我们从Python开源项目中，提取了以下10个代码示例，用于说明如何使用sympy.acos()。
def real(self, nested_scope=None): """Return the correspond floating point number.""" op = self.children[0].name expr = self.children[1] dispatch = { 'sin': sympy.sin, 'cos': sympy.cos, 'tan': sympy.tan, 'asin': sympy.asin, 'acos': sympy.acos, 'atan': sympy.atan, 'exp': sympy.exp, 'ln': sympy.log, 'sqrt': sympy.sqrt } if op in dispatch: arg = expr.real(nested_scope) return dispatch[op](arg) else: raise NodeException("internal error: undefined external")
def sym(self, nested_scope=None): """Return the corresponding symbolic expression.""" op = self.children[0].name expr = self.children[1] dispatch = { 'sin': sympy.sin, 'cos': sympy.cos, 'tan': sympy.tan, 'asin': sympy.asin, 'acos': sympy.acos, 'atan': sympy.atan, 'exp': sympy.exp, 'ln': sympy.log, 'sqrt': sympy.sqrt } if op in dispatch: arg = expr.sym(nested_scope) return dispatch[op](arg) else: raise NodeException("internal error: undefined external")
def cartesian_to_spherical_sympy(X): vacos = numpy.vectorize(sympy.acos) return numpy.stack([ _atan2_0(X), vacos(X[:, 2]) ], axis=1)
def eval_trigsubstitution(theta, func, rewritten, substep, integrand, symbol): func = func.subs(sympy.sec(theta), 1/sympy.cos(theta)) trig_function = list(func.find(TrigonometricFunction)) assert len(trig_function) == 1 trig_function = trig_function[0] relation = sympy.solve(symbol - func, trig_function) assert len(relation) == 1 numer, denom = sympy.fraction(relation[0]) if isinstance(trig_function, sympy.sin): opposite = numer hypotenuse = denom adjacent = sympy.sqrt(denom**2 - numer**2) inverse = sympy.asin(relation[0]) elif isinstance(trig_function, sympy.cos): adjacent = numer hypotenuse = denom opposite = sympy.sqrt(denom**2 - numer**2) inverse = sympy.acos(relation[0]) elif isinstance(trig_function, sympy.tan): opposite = numer adjacent = denom hypotenuse = sympy.sqrt(denom**2 + numer**2) inverse = sympy.atan(relation[0]) substitution = [ (sympy.sin(theta), opposite/hypotenuse), (sympy.cos(theta), adjacent/hypotenuse), (sympy.tan(theta), opposite/adjacent), (theta, inverse) ] return _manualintegrate(substep).subs(substitution).trigsimp()
def eval_trigsubstitution(theta, func, rewritten, substep, restriction, integrand, symbol): func = func.subs(sympy.sec(theta), 1/sympy.cos(theta)) trig_function = list(func.find(TrigonometricFunction)) assert len(trig_function) == 1 trig_function = trig_function[0] relation = sympy.solve(symbol - func, trig_function) assert len(relation) == 1 numer, denom = sympy.fraction(relation[0]) if isinstance(trig_function, sympy.sin): opposite = numer hypotenuse = denom adjacent = sympy.sqrt(denom**2 - numer**2) inverse = sympy.asin(relation[0]) elif isinstance(trig_function, sympy.cos): adjacent = numer hypotenuse = denom opposite = sympy.sqrt(denom**2 - numer**2) inverse = sympy.acos(relation[0]) elif isinstance(trig_function, sympy.tan): opposite = numer adjacent = denom hypotenuse = sympy.sqrt(denom**2 + numer**2) inverse = sympy.atan(relation[0]) substitution = [ (sympy.sin(theta), opposite/hypotenuse), (sympy.cos(theta), adjacent/hypotenuse), (sympy.tan(theta), opposite/adjacent), (theta, inverse) ] return sympy.Piecewise( (_manualintegrate(substep).subs(substitution).trigsimp(), restriction) )
def test_angle_between(): a = Point(1, 2, 3, 4) b = a.orthogonal_direction o = a.origin assert feq(Line.angle_between(Line(Point(0, 0), Point(1, 1)), Line(Point(0, 0), Point(5, 0))).evalf(), pi.evalf() / 4) assert Line(a, o).angle_between(Line(b, o)) == pi / 2 assert Line3D.angle_between(Line3D(Point3D(0, 0, 0), Point3D(1, 1, 1)), Line3D(Point3D(0, 0, 0), Point3D(5, 0, 0))), acos(sqrt(3) / 3)
def find_substitutions(integrand, symbol, u_var): results = [] def test_subterm(u, u_diff): substituted = integrand / u_diff if symbol not in substituted.free_symbols: # replaced everything already return False substituted = substituted.subs(u, u_var).cancel() if symbol not in substituted.free_symbols: return substituted.as_independent(u_var, as_Add=False) return False def possible_subterms(term): if isinstance(term, (TrigonometricFunction, sympy.asin, sympy.acos, sympy.atan, sympy.exp, sympy.log, sympy.Heaviside)): return [term.args[0]] elif isinstance(term, sympy.Mul): r = [] for u in term.args: r.append(u) r.extend(possible_subterms(u)) return r elif isinstance(term, sympy.Pow): if term.args[1].is_constant(symbol): return [term.args[0]] elif term.args[0].is_constant(symbol): return [term.args[1]] elif isinstance(term, sympy.Add): r = [] for arg in term.args: r.append(arg) r.extend(possible_subterms(arg)) return r return [] for u in possible_subterms(integrand): if u == symbol: continue u_diff = manual_diff(u, symbol) new_integrand = test_subterm(u, u_diff) if new_integrand is not False: constant, new_integrand = new_integrand substitution = (u, constant, new_integrand) if substitution not in results: results.append(substitution) return results
