n == 1 or dmp_zero_p(f, u) or dmp_one_p(f, u, K):
|
|
return f
|
|
|
|
g = dmp_one(u, K)
|
|
|
|
while True:
|
|
n, m = n//2, n
|
|
|
|
if m & 1:
|
|
g = dmp_mul(g, f, u, K)
|
|
|
|
if not n:
|
|
break
|
|
|
|
f = dmp_sqr(f, u, K)
|
|
|
|
return g
|
|
|
|
|
|
def dup_pdiv(f, g, K):
|
|
"""
|
|
Polynomial pseudo-division in ``K[x]``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.polys import ring, ZZ
|
|
>>> R, x = ring("x", ZZ)
|
|
|
|
>>> R.dup_pdiv(x**2 + 1, 2*x - 4)
|
|
(2*x + 4, 20)
|
|
|
|
"""
|
|
df = dup_degree(f)
|
|
dg = dup_degree(g)
|
|
|
|
q, r, dr = [], f, df
|
|
|
|
if not g:
|
|
raise ZeroDivisionError("polynomial division")
|
|
elif df < dg:
|
|
return q, r
|
|
|
|
N = df - dg + 1
|
|
lc_g = dup_LC(g, K)
|
|
|
|
while True:
|
|
lc_r = dup_LC(r, K)
|
|
j, N = dr - dg, N - 1
|
|
|
|
Q = dup_mul_ground(q, lc_g, K)
|
|
q = dup_add_term(Q, lc_r, j, K)
|
|
|
|
R = dup_mul_ground(r, lc_g, K)
|
|
G = dup_mul_term(g, lc_r, j, K)
|
|
r = dup_sub(R, G, K)
|
|
|
|
_dr, dr = dr, dup_degree(r)
|
|
|
|
if dr < dg:
|
|
break
|
|
elif not (dr < _dr):
|
|
raise PolynomialDivisionFailed(f, g, K)
|
|
|
|
c = lc_g**N
|
|
|
|
q = dup_mul_ground(q, c, K)
|
|
r = dup_mul_ground(r, c, K)
|
|
|
|
return q, r
|
|
|
|
|
|
def dup_prem(f, g, K):
|
|
"""
|
|
Polynomial pseudo-remainder in ``K[x]``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.polys import ring, ZZ
|
|
>>> R, x = ring("x", ZZ)
|
|
|
|
>>> R.dup_prem(x**2 + 1, 2*x - 4)
|
|
20
|
|
|
|
"""
|
|
df = dup_degree(f)
|
|
dg = dup_degree(g)
|
|
|
|
r, dr = f, df
|
|
|
|
if not g:
|
|
raise ZeroDivisionError("polynomial division")
|
|
elif df < dg:
|
|
return r
|
|
|
|
N = df - dg + 1
|
|
lc_g = dup_LC(g, K)
|
|
|
|
while True:
|
|
lc_r = dup_LC(r, K)
|
|
j, N = dr - dg, N - 1
|
|
|
|
R = dup_mul_ground(r, lc_g, K)
|
|
G = dup_mul_term(g, lc_r, j, K)
|
|
r = dup_sub(R, G, K)
|
|
|
|
_dr, dr = dr, dup_degree(r)
|
|
|
|
if dr < dg:
|
|
break
|
|
elif not (dr < _dr):
|
|
raise PolynomialDivisionFailed(f, g, K)
|
|
|
|
return dup_mul_ground(r, lc_g**N, K)
|
|
|
|
|
|
def dup_pquo(f, g, K):
|
|
"""
|
|
Polynomial exact pseudo-quotient in ``K[X]``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.polys import ring, ZZ
|
|
>>> R, x = ring("x", ZZ)
|
|
|
|
>>> R.dup_pquo(x**2 - 1, 2*x - 2)
|
|
2*x + 2
|
|
|
|
>>> R.dup_pquo(x**2 + 1, 2*x - 4)
|
|
2*x + 4
|
|
|
|
"""
|
|
return dup_pdiv(f, g, K)[0]
|
|
|
|
|
|
def dup_pexquo(f, g, K):
|
|
"""
|
|
Polynomial pseudo-quotient in ``K[x]``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.polys import ring, ZZ
|
|
>>> R, x = ring("x", ZZ)
|
|
|
|
>>> R.dup_pexquo(x**2 - 1, 2*x - 2)
|
|
2*x + 2
|
|
|
|
>>> R.dup_pexquo(x**2 + 1, 2*x - 4)
|
|
Traceback (most recent call last):
|
|
...
