Source code for diofant.series.order

from ..core import (Add, Dummy, Expr, Mul, S, Symbol, Tuple, cacheit,
                    expand_log, expand_power_base, nan, oo, sympify)
from ..core.compatibility import default_sort_key, is_sequence
from ..utilities.iterables import uniq


[docs]class Order(Expr): r"""Represents the limiting behavior of function. The formal definition for order symbol `O(f(x))` (Big O) is that `g(x) \in O(f(x))` as `x\to a` iff .. math:: \lim\limits_{x \rightarrow a} \sup \left|\frac{g(x)}{f(x)}\right| < \infty Parameters ========== expr : Expr an expression args : sequence of Symbol's or pairs (Symbol, Expr), optional If only symbols are provided, i.e. no limit point are passed, then the limit point is assumed to be zero. If no symbols are passed then all symbols in the expression are used. Examples ======== The order of a function can be intuitively thought of representing all terms of powers greater than the one specified. For example, `O(x^3)` corresponds to any terms proportional to `x^3, x^4,\ldots` and any higher power. For a polynomial, this leaves terms proportional to `x^2`, `x` and constants. >>> 1 + x + x**2 + x**3 + x**4 + O(x**3) 1 + x + x**2 + O(x**3) ``O(f(x))`` is automatically transformed to ``O(f(x).as_leading_term(x))``: >>> O(x + x**2) O(x) >>> O(cos(x)) O(1) Some arithmetic operations: >>> O(x)*x O(x**2) >>> O(x) - O(x) O(x) The Big O symbol is a set, so we support membership test: >>> x in O(x) True >>> O(1) in O(1, x) True >>> O(1, x) in O(1) False >>> O(x) in O(1, x) True >>> O(x**2) in O(x) True Limit points other then zero and multivariate Big O are also supported: >>> O(x) == O(x, (x, 0)) True >>> O(x + x**2, (x, oo)) O(x**2, (x, oo)) >>> O(cos(x), (x, pi/2)) O(x - pi/2, (x, pi/2)) >>> O(1 + x*y) O(1, x, y) >>> O(1 + x*y, (x, 0), (y, 0)) O(1, x, y) >>> O(1 + x*y, (x, oo), (y, oo)) O(x*y, (x, oo), (y, oo)) References ========== * https//en.wikipedia.org/wiki/Big_O_notation """ is_Order = True @cacheit def __new__(cls, expr, *args, **kwargs): expr = sympify(expr) if not args: if expr.is_Order: variables = expr.variables point = expr.point else: variables = list(expr.free_symbols) point = [S.Zero]*len(variables) else: args = list(args if is_sequence(args) else [args]) variables, point = [], [] if is_sequence(args[0]): for a in args: v, p = list(map(sympify, a)) variables.append(v) point.append(p) else: variables = list(map(sympify, args)) point = [S.Zero]*len(variables) if not all(isinstance(v, (Dummy, Symbol)) for v in variables): raise TypeError('Variables are not symbols, got %s' % variables) if len(list(uniq(variables))) != len(variables): raise ValueError('Variables are supposed to be unique symbols, got %s' % variables) if expr.is_Order: expr_vp = dict(expr.args[1:]) new_vp = dict(expr_vp) vp = dict(zip(variables, point)) for v, p in vp.items(): if v in new_vp: if p != new_vp[v]: raise NotImplementedError( "Mixing Order at different points is not supported.") else: new_vp[v] = p if set(expr_vp) == set(new_vp): return expr else: variables = list(new_vp) point = [new_vp[v] for v in variables] if expr is nan: return nan if any(x in p.free_symbols for x in variables for p in point): raise ValueError('Got %s as a point.' % point) if variables: if any(p != point[0] for p in point): raise NotImplementedError if point[0] in [oo, -oo]: s = {k: 1/Dummy() for k in variables} rs = {1/v: 1/k for k, v in s.items()} elif point[0] is not S.Zero: s = {k: Dummy() + point[0] for k in variables} rs = {v - point[0]: k - point[0] for k, v in s.items()} else: s = () rs = () expr = expr.subs(s) if expr.is_Add: from ..core import expand_multinomial expr = expand_multinomial(expr) if s: args = tuple(r[0] for r in rs.items()) else: args = tuple(variables) if len(variables) > 1: # XXX: better way? We need this expand() to # workaround e.g: expr = x*(x + y). # (x*(x + y)).as_leading_term(x, y) currently returns # x*y (wrong order term!). That's why we want to deal with # expand()'ed expr (handled in "if expr.is_Add" branch below). expr = expr.expand() if expr.is_Add: lst = expr.extract_leading_order(args) expr = Add(*[f.expr for (e, f) in lst]) elif expr: expr = expr.as_leading_term(*args) expr = expr.as_independent(*args, as_Add=False)[1] expr = expand_power_base(expr) expr = expand_log(expr) if len(args) == 1: # The definition of O(f(x)) symbol explicitly stated that # the argument of f(x) is irrelevant. That's why we can # combine some power exponents (only "on top" of the # expression tree for f(x)), e.g.: # x**p * (-x)**q -> x**(p+q) for real p, q. x = args[0] margs = list(Mul.make_args( expr.as_independent(x, as_Add=False)[1])) for i, t in enumerate(margs): if t.is_Pow: b, q = t.args if b in (x, -x) and q.is_extended_real and not q.has(x): margs[i] = x**q elif b.is_Pow and not b.exp.has(x): b, r = b.args if b in (x, -x) and r.is_extended_real: margs[i] = x**(r*q) elif b.is_Mul and b.args[0] is S.NegativeOne: b = -b if b.is_Pow and not b.exp.has(x): b, r = b.args if b in (x, -x) and r.is_extended_real: margs[i] = x**(r*q) expr = Mul(*margs) expr = expr.subs(rs) if expr is S.Zero: return expr if expr.is_Order: expr = expr.expr if not expr.has(*variables): expr = S.One # create Order instance: vp = dict(zip(variables, point)) variables.sort(key=default_sort_key) point = [vp[v] for v in variables] args = (expr,) + Tuple(*zip(variables, point)) obj = Expr.