image-handler/image-handler10.0/venv/lib/site-packages/sympy/polys/polyutils.py

"""Useful utilities for higher level polynomial classes. """


from sympy.core import (S, Add, Mul, Pow, Eq, Expr,
    expand_mul, expand_multinomial)
from sympy.core.exprtools import decompose_power, decompose_power_rat
from sympy.core.numbers import _illegal
from sympy.polys.polyerrors import PolynomialError, GeneratorsError
from sympy.polys.polyoptions import build_options


import re

_gens_order = {
    'a': 301, 'b': 302, 'c': 303, 'd': 304,
    'e': 305, 'f': 306, 'g': 307, 'h': 308,
    'i': 309, 'j': 310, 'k': 311, 'l': 312,
    'm': 313, 'n': 314, 'o': 315, 'p': 216,
    'q': 217, 'r': 218, 's': 219, 't': 220,
    'u': 221, 'v': 222, 'w': 223, 'x': 124,
    'y': 125, 'z': 126,
}

_max_order = 1000
_re_gen = re.compile(r"^(.*?)(\d*)$", re.MULTILINE)


def _nsort(roots, separated=False):
    """Sort the numerical roots putting the real roots first, then sorting
    according to real and imaginary parts. If ``separated`` is True, then
    the real and imaginary roots will be returned in two lists, respectively.

    This routine tries to avoid issue 6137 by separating the roots into real
    and imaginary parts before evaluation. In addition, the sorting will raise
    an error if any computation cannot be done with precision.
    """
    if not all(r.is_number for r in roots):
        raise NotImplementedError
    # see issue 6137:
    # get the real part of the evaluated real and imaginary parts of each root
    key = [[i.n(2).as_real_imag()[0] for i in r.as_real_imag()] for r in roots]
    # make sure the parts were computed with precision
    if len(roots) > 1 and any(i._prec == 1 for k in key for i in k):
        raise NotImplementedError("could not compute root with precision")
    # insert a key to indicate if the root has an imaginary part
    key = [(1 if i else 0, r, i) for r, i in key]
    key = sorted(zip(key, roots))
    # return the real and imaginary roots separately if desired
    if separated:
        r = []
        i = []
        for (im, _, _), v in key:
            if im:
                i.append(v)
            else:
                r.append(v)
        return r, i
    _, roots = zip(*key)
    return list(roots)


def _sort_gens(gens, **args):
    """Sort generators in a reasonably intelligent way. """
    opt = build_options(args)

    gens_order, wrt = {}, None

    if opt is not None:
        gens_order, wrt = {}, opt.wrt

        for i, gen in enumerate(opt.sort):
            gens_order[gen] = i + 1

    def order_key(gen):
        gen = str(gen)

        if wrt is not None:
            try:
                return (-len(wrt) + wrt.index(gen), gen, 0)
            except ValueError:
                pass

        name, index = _re_gen.match(gen).groups()

        if index:
            index = int(index)
        else:
            index = 0

        try:
            return ( gens_order[name], name, index)
        except KeyError:
            pass

        try:
            return (_gens_order[name], name, index)
        except KeyError:
            pass

        return (_max_order, name, index)

    try:
        gens = sorted(gens, key=order_key)
    except TypeError:  # pragma: no cover
        pass

    return tuple(gens)


def _unify_gens(f_gens, g_gens):
    """Unify generators in a reasonably intelligent way. """
    f_gens = list(f_gens)
    g_gens = list(g_gens)

    if f_gens == g_gens:
        return tuple(f_gens)

    gens, common, k = [], [], 0

    for gen in f_gens:
        if gen in g_gens:
            common.append(gen)

    for i, gen in enumerate(g_gens):
        if gen in common:
            g_gens[i], k = common[k], k + 1

    for gen in common:
        i = f_gens.index(gen)

        gens.extend(f_gens[:i])
        f_gens = f_gens[i + 1:]