|
|
ExactQuotientFailed: [2, -4] does not divide [1, 0, 1]
|
|
|
|
"""
|
|
q, r = dup_pdiv(f, g, K)
|
|
|
|
if not r:
|
|
return q
|
|
else:
|
|
raise ExactQuotientFailed(f, g)
|
|
|
|
|
|
def dmp_pdiv(f, g, u, K):
|
|
"""
|
|
Polynomial pseudo-division in ``K[X]``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.polys import ring, ZZ
|
|
>>> R, x,y = ring("x,y", ZZ)
|
|
|
|
>>> R.dmp_pdiv(x**2 + x*y, 2*x + 2)
|
|
(2*x + 2*y - 2, -4*y + 4)
|
|
|
|
"""
|
|
if not u:
|
|
return dup_pdiv(f, g, K)
|
|
|
|
df = dmp_degree(f, u)
|
|
dg = dmp_degree(g, u)
|
|
|
|
if dg < 0:
|
|
raise ZeroDivisionError("polynomial division")
|
|
|
|
q, r, dr = dmp_zero(u), f, df
|
|
|
|
if df < dg:
|
|
return q, r
|
|
|
|
N = df - dg + 1
|
|
lc_g = dmp_LC(g, K)
|
|
|
|
while True:
|
|
lc_r = dmp_LC(r, K)
|
|
j, N = dr - dg, N - 1
|
|
|
|
Q = dmp_mul_term(q, lc_g, 0, u, K)
|
|
q = dmp_add_term(Q, lc_r, j, u, K)
|
|
|
|
R = dmp_mul_term(r, lc_g, 0, u, K)
|
|
G = dmp_mul_term(g, lc_r, j, u, K)
|
|
r = dmp_sub(R, G, u, K)
|
|
|
|
_dr, dr = dr, dmp_degree(r, u)
|
|
|
|
if dr < dg:
|
|
break
|
|
elif not (dr < _dr):
|
|
raise PolynomialDivisionFailed(f, g, K)
|
|
|
|
c = dmp_pow(lc_g, N, u - 1, K)
|
|
|
|
q = dmp_mul_term(q, c, 0, u, K)
|
|
r = dmp_mul_term(r, c, 0, u, K)
|
|
|
|
return q, r
|
|
|
|
|
|
def dmp_prem(f, g, u, K):
|
|
"""
|
|
Polynomial pseudo-remainder in ``K[X]``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.polys import ring, ZZ
|
|
>>> R, x,y = ring("x,y", ZZ)
|
|
|
|
>>> R.dmp_prem(x**2 + x*y, 2*x + 2)
|
|
-4*y + 4
|
|
|
|
"""
|
|
if not u:
|
|
return dup_prem(f, g, K)
|
|
|
|
df = dmp_degree(f, u)
|
|
dg = dmp_degree(g, u)
|
|
|
|
if dg < 0:
|
|
raise ZeroDivisionError("polynomial division")
|
|
|
|
r, dr = f, df
|
|
|
|
if df < dg:
|
|
return r
|
|
|
|
N = df - dg + 1
|
|
lc_g = dmp_LC(g, K)
|
|
|
|
while True:
|
|
lc_r = dmp_LC(r, K)
|
|
j, N = dr - dg, N - 1
|
|
|
|
R = dmp_mul_term(r, lc_g, 0, u, K)
|
|
G = dmp_mul_term(g, lc_r, j, u, K)
|
|
r = dmp_sub(R, G, u, K)
|
|
|
|
_dr, dr = dr, dmp_degree(r, u)
|
|
|
|
if dr < dg:
|
|
break
|
|
elif not (dr < _dr):
|
|
raise PolynomialDivisionFailed(f, g, K)
|
|
|
|
c = dmp_pow(lc_g, N, u - 1, K)
|
|
|
|
return dmp_mul_term(r, c, 0, u, K)
|
|
|
|
|
|
def dmp_pquo(f, g, u, K):
|
|
"""
|
|
Polynomial exact pseudo-quotient in ``K[X]``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.polys import ring, ZZ
|
|
>>> R, x,y = ring("x,y", ZZ)
|
|
|
|
>>> f = x**2 + x*y
|
|
>>> g = 2*x + 2*y
|
|
>>> h = 2*x + 2
|
|
|
|
>>> R.dmp_pquo(f, g)
|
|
2*x
|
|
|
|
>>> R.dmp_pquo(f, h)
|
|
2*x + 2*y - 2
|
|
|
|
"""
|
|
return dmp_pdiv(f, g, u, K)[0]
|
|
|
|
|
|
def dmp_pexquo(f, g, u, K):
|
|
"""
|
|
Polynomial pseudo-quotient in ``K[X]``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.polys import ring, ZZ
|
|
>>> R, x,y = ring("x,y", ZZ)
|
|
|
|
>>> f = x**2 + x*y
|
|
>>> g = 2*x + 2*y
|
|
>>> h = 2*x + 2
|
|
|
|
>>> R.dmp_pexquo(f, g)
|
|
2*x
|
|
|
|
>>> R.dmp_pexquo(f, h)
|
|
Traceback (most recent call last):
|
|
...