__new__(cls, *args) return obj def _eval_nseries(self, x, n, logx): return self @property def expr(self): return self.args[0] @property def variables(self): if self.args[1:]: return tuple(x[0] for x in self.args[1:]) else: return () @property def point(self): if self.args[1:]: return tuple(x[1] for x in self.args[1:]) else: return () @property def free_symbols(self): return self.expr.free_symbols | set(self.variables) def _eval_power(self, e): if e.is_Number and e.is_nonnegative: return self.func(self.expr ** e, *self.args[1:]) if e == O(1): return self return def as_expr_variables(self, order_symbols): if order_symbols is None: order_symbols = self.args[1:] else: if (not all(o[1] == order_symbols[0][1] for o in order_symbols) and not all(p == self.point[0] for p in self.point)): # pragma: no cover raise NotImplementedError('Order at points other than 0 ' 'or oo not supported, got %s as a point.' % point) if order_symbols and order_symbols[0][1] != self.point[0]: raise NotImplementedError( "Multiplying Order at different points is not supported.") order_symbols = dict(order_symbols) for s, p in dict(self.args[1:]).items(): if s not in order_symbols: order_symbols[s] = p order_symbols = sorted(order_symbols.items(), key=lambda x: default_sort_key(x[0])) return self.expr, tuple(order_symbols) def removeO(self): return S.Zero def getO(self): return self
[docs] @cacheit def contains(self, expr): """Membership test. Returns ======= Boolean or None Return True if ``expr`` belongs to ``self``. Return False if ``self`` belongs to ``expr``. Return None if the inclusion relation cannot be determined. """ from ..simplify import powsimp from .limits import Limit if expr is S.Zero: return True if expr is nan: return False if expr.is_Order: if (not all(p == expr.point[0] for p in expr.point) and not all(p == self.point[0] for p in self.point)): # pragma: no cover raise NotImplementedError('Order at points other than 0 ' 'or oo not supported, got %s as a point.' % point) else: # self and/or expr is O(1): if any(not p for p in [expr.point, self.point]): point = self.point + expr.point if point: point = point[0] else: point = S.Zero else: point = self.point[0] if expr.expr == self.expr: # O(1) + O(1), O(1) + O(1, x), etc. return all(x in self.args[1:] for x in expr.args[1:]) if expr.expr.is_Add: return all(self.contains(x) for x in expr.expr.args) if self.expr.is_Add and point == 0: return any(self.func(x, *self.args[1:]).contains(expr) for x in self.expr.args) if self.variables and expr.variables: common_symbols = tuple(s for s in self.variables if s in expr.variables) elif self.variables: common_symbols = self.variables else: common_symbols = expr.variables if not common_symbols: return r = None ratio = self.expr/expr.expr ratio = powsimp(ratio, deep=True, combine='exp') for s in common_symbols: l = Limit(ratio, s, point).doit(heuristics=False) if not isinstance(l, Limit): l = l != 0 else: l = None if r is None: r = l else: if r != l: return return r obj = self.func(expr, *self.args[1:]) return self.contains(obj)
def __contains__(self, other): result = self.contains(other) if result is None: raise TypeError('contains did not evaluate to a bool') return result def _eval_subs(self, old, new): if old in self.variables: newexpr = self.expr.subs({old: new}) i = self.variables.index(old) newvars = list(self.variables) newpt = list(self.point) if new.is_Symbol: newvars[i] = new else: syms = new.free_symbols if len(syms) == 1 or old in syms: if old in syms: var = self.variables[i] else: var = syms.pop() # First, try to substitute self.point in the "new" # expr to see if this is a fixed point. # E.g. O(y).subs({y: sin(x)}) point = new.subs({var: self.point[i]}) if point != self.point[i]: from ..solvers import solve d = Dummy() res = solve(old - new.subs({var: d}), d) point = d.subs(res[0]).limit(old, self.point[i]) newvars[i] = var newpt[i] = point else: del newvars[i], newpt[i] if not syms and new == self.point[i]: newvars.extend(syms) newpt.extend([S.Zero]*len(syms)) return Order(newexpr, *zip(newvars, newpt)) def _eval_conjugate(self): expr = self.expr._eval_conjugate() if expr is not None: return self.func(expr, *self.args[1:]) def _eval_transpose(self): expr = self.expr._eval_transpose() if expr is not None: return self.func(expr, *self.args[1:]) def _eval_is_commutative(self): return self.expr.is_commutative
O = Order