632 lines
21 KiB
Python
632 lines
21 KiB
Python
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"""Sparse rational function fields. """
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from __future__ import annotations
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from typing import Any
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from functools import reduce
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from operator import add, mul, lt, le, gt, ge
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from sympy.core.expr import Expr
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from sympy.core.mod import Mod
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from sympy.core.numbers import Exp1
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from sympy.core.singleton import S
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from sympy.core.symbol import Symbol
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from sympy.core.sympify import CantSympify, sympify
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from sympy.functions.elementary.exponential import ExpBase
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from sympy.polys.domains.domainelement import DomainElement
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from sympy.polys.domains.fractionfield import FractionField
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from sympy.polys.domains.polynomialring import PolynomialRing
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from sympy.polys.constructor import construct_domain
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from sympy.polys.orderings import lex
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from sympy.polys.polyerrors import CoercionFailed
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from sympy.polys.polyoptions import build_options
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from sympy.polys.polyutils import _parallel_dict_from_expr
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from sympy.polys.rings import PolyElement
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from sympy.printing.defaults import DefaultPrinting
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from sympy.utilities import public
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from sympy.utilities.iterables import is_sequence
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from sympy.utilities.magic import pollute
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@public
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def field(symbols, domain, order=lex):
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"""Construct new rational function field returning (field, x1, ..., xn). """
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_field = FracField(symbols, domain, order)
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return (_field,) + _field.gens
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@public
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def xfield(symbols, domain, order=lex):
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"""Construct new rational function field returning (field, (x1, ..., xn)). """
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_field = FracField(symbols, domain, order)
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return (_field, _field.gens)
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@public
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def vfield(symbols, domain, order=lex):
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"""Construct new rational function field and inject generators into global namespace. """
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_field = FracField(symbols, domain, order)
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pollute([ sym.name for sym in _field.symbols ], _field.gens)
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return _field
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@public
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def sfield(exprs, *symbols, **options):
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"""Construct a field deriving generators and domain
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from options and input expressions.
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Parameters
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==========
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exprs : py:class:`~.Expr` or sequence of :py:class:`~.Expr` (sympifiable)
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symbols : sequence of :py:class:`~.Symbol`/:py:class:`~.Expr`
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options : keyword arguments understood by :py:class:`~.Options`
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Examples
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========
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>>> from sympy import exp, log, symbols, sfield
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>>> x = symbols("x")
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>>> K, f = sfield((x*log(x) + 4*x**2)*exp(1/x + log(x)/3)/x**2)
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>>> K
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Rational function field in x, exp(1/x), log(x), x**(1/3) over ZZ with lex order
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>>> f
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(4*x**2*(exp(1/x)) + x*(exp(1/x))*(log(x)))/((x**(1/3))**5)
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"""
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single = False
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if not is_sequence(exprs):
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exprs, single = [exprs], True
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exprs = list(map(sympify, exprs))
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opt = build_options(symbols, options)
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numdens = []
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for expr in exprs:
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numdens.extend(expr.as_numer_denom())
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reps, opt = _parallel_dict_from_expr(numdens, opt)
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if opt.domain is None:
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# NOTE: this is inefficient because construct_domain() automatically
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# performs conversion to the target domain. It shouldn't do this.