|
|
ExactQuotientFailed: [[2], [2]] does not divide [[1], [1, 0], []]
|
|
|
|
"""
|
|
q, r = dmp_pdiv(f, g, u, K)
|
|
|
|
if dmp_zero_p(r, u):
|
|
return q
|
|
else:
|
|
raise ExactQuotientFailed(f, g)
|
|
|
|
|
|
def dup_rr_div(f, g, K):
|
|
"""
|
|
Univariate division with remainder over a ring.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.polys import ring, ZZ
|
|
>>> R, x = ring("x", ZZ)
|
|
|
|
>>> R.dup_rr_div(x**2 + 1, 2*x - 4)
|
|
(0, x**2 + 1)
|
|
|
|
"""
|
|
df = dup_degree(f)
|
|
dg = dup_degree(g)
|
|
|
|
q, r, dr = [], f, df
|
|
|
|
if not g:
|
|
raise ZeroDivisionError("polynomial division")
|
|
elif df < dg:
|
|
return q, r
|
|
|
|
lc_g = dup_LC(g, K)
|
|
|
|
while True:
|
|
lc_r = dup_LC(r, K)
|
|
|
|
if lc_r % lc_g:
|
|
break
|
|
|
|
c = K.exquo(lc_r, lc_g)
|
|
j = dr - dg
|
|
|
|
q = dup_add_term(q, c, j, K)
|
|
h = dup_mul_term(g, c, j, K)
|
|
r = dup_sub(r, h, K)
|
|
|
|
_dr, dr = dr, dup_degree(r)
|
|
|
|
if dr < dg:
|
|
break
|
|
elif not (dr < _dr):
|
|
raise PolynomialDivisionFailed(f, g, K)
|
|
|
|
return q, r
|
|
|
|
|
|
def dmp_rr_div(f, g, u, K):
|
|
"""
|
|
Multivariate division with remainder over a ring.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.polys import ring, ZZ
|
|
>>> R, x,y = ring("x,y", ZZ)
|
|
|
|
>>> R.dmp_rr_div(x**2 + x*y, 2*x + 2)
|
|
(0, x**2 + x*y)
|
|
|
|
"""
|
|
if not u:
|
|
return dup_rr_div(f, g, K)
|
|
|
|
df = dmp_degree(f, u)
|
|
dg = dmp_degree(g, u)
|
|
|
|
if dg < 0:
|
|
raise ZeroDivisionError("polynomial division")
|
|
|
|
q, r, dr = dmp_zero(u), f, df
|
|
|
|
if df < dg:
|
|
return q, r
|
|
|
|
lc_g, v = dmp_LC(g, K), u - 1
|
|
|
|
while True:
|
|
lc_r = dmp_LC(r, K)
|
|
c, R = dmp_rr_div(lc_r, lc_g, v, K)
|
|
|
|
if not dmp_zero_p(R, v):
|
|
break
|
|
|
|
j = dr - dg
|
|
|
|
q = dmp_add_term(q, c, j, u, K)
|
|
h = dmp_mul_term(g, c, j, u, K)
|
|
r = dmp_sub(r, h, u, K)
|
|
|
|
_dr, dr = dr, dmp_degree(r, u)
|
|
|
|
if dr < dg:
|
|
break
|
|
elif not (dr < _dr):
|
|
raise PolynomialDivisionFailed(f, g, K)
|
|
|
|
return q, r
|
|
|
|
|
|
def dup_ff_div(f, g, K):
|
|
"""
|
|
Polynomial division with remainder over a field.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.polys import ring, QQ
|
|
>>> R, x = ring("x", QQ)
|
|
|
|
>>> R.dup_ff_div(x**2 + 1, 2*x - 4)
|
|
(1/2*x + 1, 5)
|
|
|
|
"""
|
|
df = dup_degree(f)
|
|
dg = dup_degree(g)
|
|
|
|
q, r, dr = [], f, df
|
|
|
|
if not g:
|
|
raise ZeroDivisionError("polynomial division")
|
