예제 #1
0
    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(v.is_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.keys():
                    if p != new_vp[v]:
                        raise NotImplementedError(
                            "Mixing Order at different points is not supported."
                        )
                else:
                    new_vp[v] = p
            if set(expr_vp.keys()) == set(new_vp.keys()):
                return expr
            else:
                variables = list(new_vp.keys())
                point = [new_vp[v] for v in variables]

        if expr is S.NaN:
            return S.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(
                    "Multivariable orders at different points are not supported."
                )
            if point[0] is S.Infinity:
                s = {k: 1 / Dummy() for k in variables}
                rs = {1 / v: 1 / k for k, v in s.items()}
            elif point[0] is S.NegativeInfinity:
                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 = dict((k, Dummy() + point[0]) for k in variables)
                rs = dict((v - point[0], k - point[0]) for k, v in s.items())
            else:
                s = ()
                rs = ()

            expr = expr.subs(s)

            if expr.is_Add:
                from sympy 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_Power:
                            b, q = t.args
                            if b in (x, -x) and q.is_real and not q.has(x):
                                margs[i] = x**q
                            elif b.is_Power and not b.exp.has(x):
                                b, r = b.args
                                if b in (x, -x) and r.is_real:
                                    margs[i] = x**(r * q)
                            elif b.is_Mul and b.args[0] is S.NegativeOne:
                                b = -b
                                if b.is_Power and not b.exp.has(x):
                                    b, r = b.args
                                    if b in (x, -x) and r.is_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
예제 #2
0
    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, 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.keys():
                    if p != new_vp[v]:
                        raise NotImplementedError(
                            "Mixing Order at different points is not supported.")
                else:
                    new_vp[v] = p
            if set(expr_vp.keys()) == set(new_vp.keys()):
                return expr
            else:
                variables = list(new_vp.keys())
                point = [new_vp[v] for v in variables]

        if expr is S.NaN:
            return S.NaN

        if not all(p is S.Zero for p in point) and \
           not all(p is S.Infinity for p in point):
            raise NotImplementedError('Order at points other than 0 '
                'or oo not supported, got %s as a point.' % point)

        if 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(variables, point)
                expr = Add(*[f.expr for (e, f) in lst])

            elif expr:
                if point[0] == S.Zero:
                    expr = expr.as_leading_term(*variables)
                expr = expr.as_independent(*variables, as_Add=False)[1]

                expr = expand_power_base(expr)
                expr = expand_log(expr)

                if len(variables) == 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 = variables[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_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_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_real:
                                        margs[i] = x**(r*q)

                    expr = Mul(*margs)

        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:
        variables.sort(key=default_sort_key)
        args = (expr,) + Tuple(*zip(variables, point))
        obj = Expr.__new__(cls, *args)
        return obj
예제 #3
0
파일: order.py 프로젝트: QuaBoo/sympy
    def __new__(cls, expr, *symbols):

        expr = sympify(expr)
        if expr is S.NaN:
            return S.NaN

        point = S.Zero
        if symbols:
            symbols = list(map(sympify, symbols))
            if symbols[-1] in (S.Infinity, S.Zero):
                point = symbols[-1]
                symbols = symbols[:-1]
            if not all(isinstance(s, Symbol) for s in symbols):
                raise NotImplementedError(
                    'Order at points other than 0 or oo not supported.')
        if not symbols:
            symbols = list(expr.free_symbols)

        if expr.is_Order:
            v = set(expr.variables)
            symbols = v | set(symbols)
            if symbols == v:
                return expr
            symbols = list(symbols)

        elif symbols:

            symbols = list(set(symbols))
            args = tuple(symbols) + (point,)

            if len(symbols) > 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(*symbols)
                expr = expr.as_independent(*symbols, as_Add=False)[1]

                expr = expand_power_base(expr)
                expr = expand_log(expr)

                if len(symbols) == 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 = symbols[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_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_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_real:
                                        margs[i] = x**(r*q)

                    expr = Mul(*margs)

        if expr is S.Zero:
            return expr

        if not expr.has(*symbols):
            expr = S.One

        # create Order instance:
        symbols.sort(key=default_sort_key)
        args = (expr,) + tuple(symbols) + (point,)
        obj = Expr.__new__(cls, *args)
        return obj
예제 #4
0
파일: order.py 프로젝트: mattpap/sympy
    def __new__(cls, expr, *symbols, **assumptions):

        expr = sympify(expr)
        if expr is S.NaN:
            return S.NaN

        if symbols:
            symbols = map(sympify, symbols)
            if not all(isinstance(s, Symbol) for s in symbols):
                raise NotImplementedError(
                    'Order at points other than 0 not supported.')
        else:
            symbols = list(expr.free_symbols)

        if expr.is_Order:
            v = set(expr.variables)
            symbols = v | set(symbols)
            if symbols == v:
                return expr
            symbols = list(symbols)

        elif symbols:

            symbols = list(set(symbols))

            if len(symbols) > 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(*symbols)
                expr = Add(*[f.expr for (e, f) in lst])

            elif expr:
                expr = expr.as_leading_term(*symbols)
                expr = expr.as_independent(*symbols, as_Add=False)[1]

                expr = expand_power_base(expr)
                expr = expand_log(expr)