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coeffs = sum([list(rep.values()) for rep in reps], [])
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opt.domain, _ = construct_domain(coeffs, opt=opt)
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_field = FracField(opt.gens, opt.domain, opt.order)
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fracs = []
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for i in range(0, len(reps), 2):
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fracs.append(_field(tuple(reps[i:i+2])))
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if single:
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return (_field, fracs[0])
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else:
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return (_field, fracs)
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_field_cache: dict[Any, Any] = {}
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class FracField(DefaultPrinting):
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"""Multivariate distributed rational function field. """
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def __new__(cls, symbols, domain, order=lex):
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from sympy.polys.rings import PolyRing
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ring = PolyRing(symbols, domain, order)
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symbols = ring.symbols
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ngens = ring.ngens
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domain = ring.domain
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order = ring.order
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_hash_tuple = (cls.__name__, symbols, ngens, domain, order)
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obj = _field_cache.get(_hash_tuple)
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if obj is None:
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obj = object.__new__(cls)
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obj._hash_tuple = _hash_tuple
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obj._hash = hash(_hash_tuple)
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obj.ring = ring
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obj.dtype = type("FracElement", (FracElement,), {"field": obj})
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obj.symbols = symbols
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obj.ngens = ngens
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obj.domain = domain
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obj.order = order
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obj.zero = obj.dtype(ring.zero)
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obj.one = obj.dtype(ring.one)
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obj.gens = obj._gens()
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for symbol, generator in zip(obj.symbols, obj.gens):
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if isinstance(symbol, Symbol):
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name = symbol.name
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if not hasattr(obj, name):
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setattr(obj, name, generator)
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_field_cache[_hash_tuple] = obj
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return obj
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def _gens(self):
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"""Return a list of polynomial generators. """
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return tuple([ self.dtype(gen) for gen in self.ring.gens ])
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def __getnewargs__(self):
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return (self.symbols, self.domain, self.order)
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def __hash__(self):
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return self._hash
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def index(self, gen):
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if isinstance(gen, self.dtype):
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return self.ring.index(gen.to_poly())
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else:
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raise ValueError("expected a %s, got %s instead" % (self.dtype,gen))
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def __eq__(self, other):
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return isinstance(other, FracField) and \
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(self.symbols, self.ngens, self.domain, self.order) == \
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(other.symbols, other.ngens, other.domain, other.order)
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def __ne__(self, other):
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return not self == other
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def raw_new(self, numer, denom=None):
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return self.dtype(numer, denom)
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def new(self, numer, denom=None):
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if denom is None: denom = self.ring.one
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numer, denom = numer.cancel(denom)