|
elif df < dg:
|
|
return q, r
|
|
|
|
lc_g = dup_LC(g, K)
|
|
|
|
while True:
|
|
lc_r = dup_LC(r, K)
|
|
|
|
c = K.exquo(lc_r, lc_g)
|
|
j = dr - dg
|
|
|
|
q = dup_add_term(q, c, j, K)
|
|
h = dup_mul_term(g, c, j, K)
|
|
r = dup_sub(r, h, K)
|
|
|
|
_dr, dr = dr, dup_degree(r)
|
|
|
|
if dr < dg:
|
|
break
|
|
elif dr == _dr and not K.is_Exact:
|
|
# remove leading term created by rounding error
|
|
r = dup_strip(r[1:])
|
|
dr = dup_degree(r)
|
|
if dr < dg:
|
|
break
|
|
elif not (dr < _dr):
|
|
raise PolynomialDivisionFailed(f, g, K)
|
|
|
|
return q, r
|
|
|
|
|
|
def dmp_ff_div(f, g, u, K):
|
|
"""
|
|
Polynomial division with remainder over a field.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.polys import ring, QQ
|
|
>>> R, x,y = ring("x,y", QQ)
|
|
|
|
>>> R.dmp_ff_div(x**2 + x*y, 2*x + 2)
|
|
(1/2*x + 1/2*y - 1/2, -y + 1)
|
|
|
|
"""
|
|
if not u:
|
|
return dup_ff_div(f, g, K)
|
|
|
|
df = dmp_degree(f, u)
|
|
dg = dmp_degree(g, u)
|
|
|
|
if dg < 0:
|
|
raise ZeroDivisionError("polynomial division")
|
|
|
|
q, r, dr = dmp_zero(u), f, df
|
|
|
|
if df < dg:
|
|
return q, r
|
|
|
|
lc_g, v = dmp_LC(g, K), u - 1
|
|
|
|
while True:
|
|
lc_r = dmp_LC(r, K)
|
|
c, R = dmp_ff_div(lc_r, lc_g, v, K)
|
|
|
|
if not dmp_zero_p(R, v):
|
|
break
|
|
|
|
j = dr - dg
|
|
|
|
q = dmp_add_term(q, c, j, u, K)
|
|
h = dmp_mul_term(g, c, j, u, K)
|
|
r = dmp_sub(r, h, u, K)
|
|
|
|
_dr, dr = dr, dmp_degree(r, u)
|
|
|
|
if dr < dg:
|
|
break
|
|
elif not (dr < _dr):
|
|
raise PolynomialDivisionFailed(f, g, K)
|
|
|
|
return q, r
|
|
|
|
|
|
def dup_div(f, g, K):
|
|
"""
|
|
Polynomial division with remainder in ``K[x]``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.polys import ring, ZZ, QQ
|
|
|
|
>>> R, x = ring("x", ZZ)
|
|
>>> R.dup_div(x**2 + 1, 2*x - 4)
|
|
(0, x**2 + 1)
|
|
|
|
>>> R, x = ring("x", QQ)
|
|
>>> R.dup_div(x**2 + 1, 2*x - 4)
|
|
(1/2*x + 1, 5)
|
|
|
|
"""
|
|
if K.is_Field:
|
|
return dup_ff_div(f, g, K)
|
|
else:
|
|
return dup_rr_div(f, g, K)
|
|
|
|
|
|
def dup_rem(f, g, K):
|
|
"""
|
|
Returns polynomial remainder in ``K[x]``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.polys import ring, ZZ, QQ
|
|
|
|
>>> R, x = ring("x", ZZ)
|
|
>>> R.dup_rem(x**2 + 1, 2*x - 4)
|
|
x**2 + 1
|
|
|
|
>>> R, x = ring("x", QQ)
|
|
>>> R.dup_rem(x**2 + 1, 2*x - 4)
|
|
5
|
|
|
|
"""
|
|
return dup_div(f, g, K)[1]
|
|
|
|
|
|
def dup_quo(f, g, K):
|
|
"""
|
|