                if len(symbols) == 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 = symbols[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_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_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_real:
                                        margs[i] = x**(r * q)

                    expr = Mul(*margs)

        if expr is S.Zero:
            return expr

        if not expr.has(*symbols):
            expr = S.One

        # create Order instance:
        symbols.sort(key=cmp_to_key(Basic.compare))
        obj = Expr.__new__(cls, expr, *symbols, **assumptions)

        return obj
예제 #5
0
    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(v.is_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.keys():
                    if p != new_vp[v]:
                        raise NotImplementedError(
                            "Mixing Order at different points is not supported.")
                else:
                    new_vp[v] = p
            if set(expr_vp.keys()) == set(new_vp.keys()):
                return expr
            else:
                variables = list(new_vp.keys())
                point = [new_vp[v] for v in variables]

        if expr is S.NaN:
            return S.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(
                    "Multivariable orders at different points are not supported.")
            if point[0] is S.Infinity:
                s = {k: 1/Dummy() for k in variables}
                rs = {1/v: 1/k for k, v in s.items()}
                ps = [S.Zero for p in point]
            elif point[0] is S.NegativeInfinity:
                s = {k: -1/Dummy() for k in variables}
                rs = {-1/v: -1/k for k, v in s.items()}
                ps = [S.Zero for p in point]
            elif point[0] is not S.Zero:
                s = {k: Dummy() + point[0] for k in variables}
                rs = {(v - point[0]).together(): k - point[0] for k, v in s.items()}
                ps = [S.Zero for p in point]
            else:
                s = ()
                rs = ()
                ps = list(point)

            expr = expr.subs(s)

            if expr.is_Add:
                expr = expr.factor()

            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()

            old_expr = None
            while old_expr != expr:
                old_expr = expr
                if expr.is_Add:
                    lst = expr.extract_leading_order(args)
                    expr = Add(*[f.expr for (e, f) in lst])

                elif expr:
                    try:
                        expr = expr.as_leading_term(*args)
                    except PoleError:
                        if isinstance(expr, Function) or\
                                all(isinstance(arg, Function) for arg in expr.args):
                            # It is not possible to simplify an expression
                            # containing only functions (which raise error on
                            # call to leading term) further
                            pass
                        else:
                            orders = []
                            pts = tuple(zip(args, ps))
                            for arg in expr.args:
                                try:
                                    lt = arg.as_leading_term(*args)
                                except PoleError:
                                    lt = arg
                                if lt not in args:
                                    order = Order(lt)
                                else:
                                    order = Order(lt, *pts)
                                orders.append(order)
                            if expr.is_Add:
                                new_expr = Order(Add(*orders), *pts)
                                if new_expr.is_Add:
                                    new_expr = Order(Add(*[a.expr for a in new_expr.args]), *pts)
                                expr = new_expr.expr
                            elif expr.is_Mul:
                                expr = Mul(*[a.expr for a in orders])
                            elif expr.is_Pow:
                                e = expr.exp
                                b = expr.base
                                expr = exp(e * log(b))

                    # It would probably be better to handle this somewhere
                    # else. This is needed for a testcase in which there is a
                    # symbol with the assumptions zero=True.
                    if expr.is_zero:
                        expr = S.Zero
                    else:
                        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_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_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_real:
                                            margs[i] = x**(r*q)

                        expr = Mul(*margs)

            expr = expr.subs(rs)

        if expr.is_Order:
            expr = expr.expr

        if not expr.has(*variables) and not expr.is_zero:
            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
예제 #6
0
    def __new__(cls, expr, *symbols, **assumptions):

        expr = sympify(expr).expand()
        if expr is S.NaN:
            return S.NaN

        if symbols:
            symbols = map(sympify, symbols)
            if not all(isinstance(s, Symbol) for s in symbols):
                raise NotImplementedError(
                    'Order at points other than 0 not supported.')
        else:
            symbols = list(expr.free_symbols)

        if expr.is_Order:
            v = set(expr.variables)
            symbols = v | set(symbols)
            if symbols == v:
                return expr
            symbols = list(symbols)

        elif symbols:

            symbols = list(set(symbols))

            if expr.is_Add:
                lst = expr.extract_leading_order(*symbols)
                expr = Add(*[f.expr for (e, f) in lst])

            elif expr:
                if len(symbols) > 1 or expr.is_commutative is False:
                    # TODO
                    # We cannot use compute_leading_term because that only
                    # works in one symbol.
                    expr = expr.as_leading_term(*symbols)
                else:
                    expr = expr.compute_leading_term(symbols[0])

                margs = list(Mul.make_args(expr.as_independent(*symbols)[1]))

                if len(symbols) == 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 = symbols[0]

                    for i, t in enumerate(margs):
                        if t.is_Pow:
                            b, q = t.args
                            if b in (x, -x) and q.is_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_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_real:
                                        margs[i] = x**(r*q)

                expr = Mul(*margs)

        if expr is S.Zero:
            return expr

        if not expr.has(*symbols):
            expr = S.One

        # create Order instance:
        symbols.sort(key=cmp_to_key(Basic.compare))
        obj = Expr.__new__(cls, expr, *symbols, **assumptions)

        return obj