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return self.raw_new(numer, denom)
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def domain_new(self, element):
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return self.domain.convert(element)
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def ground_new(self, element):
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try:
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return self.new(self.ring.ground_new(element))
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except CoercionFailed:
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domain = self.domain
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if not domain.is_Field and domain.has_assoc_Field:
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ring = self.ring
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ground_field = domain.get_field()
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element = ground_field.convert(element)
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numer = ring.ground_new(ground_field.numer(element))
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denom = ring.ground_new(ground_field.denom(element))
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return self.raw_new(numer, denom)
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else:
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raise
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def field_new(self, element):
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if isinstance(element, FracElement):
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if self == element.field:
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return element
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if isinstance(self.domain, FractionField) and \
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self.domain.field == element.field:
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return self.ground_new(element)
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elif isinstance(self.domain, PolynomialRing) and \
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self.domain.ring.to_field() == element.field:
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return self.ground_new(element)
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else:
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raise NotImplementedError("conversion")
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elif isinstance(element, PolyElement):
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denom, numer = element.clear_denoms()
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if isinstance(self.domain, PolynomialRing) and \
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numer.ring == self.domain.ring:
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numer = self.ring.ground_new(numer)
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elif isinstance(self.domain, FractionField) and \
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numer.ring == self.domain.field.to_ring():
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numer = self.ring.ground_new(numer)
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else:
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numer = numer.set_ring(self.ring)
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denom = self.ring.ground_new(denom)
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return self.raw_new(numer, denom)
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elif isinstance(element, tuple) and len(element) == 2:
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numer, denom = list(map(self.ring.ring_new, element))
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return self.new(numer, denom)
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elif isinstance(element, str):
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raise NotImplementedError("parsing")
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elif isinstance(element, Expr):
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return self.from_expr(element)
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else:
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return self.ground_new(element)
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__call__ = field_new
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def _rebuild_expr(self, expr, mapping):
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domain = self.domain
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powers = tuple((gen, gen.as_base_exp()) for gen in mapping.keys()
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if gen.is_Pow or isinstance(gen, ExpBase))
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def _rebuild(expr):
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generator = mapping.get(expr)
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if generator is not None:
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return generator
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elif expr.is_Add:
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return reduce(add, list(map(_rebuild, expr.args)))
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elif expr.is_Mul:
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return reduce(mul, list(map(_rebuild, expr.args)))
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elif expr.is_Pow or isinstance(expr, (ExpBase, Exp1)):
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b, e = expr.as_base_exp()