        i = g_gens.index(gen)

        gens.extend(g_gens[:i])
        g_gens = g_gens[i + 1:]

        gens.append(gen)

    gens.extend(f_gens)
    gens.extend(g_gens)

    return tuple(gens)


def _analyze_gens(gens):
    """Support for passing generators as `*gens` and `[gens]`. """
    if len(gens) == 1 and hasattr(gens[0], '__iter__'):
        return tuple(gens[0])
    else:
        return tuple(gens)


def _sort_factors(factors, **args):
    """Sort low-level factors in increasing 'complexity' order. """
    def order_if_multiple_key(factor):
        (f, n) = factor
        return (len(f), n, f)

    def order_no_multiple_key(f):
        return (len(f), f)

    if args.get('multiple', True):
        return sorted(factors, key=order_if_multiple_key)
    else:
        return sorted(factors, key=order_no_multiple_key)

illegal_types = [type(obj) for obj in _illegal]
finf = [float(i) for i in _illegal[1:3]]
def _not_a_coeff(expr):
    """Do not treat NaN and infinities as valid polynomial coefficients. """
    if type(expr) in illegal_types or expr in finf:
        return True
    if isinstance(expr, float) and float(expr) != expr:
        return True  # nan
    return  # could be


def _parallel_dict_from_expr_if_gens(exprs, opt):
    """Transform expressions into a multinomial form given generators. """
    k, indices = len(opt.gens), {}

    for i, g in enumerate(opt.gens):
        indices[g] = i

    polys = []

    for expr in exprs:
        poly = {}

        if expr.is_Equality:
            expr = expr.lhs - expr.rhs

        for term in Add.make_args(expr):
            coeff, monom = [], [0]*k

            for factor in Mul.make_args(term):
                if not _not_a_coeff(factor) and factor.is_Number:
                    coeff.append(factor)
                else:
                    try:
                        if opt.series is False:
                            base, exp = decompose_power(factor)

                            if exp < 0:
                                exp, base = -exp, Pow(base, -S.One)
                        else:
                            base, exp = decompose_power_rat(factor)

                        monom[indices[base]] = exp
                    except KeyError:
                        if not factor.has_free(*opt.gens):
                            coeff.append(factor)
                        else:
                            raise PolynomialError("%s contains an element of "
                                                  "the set of generators." % factor)

            monom = tuple(monom)

            if monom in poly:
                poly[monom] += Mul(*coeff)
            else:
                poly[monom] = Mul(*coeff)

        polys.append(poly)

    return polys, opt.gens


def _parallel_dict_from_expr_no_gens(exprs, opt):
    """Transform expressions into a multinomial form and figure out generators. """
    if opt.domain is not None:
        def _is_coeff(factor):
            return factor in opt.domain
    elif opt.extension is True:
        def _is_coeff(factor):
            return factor.is_algebraic
    elif opt.greedy is not False:
        def _is_coeff(factor):
            return factor is S.ImaginaryUnit
    else:
        def _is_coeff(factor):
            return factor.is_number

    gens, reprs = set(), []

    for expr in exprs:
        terms = []

        if expr.is_Equality:
            expr = expr.lhs - expr.rhs

        for term in Add.make_args(expr):
            coeff, elements = [], {}

            for factor in Mul.make_args(term):
                if not _not_a_coeff(factor) and (factor.is_Number or _is_coeff(factor)):
                    coeff.append(factor)
                else:
                    if opt.series is False:
                        base, exp = decompose_power(factor)