Returns exact polynomial quotient in ``K[x]``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.polys import ring, ZZ, QQ
|
|
|
|
>>> R, x = ring("x", ZZ)
|
|
>>> R.dup_quo(x**2 + 1, 2*x - 4)
|
|
0
|
|
|
|
>>> R, x = ring("x", QQ)
|
|
>>> R.dup_quo(x**2 + 1, 2*x - 4)
|
|
1/2*x + 1
|
|
|
|
"""
|
|
return dup_div(f, g, K)[0]
|
|
|
|
|
|
def dup_exquo(f, g, K):
|
|
"""
|
|
Returns polynomial quotient in ``K[x]``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.polys import ring, ZZ
|
|
>>> R, x = ring("x", ZZ)
|
|
|
|
>>> R.dup_exquo(x**2 - 1, x - 1)
|
|
x + 1
|
|
|
|
>>> R.dup_exquo(x**2 + 1, 2*x - 4)
|
|
Traceback (most recent call last):
|
|
...
|
|
ExactQuotientFailed: [2, -4] does not divide [1, 0, 1]
|
|
|
|
"""
|
|
q, r = dup_div(f, g, K)
|
|
|
|
if not r:
|
|
return q
|
|
else:
|
|
raise ExactQuotientFailed(f, g)
|
|
|
|
|
|
def dmp_div(f, g, u, K):
|
|
"""
|
|
Polynomial division with remainder in ``K[X]``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.polys import ring, ZZ, QQ
|
|
|
|
>>> R, x,y = ring("x,y", ZZ)
|
|
>>> R.dmp_div(x**2 + x*y, 2*x + 2)
|
|
(0, x**2 + x*y)
|
|
|
|
>>> R, x,y = ring("x,y", QQ)
|
|
>>> R.dmp_div(x**2 + x*y, 2*x + 2)
|
|
(1/2*x + 1/2*y - 1/2, -y + 1)
|
|
|
|
"""
|
|
if K.is_Field:
|
|
return dmp_ff_div(f, g, u, K)
|
|
else:
|
|
return dmp_rr_div(f, g, u, K)
|
|
|
|
|
|
def dmp_rem(f, g, u, K):
|
|
"""
|
|
Returns polynomial remainder in ``K[X]``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.polys import ring, ZZ, QQ
|
|
|
|
>>> R, x,y = ring("x,y", ZZ)
|
|
>>> R.dmp_rem(x**2 + x*y, 2*x + 2)
|
|
x**2 + x*y
|
|
|
|
>>> R, x,y = ring("x,y", QQ)
|
|
>>> R.dmp_rem(x**2 + x*y, 2*x + 2)
|
|
-y + 1
|
|
|
|
"""
|
|
return dmp_div(f, g, u, K)[1]
|
|
|
|
|
|
def dmp_quo(f, g, u, K):
|
|
"""
|
|
Returns exact polynomial quotient in ``K[X]``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.polys import ring, ZZ, QQ
|
|
|
|
>>> R, x,y = ring("x,y", ZZ)
|
|
>>> R.dmp_quo(x**2 + x*y, 2*x + 2)
|
|
0
|
|
|
|
>>> R, x,y = ring("x,y", QQ)
|
|
>>> R.dmp_quo(x**2 + x*y, 2*x + 2)
|
|
1/2*x + 1/2*y - 1/2
|
|
|
|
"""
|
|
return dmp_div(f, g, u, K)[0]
|
|
|
|
|
|
def dmp_exquo(f, g, u, K):
|
|
"""
|
|
Returns polynomial quotient in ``K[X]``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.polys import ring, ZZ
|
|
>>> R, x,y = ring("x,y", ZZ)
|
|
|
|
>>> f = x**2 + x*y
|
|
>>> g = x + y
|
|
>>> h = 2*x + 2
|
|
|
|
>>> R.dmp_exquo(f, g)
|
|
x
|
|
|
|
>>> R.dmp_exquo(f, h)
|
|
Traceback (most recent call last):
|
|
...