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# look for bg**eg whose integer power may be b**e
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for gen, (bg, eg) in powers:
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if bg == b and Mod(e, eg) == 0:
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return mapping.get(gen)**int(e/eg)
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if e.is_Integer and e is not S.One:
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return _rebuild(b)**int(e)
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elif mapping.get(1/expr) is not None:
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return 1/mapping.get(1/expr)
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try:
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return domain.convert(expr)
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except CoercionFailed:
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if not domain.is_Field and domain.has_assoc_Field:
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return domain.get_field().convert(expr)
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else:
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raise
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return _rebuild(expr)
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def from_expr(self, expr):
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mapping = dict(list(zip(self.symbols, self.gens)))
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try:
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frac = self._rebuild_expr(sympify(expr), mapping)
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except CoercionFailed:
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raise ValueError("expected an expression convertible to a rational function in %s, got %s" % (self, expr))
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else:
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return self.field_new(frac)
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def to_domain(self):
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return FractionField(self)
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def to_ring(self):
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from sympy.polys.rings import PolyRing
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return PolyRing(self.symbols, self.domain, self.order)
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class FracElement(DomainElement, DefaultPrinting, CantSympify):
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"""Element of multivariate distributed rational function field. """
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def __init__(self, numer, denom=None):
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if denom is None:
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denom = self.field.ring.one
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elif not denom:
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raise ZeroDivisionError("zero denominator")
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self.numer = numer
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self.denom = denom
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def raw_new(f, numer, denom):
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return f.__class__(numer, denom)
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def new(f, numer, denom):
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return f.raw_new(*numer.cancel(denom))
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def to_poly(f):
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if f.denom != 1:
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raise ValueError("f.denom should be 1")
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return f.numer
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def parent(self):
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return self.field.to_domain()
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def __getnewargs__(self):
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return (self.field, self.numer, self.denom)
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_hash = None
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def __hash__(self):
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_hash = self._hash
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if _hash is None:
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self._hash = _hash = hash((self.field, self.numer, self.denom))
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return _hash
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def copy(self):
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return self.raw_new(self.numer.copy(), self.denom.copy())
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def set_field(self, new_field):
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if self.field == new_field:
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return self
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else:
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new_ring = new_field.ring
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numer = self.numer.set_ring(new_ring)
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denom = self.denom.set_ring(new_ring)
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return new_field.new(numer, denom)
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def as_expr(self, *symbols):