                        if exp < 0:
                            exp, base = -exp, Pow(base, -S.One)
                    else:
                        base, exp = decompose_power_rat(factor)

                    elements[base] = elements.setdefault(base, 0) + exp
                    gens.add(base)

            terms.append((coeff, elements))

        reprs.append(terms)

    gens = _sort_gens(gens, opt=opt)
    k, indices = len(gens), {}

    for i, g in enumerate(gens):
        indices[g] = i

    polys = []

    for terms in reprs:
        poly = {}

        for coeff, term in terms:
            monom = [0]*k

            for base, exp in term.items():
                monom[indices[base]] = exp

            monom = tuple(monom)

            if monom in poly:
                poly[monom] += Mul(*coeff)
            else:
                poly[monom] = Mul(*coeff)

        polys.append(poly)

    return polys, tuple(gens)


def _dict_from_expr_if_gens(expr, opt):
    """Transform an expression into a multinomial form given generators. """
    (poly,), gens = _parallel_dict_from_expr_if_gens((expr,), opt)
    return poly, gens


def _dict_from_expr_no_gens(expr, opt):
    """Transform an expression into a multinomial form and figure out generators. """
    (poly,), gens = _parallel_dict_from_expr_no_gens((expr,), opt)
    return poly, gens


def parallel_dict_from_expr(exprs, **args):
    """Transform expressions into a multinomial form. """
    reps, opt = _parallel_dict_from_expr(exprs, build_options(args))
    return reps, opt.gens


def _parallel_dict_from_expr(exprs, opt):
    """Transform expressions into a multinomial form. """
    if opt.expand is not False:
        exprs = [ expr.expand() for expr in exprs ]

    if any(expr.is_commutative is False for expr in exprs):
        raise PolynomialError('non-commutative expressions are not supported')

    if opt.gens:
        reps, gens = _parallel_dict_from_expr_if_gens(exprs, opt)
    else:
        reps, gens = _parallel_dict_from_expr_no_gens(exprs, opt)

    return reps, opt.clone({'gens': gens})


def dict_from_expr(expr, **args):
    """Transform an expression into a multinomial form. """
    rep, opt = _dict_from_expr(expr, build_options(args))
    return rep, opt.gens


def _dict_from_expr(expr, opt):
    """Transform an expression into a multinomial form. """
    if expr.is_commutative is False:
        raise PolynomialError('non-commutative expressions are not supported')

    def _is_expandable_pow(expr):
        return (expr.is_Pow and expr.exp.is_positive and expr.exp.is_Integer
                and expr.base.is_Add)

    if opt.expand is not False:
        if not isinstance(expr, (Expr, Eq)):
            raise PolynomialError('expression must be of type Expr')
        expr = expr.expand()
        # TODO: Integrate this into expand() itself
        while any(_is_expandable_pow(i) or i.is_Mul and
            any(_is_expandable_pow(j) for j in i.args) for i in
                Add.make_args(expr)):

            expr = expand_multinomial(expr)
        while any(i.is_Mul and any(j.is_Add for j in i.args) for i in Add.make_args(expr)):
            expr = expand_mul(expr)

    if opt.gens:
        rep, gens = _dict_from_expr_if_gens(expr, opt)
    else:
        rep, gens = _dict_from_expr_no_gens(expr, opt)

    return rep, opt.clone({'gens': gens})


def expr_from_dict(rep, *gens):
    """Convert a multinomial form into an expression. """
    result = []

    for monom, coeff in rep.items():
        term = [coeff]
        for g, m in zip(gens, monom):
            if m:
                term.append(Pow(g, m))

        result.append(Mul(*term))

    return Add(*result)

parallel_dict_from_basic = parallel_dict_from_expr
dict_from_basic = dict_from_expr
basic_from_dict = expr_from_dict


def _dict_reorder(rep, gens, new_gens):
    """Reorder levels using dict representation. """
    gens = list(gens)

    monoms = rep.keys()
    coeffs = rep.values()

    new_monoms = [ [] for _ in range(len(rep)) ]
    used_indices = set()

    for gen in new_gens:
        try:
            j = gens.index(gen)
            used_indices.add(j)

            for M, new_M in zip(monoms, new_monoms):
                new_M.append(M[j])
        except ValueError:
            for new_M in new_monoms:
                new_M.append(0)

    for i, _ in enumerate(gens):
        if i not in used_indices:
            for monom in monoms:
                if monom[i]:
                    raise GeneratorsError("unable to drop generators")

    return map(tuple, new_monoms), coeffs


class PicklableWithSlots:
    """
    Mixin class that allows to pickle objects with ``__slots__``.