|
|
ExactQuotientFailed: [[2], [2]] does not divide [[1], [1, 0], []]
|
|
|
|
"""
|
|
q, r = dmp_div(f, g, u, K)
|
|
|
|
if dmp_zero_p(r, u):
|
|
return q
|
|
else:
|
|
raise ExactQuotientFailed(f, g)
|
|
|
|
|
|
def dup_max_norm(f, K):
|
|
"""
|
|
Returns maximum norm of a polynomial in ``K[x]``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.polys import ring, ZZ
|
|
>>> R, x = ring("x", ZZ)
|
|
|
|
>>> R.dup_max_norm(-x**2 + 2*x - 3)
|
|
3
|
|
|
|
"""
|
|
if not f:
|
|
return K.zero
|
|
else:
|
|
return max(dup_abs(f, K))
|
|
|
|
|
|
def dmp_max_norm(f, u, K):
|
|
"""
|
|
Returns maximum norm of a polynomial in ``K[X]``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.polys import ring, ZZ
|
|
>>> R, x,y = ring("x,y", ZZ)
|
|
|
|
>>> R.dmp_max_norm(2*x*y - x - 3)
|
|
3
|
|
|
|
"""
|
|
if not u:
|
|
return dup_max_norm(f, K)
|
|
|
|
v = u - 1
|
|
|
|
return max([ dmp_max_norm(c, v, K) for c in f ])
|
|
|
|
|
|
def dup_l1_norm(f, K):
|
|
"""
|
|
Returns l1 norm of a polynomial in ``K[x]``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.polys import ring, ZZ
|
|
>>> R, x = ring("x", ZZ)
|
|
|
|
>>> R.dup_l1_norm(2*x**3 - 3*x**2 + 1)
|
|
6
|
|
|
|
"""
|
|
if not f:
|
|
return K.zero
|
|
else:
|
|
return sum(dup_abs(f, K))
|
|
|
|
|
|
def dmp_l1_norm(f, u, K):
|
|
"""
|
|
Returns l1 norm of a polynomial in ``K[X]``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.polys import ring, ZZ
|
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>>> R, x,y = ring("x,y", ZZ)
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>>> R.dmp_l1_norm(2*x*y - x - 3)
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6
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"""
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if not u:
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return dup_l1_norm(f, K)
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v = u - 1
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return sum([ dmp_l1_norm(c, v, K) for c in f ])
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def dup_l2_norm_squared(f, K):
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"""
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Returns squared l2 norm of a polynomial in ``K[x]``.
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Examples
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========
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>>> from sympy.polys import ring, ZZ
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>>> R, x = ring("x", ZZ)
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>>> R.dup_l2_norm_squared(2*x**3 - 3*x**2 + 1)
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14
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"""
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return sum([coeff**2 for coeff in f], K.zero)
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def dmp_l2_norm_squared(f, u, K):
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"""
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Returns squared l2 norm of a polynomial in ``K[X]``.
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Examples
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========
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>>> from sympy.polys import ring, ZZ
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>>> R, x,y = ring("x,y", ZZ)
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>>> R.dmp_l2_norm_squared(2*x*y - x - 3)
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14
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"""
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if not u:
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return dup_l2_norm_squared(f, K)
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v = u - 1
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return sum([ dmp_l2_norm_squared(c, v, K) for c in f ])
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def dup_expand(polys, K):
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"""
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Multiply together several polynomials in ``K[x]``.
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Examples
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========
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>>> from sympy.polys import ring, ZZ
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>>> R, x = ring("x", ZZ)
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>>> R.dup_expand([x**2 - 1, x, 2])
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2*x**3 - 2*x
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"""
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if not polys:
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return [K.one]
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f = polys[0]
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for g in polys[1:]:
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f = dup_mul(f, g, K)
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return f
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def dmp_expand(polys, u, K):
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"""
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Multiply together several polynomials in ``K[X]``.
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Examples
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========
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>>> from sympy.polys import ring, ZZ
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>>> R, x,y = ring("x,y", ZZ)
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>>> R.dmp_expand([x**2 + y**2, x + 1])
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x**3 + x**2 + x*y**2 + y**2
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"""
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if not polys:
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return dmp_one(u, K)
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f = polys[0]
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for g in polys[1:]:
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f = dmp_mul(f, g, u, K)
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return f
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