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return self.numer.as_expr(*symbols)/self.denom.as_expr(*symbols)
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def __eq__(f, g):
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if isinstance(g, FracElement) and f.field == g.field:
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return f.numer == g.numer and f.denom == g.denom
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else:
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return f.numer == g and f.denom == f.field.ring.one
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def __ne__(f, g):
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return not f == g
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def __bool__(f):
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return bool(f.numer)
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def sort_key(self):
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return (self.denom.sort_key(), self.numer.sort_key())
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def _cmp(f1, f2, op):
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if isinstance(f2, f1.field.dtype):
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return op(f1.sort_key(), f2.sort_key())
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else:
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return NotImplemented
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def __lt__(f1, f2):
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return f1._cmp(f2, lt)
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def __le__(f1, f2):
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return f1._cmp(f2, le)
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def __gt__(f1, f2):
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return f1._cmp(f2, gt)
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def __ge__(f1, f2):
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return f1._cmp(f2, ge)
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def __pos__(f):
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"""Negate all coefficients in ``f``. """
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return f.raw_new(f.numer, f.denom)
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def __neg__(f):
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"""Negate all coefficients in ``f``. """
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||
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|
return f.raw_new(-f.numer, f.denom)
|
||
|
|
|
||
|
|
def _extract_ground(self, element):
|
||
|
|
domain = self.field.domain
|
||
|
|
|
||
|
|
try:
|
||
|
|
element = domain.convert(element)
|
||
|
|
except CoercionFailed:
|
||
|
|
if not domain.is_Field and domain.has_assoc_Field:
|
||
|
|
ground_field = domain.get_field()
|
||
|
|
|
||
|
|
try:
|
||
|
|
element = ground_field.convert(element)
|
||
|
|
except CoercionFailed:
|
||
|
|
pass
|
||
|
|
else:
|
||
|
|
return -1, ground_field.numer(element), ground_field.denom(element)
|
||
|
|
|
||
|
|
return 0, None, None
|
||
|
|
else:
|
||
|
|
return 1, element, None
|
||
|
|
|
||
|
|
def __add__(f, g):
|
||
|
|
"""Add rational functions ``f`` and ``g``. """
|
||
|
|
field = f.field
|
||
|
|
|
||
|
|
if not g:
|
||
|
|
return f
|
||
|
|
elif not f:
|
||
|
|
return g
|
||
|
|
elif isinstance(g, field.dtype):
|
||
|
|
if f.denom == g.denom:
|
||
|
|
return f.new(f.numer + g.numer, f.denom)
|
||
|
|
else:
|
||
|
|
return f.new(f.numer*g.denom + f.denom*g.numer, f.denom*g.denom)
|
||
|
|
elif isinstance(g, field.ring.dtype):
|
||
|
|
return f.new(f.numer + f.denom*g, f.denom)
|
||
|
|
else:
|
||
|
|
if isinstance(g, FracElement):
|
||
|
|
if isinstance(field.domain, FractionField) and field.domain.field == g.field:
|
||
|
|
pass
|
||
|
|
elif isinstance(g.field.domain, FractionField) and g.field.domain.field == field:
|
||
|
|
return g.__radd__(f)
|
||
|
|
else:
|
||
|
|
return NotImplemented
|
||
|
|
elif isinstance(g, PolyElement):
|
||
|
|
if isinstance(field.domain, PolynomialRing) and field.domain.ring == g.ring:
|
||
|
|
pass
|
||
|
|
else:
|
||
|
|
return g.__radd__(f)
|
||
|
|
|
||
|
|
return f.__radd__(g)
|
||
|
|
|
||
|
|
def __radd__(f, c):
|
||
|
|
if isinstance(c, f.field.ring.dtype):
|
||
|
|
return f.new(f.numer + f.denom*c, f.denom)
|
||
|
|
|
||
|
|
op, g_numer, g_denom = f._extract_ground(c)
|
||
|
|
|
||
|
|
if op == 1:
|
||
|
|
return f.new(f.numer + f.denom*g_numer, f.denom)
|
||
|
|
elif not op:
|
||
|
|
return NotImplemented
|
||
|
|
else:
|
||
|
|
return f.new(f.numer*g_denom + f.denom*g_numer, f.denom*g_denom)
|
||
|
|
|
||
|
|
def __sub__(f, g):
|
||
|
|
"""Subtract rational functions ``f`` and ``g``. """
|
||
|
|
field = f.field
|
||
|
|
|
||
|
|
if not g:
|
||
|
|
return f
|
||
|
|
elif not f:
|
||
|
|
return -g
|
||
|
|
elif isinstance(g, field.dtype):
|
||
|
|
if f.denom == g.denom:
|
||
|
|
return f.new(f.numer - g.numer, f.denom)
|
||
|
|
else:
|
||
|
|
return f.new(f.numer*g.denom - f.denom*g.numer, f.denom*g.denom)
|
||
|
|
elif isinstance(g, field.ring.dtype):
|
||
|
|
return f.new(f.numer - f.denom*g, f.denom)
|
||
|
|
else:
|
||
|
|
if isinstance(g, FracElement):
|
||
|
|
if isinstance(field.domain, FractionField) and field.domain.field == g.field:
|
||
|
|
pass
|
||
|
|
elif isinstance(g.field.domain, FractionField) and g.field.domain.field == field:
|
||
|
|
return g.__rsub__(f)
|
||
|
|
else:
|
||
|
|
return NotImplemented
|
||
|
|
elif isinstance(g, PolyElement):
|
||
|
|
if isinstance(field.domain, PolynomialRing) and field.domain.ring == g.ring:
|
||
|
|
pass
|
||
|
|
else:
|
||
|
|
return g.__rsub__(f)
|
||
|
|
|
||
|
|
op, g_numer, g_denom = f._extract_ground(g)
|
||
|
|
|
||
|
|
if op == 1:
|
||
|
|
return f.new(f.numer - f.denom*g_numer, f.denom)
|
||
|
|
elif not op:
|
||
|
|
return NotImplemented
|
||
|
|
else:
|
||
|
|
return f.new(f.numer*g_denom - f.denom*g_numer, f.denom*g_denom)
|
||
|
|
|
||
|
|
def __rsub__(f, c):
|
||
|
|
if isinstance(c, f.field.ring.dtype):
|
||
|
|
return f.new(-f.numer + f.denom*c, f.denom)
|
||
|
|
|
||
|
|
op, g_numer, g_denom = f._extract_ground(c)
|
||
|
|
|
||
|
|
if op == 1:
|
||
|
|
return f.new(-f.numer + f.denom*g_numer, f.denom)
|
||
|
|
elif not op:
|
||
|
|
return NotImplemented
|
||
|
|
else:
|
||
|
|
return f.new(-f.numer*g_denom + f.denom*g_numer, f.denom*g_denom)
|
||
|
|
|
||
|
|
def __mul__(f, g):
|
||
|
|
"""Multiply rational functions ``f`` and ``g``. """
|
||
|
|
field = f.field
|
||
|
|
|
||
|
|
if not f or not g:
|
||
|
|
return field.zero
|
||
|
|
elif isinstance(g, field.dtype):
|
||
|
|
return f.new(f.numer*g.numer, f.denom*g.denom)
|
||
|
|
elif isinstance(g, field.ring.dtype):
|
||
|
|
return f.new(f.numer*g, f.denom)
|
||
|
|
else:
|
||
|
|
if isinstance(g, FracElement):
|
||
|
|
if isinstance(field.domain, FractionField) and field.domain.field == g.field:
|
||
|
|
pass
|
||
|
|
elif isinstance(g.field.domain, FractionField) and g.field.domain.field == field:
|
||
|
|
return g.__rmul__(f)
|
||
|
|
else:
|
||
|
|
return NotImplemented
|
||
|
|
elif isinstance(g, PolyElement):
|
||
|
|
if isinstance(field.domain, PolynomialRing) and field.domain.ring == g.ring:
|
||
|
|
pass
|
||
|
|
else:
|
||
|
|
return g.__rmul__(f)
|
||
|
|
|
||
|
|
return f.__rmul__(g)
|
||
|
|
|
||
|
|
def __rmul__(f, c):
|
||
|
|
if isinstance(c, f.field.ring.dtype):
|
||
|
|
return f.new(f.numer*c, f.denom)
|
||
|
|
|
||
|
|
op, g_numer, g_denom = f._extract_ground(c)
|
||
|
|
|
||
|
|
if op == 1:
|
||
|
|
return f.new(f.numer*g_numer, f.denom)
|
||
|
|
elif not op:
|
||
|
|
return NotImplemented
|
||
|
|
else:
|
||
|
|
return f.new(f.numer*g_numer, f.denom*g_denom)
|
||
|
|
|
||
|
|
def __truediv__(f, g):
|
||
|
|
"""Computes quotient of fractions ``f`` and ``g``. """
|
||
|
|
field = f.field
|
||
|
|
|
||
|
|
if not g:
|
||
|
|
raise ZeroDivisionError
|
||
|
|
elif isinstance(g, field.dtype):
|
||
|
|
return f.new(f.numer*g.denom, f.denom*g.numer)
|
||
|
|
elif isinstance(g, field.ring.dtype):
|
||
|
|
return f.new(f.numer, f.denom*g)
|
||
|
|
else:
|
||
|
|
if isinstance(g, FracElement):
|
||
|
|
if isinstance(field.domain, FractionField) and field.domain.field == g.field:
|
||
|
|
pass
|
||
|
|
elif isinstance(g.field.domain, FractionField) and g.field.domain.field == field:
|
||
|
|
return g.__rtruediv__(f)
|
||
|
|
else:
|
||
|
|
return NotImplemented
|
||
|
|
elif isinstance(g, PolyElement):
|
||
|
|
if isinstance(field.domain, PolynomialRing) and field.domain.ring == g.ring:
|
||
|
|
pass
|
||
|
|
else:
|
||
|
|
return g.__rtruediv__(f)
|
||
|
|
|
||
|
|
op, g_numer, g_denom = f._extract_ground(g)
|
||
|
|
|
||
|
|
if op == 1:
|
||
|
|
return f.new(f.numer, f.denom*g_numer)
|
||
|
|
elif not op:
|
||
|
|
return NotImplemented
|
||
|
|
else:
|
||
|
|
return f.new(f.numer*g_denom, f.denom*g_numer)
|
||
|
|
|
||
|
|
def __rtruediv__(f, c):
|
||
|
|
if not f:
|
||
|
|
raise ZeroDivisionError
|
||
|
|
elif isinstance(c, f.field.ring.dtype):
|
||
|
|
return f.new(f.denom*c, f.numer)
|
||
|
|
|
||
|
|
op, g_numer, g_denom = f._extract_ground(c)
|
||
|
|
|
||
|
|
if op == 1:
|
||
|
|
return f.new(f.denom*g_numer, f.numer)
|
||
|
|
elif not op:
|
||
|
|
return NotImplemented
|
||
|
|
else:
|
||
|
|
return f.new(f.denom*g_numer, f.numer*g_denom)
|
||
|
|
|
||
|
|
def __pow__(f, n):
|
||
|
|
"""Raise ``f`` to a non-negative power ``n``. """
|
||
|
|
if n >= 0:
|
||
|
|
return f.raw_new(f.numer**n, f.denom**n)
|
||
|
|
elif not f:
|
||
|
|
raise ZeroDivisionError
|
||
|
|
else:
|
||
|
|
return f.raw_new(f.denom**-n, f.numer**-n)
|
||
|
|
|
||
|
|
def diff(f, x):
|
||
|
|
"""Computes partial derivative in ``x``.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.polys.fields import field
|
||
|
|
>>> from sympy.polys.domains import ZZ
|
||
|
|
|
||
|
|
>>> _, x, y, z = field("x,y,z", ZZ)
|
||
|
|
>>> ((x**2 + y)/(z + 1)).diff(x)
|
||
|
|
2*x/(z + 1)
|
||
|
|
|
||
|
|
"""
|
||
|
|
x = x.to_poly()
|
||
|
|
return f.new(f.numer.diff(x)*f.denom - f.numer*f.denom.diff(x), f.denom**2)
|
||
|
|
|
||
|
|
def __call__(f, *values):
|
||
|
|
if 0 < len(values) <= f.field.ngens:
|
||
|
|
return f.evaluate(list(zip(f.field.gens, values)))
|
||
|
|
else:
|
||
|
|
raise ValueError("expected at least 1 and at most %s values, got %s" % (f.field.ngens, len(values)))
|
||
|
|
|
||
|
|
def evaluate(f, x, a=None):
|
||
|
|
if isinstance(x, list) and a is None:
|
||
|
|
x = [ (X.to_poly(), a) for X, a in x ]
|
||
|
|
numer, denom = f.numer.evaluate(x), f.denom.evaluate(x)
|
||
|
|
else:
|
||
|
|
x = x.to_poly()
|
||
|
|
numer, denom = f.numer.evaluate(x, a), f.denom.evaluate(x, a)
|
||
|
|
|
||
|
|
field = numer.ring.to_field()
|
||
|
|
return field.new(numer, denom)
|
||
|
|
|
||
|
|
def subs(f, x, a=None):
|
||
|
|
if isinstance(x, list) and a is None:
|
||
|
|
x = [ (X.to_poly(), a) for X, a in x ]
|
||
|
|
numer, denom = f.numer.subs(x), f.denom.subs(x)
|
||
|
|
else:
|
||
|
|
x = x.to_poly()
|
||
|
|
numer, denom = f.numer.subs(x, a), f.denom.subs(x, a)
|
||
|
|
|
||
|
|
return f.new(numer, denom)
|
||
|
|
|
||
|
|
def compose(f, x, a=None):
|
||
|
|
raise NotImplementedError
|