    Examples
    ========

    First define a class that mixes :class:`PicklableWithSlots` in::

        >>> from sympy.polys.polyutils import PicklableWithSlots
        >>> class Some(PicklableWithSlots):
        ...     __slots__ = ('foo', 'bar')
        ...
        ...     def __init__(self, foo, bar):
        ...         self.foo = foo
        ...         self.bar = bar

    To make :mod:`pickle` happy in doctest we have to use these hacks::

        >>> import builtins
        >>> builtins.Some = Some
        >>> from sympy.polys import polyutils
        >>> polyutils.Some = Some

    Next lets see if we can create an instance, pickle it and unpickle::

        >>> some = Some('abc', 10)
        >>> some.foo, some.bar
        ('abc', 10)

        >>> from pickle import dumps, loads
        >>> some2 = loads(dumps(some))

        >>> some2.foo, some2.bar
        ('abc', 10)

    """

    __slots__ = ()

    def __getstate__(self, cls=None):
        if cls is None:
            # This is the case for the instance that gets pickled
            cls = self.__class__

        d = {}

        # Get all data that should be stored from super classes
        for c in cls.__bases__:
            # XXX: Python 3.11 defines object.__getstate__ and it does not
            # accept any arguments so we need to make sure not to call it with
            # an argument here. To be compatible with Python < 3.11 we need to
            # be careful not to assume that c or object has a __getstate__
            # method though.
            getstate = getattr(c, "__getstate__", None)
            objstate = getattr(object, "__getstate__", None)
            if getstate is not None and getstate is not objstate:
                d.update(getstate(self, c))

        # Get all information that should be stored from cls and return the dict
        for name in cls.__slots__:
            if hasattr(self, name):
                d[name] = getattr(self, name)

        return d

    def __setstate__(self, d):
        # All values that were pickled are now assigned to a fresh instance
        for name, value in d.items():
            try:
                setattr(self, name, value)
            except AttributeError:    # This is needed in cases like Rational :> Half
                pass


class IntegerPowerable:
    r"""
    Mixin class for classes that define a `__mul__` method, and want to be
    raised to integer powers in the natural way that follows. Implements
    powering via binary expansion, for efficiency.

    By default, only integer powers $\geq 2$ are supported. To support the
    first, zeroth, or negative powers, override the corresponding methods,
    `_first_power`, `_zeroth_power`, `_negative_power`, below.
    """

    def __pow__(self, e, modulo=None):
        if e < 2:
            try:
                if e == 1:
                    return self._first_power()
                elif e == 0:
                    return self._zeroth_power()
                else:
                    return self._negative_power(e, modulo=modulo)
            except NotImplementedError:
                return NotImplemented
        else:
            bits = [int(d) for d in reversed(bin(e)[2:])]
            n = len(bits)
            p = self
            first = True
            for i in range(n):
                if bits[i]:
                    if first:
                        r = p
                        first = False
                    else:
                        r *= p
                        if modulo is not None:
                            r %= modulo
                if i < n - 1:
                    p *= p
                    if modulo is not None:
                        p %= modulo
            return r

    def _negative_power(self, e, modulo=None):
        """
        Compute inverse of self, then raise that to the abs(e) power.
        For example, if the class has an `inv()` method,
            return self.inv() ** abs(e) % modulo
        """
        raise NotImplementedError

    def _zeroth_power(self):
        """Return unity element of algebraic struct to which self belongs."""
        raise NotImplementedError

    def _first_power(self):
        """Return a copy of self."""
        raise NotImplementedError