def _parts_rule(integrand, symbol): # LIATE rule: # log, inverse trig, algebraic (polynomial), trigonometric, exponential def pull_out_polys(integrand): integrand = integrand.together() polys = [arg for arg in integrand.args if arg.is_polynomial(symbol)] if polys: u = sympy.Mul(*polys) dv = integrand / u return u, dv def pull_out_u(*functions): def pull_out_u_rl(integrand): if any([integrand.has(f) for f in functions]): args = [arg for arg in integrand.args if any(isinstance(arg, cls) for cls in functions)] if args: u = reduce(lambda a,b: a*b, args) dv = integrand / u return u, dv return pull_out_u_rl liate_rules = [pull_out_u(sympy.log), pull_out_u(sympy.atan, sympy.asin, sympy.acos), pull_out_polys, pull_out_u(sympy.sin, sympy.cos), pull_out_u(sympy.exp)] dummy = sympy.Dummy("temporary") # we can integrate log(x) and atan(x) by setting dv = 1 if isinstance(integrand, (sympy.log, sympy.atan, sympy.asin, sympy.acos)): integrand = dummy * integrand for index, rule in enumerate(liate_rules): result = rule(integrand) if result: u, dv = result # Don't pick u to be a constant if possible if symbol not in u.free_symbols and not u.has(dummy): return u = u.subs(dummy, 1) dv = dv.subs(dummy, 1) for rule in liate_rules[index + 1:]: r = rule(integrand) # make sure dv is amenable to integration if r and r[0].subs(dummy, 1) == dv: du = u.diff(symbol) v_step = integral_steps(dv, symbol) v = _manualintegrate(v_step) return u, dv, v, du, v_step
def find_substitutions(integrand, symbol, u_var): results = [] def test_subterm(u, u_diff): substituted = integrand / u_diff if symbol not in substituted.free_symbols: # replaced everything already return False substituted = substituted.subs(u, u_var).cancel() if symbol not in substituted.free_symbols: return substituted.as_independent(u_var, as_Add=False) return False def possible_subterms(term): if isinstance(term, (TrigonometricFunction, sympy.asin, sympy.acos, sympy.atan, sympy.exp, sympy.log, sympy.Heaviside)): return [term.args[0]] elif isinstance(term, sympy.Mul): r = [] for u in term.args: r.append(u) r.extend(possible_subterms(u)) return r elif isinstance(term, sympy.Pow): if term.args[1].is_constant(symbol): return [term.args[0]] elif term.args[0].is_constant(symbol): return [term.args[1]] elif isinstance(term, sympy.Add): r = [] for arg in term.args: r.append(arg) r.extend(possible_subterms(arg)) return r return [] for u in possible_subterms(integrand): if u == symbol: continue u_diff = manual_diff(u, symbol) new_integrand = test_subterm(u, u_diff) if new_integrand is not False: constant, new_integrand = new_integrand if new_integrand == integrand.subs(symbol, u_var): continue substitution = (u, constant, new_integrand) if substitution not in results: results.append(substitution) return results
def _parts_rule(integrand, symbol): # LIATE rule: # log, inverse trig, algebraic, trigonometric, exponential def pull_out_algebraic(integrand): integrand = integrand.cancel().together() algebraic = [arg for arg in integrand.args if arg.is_algebraic_expr(symbol)] if algebraic: u = sympy.Mul(*algebraic) dv = (integrand / u).cancel() if not u.is_polynomial() and isinstance(dv, sympy.exp): return return u, dv def pull_out_u(*functions): def pull_out_u_rl(integrand): if any([integrand.has(f) for f in functions]): args = [arg for arg in integrand.args if any(isinstance(arg, cls) for cls in functions)] if args: u = reduce(lambda a,b: a*b, args) dv = integrand / u return u, dv return pull_out_u_rl liate_rules = [pull_out_u(sympy.log), pull_out_u(sympy.atan, sympy.asin, sympy.acos), pull_out_algebraic, pull_out_u(sympy.sin, sympy.cos), pull_out_u(sympy.exp)] dummy = sympy.Dummy("temporary") # we can integrate log(x) and atan(x) by setting dv = 1 if isinstance(integrand, (sympy.log, sympy.atan, sympy.asin, sympy.acos)): integrand = dummy * integrand for index, rule in enumerate(liate_rules): result = rule(integrand) if result: u, dv = result # Don't pick u to be a constant if possible if symbol not in u.free_symbols and not u.has(dummy): return u = u.subs(dummy, 1) dv = dv.subs(dummy, 1) for rule in liate_rules[index + 1:]: r = rule(integrand) # make sure dv is amenable to integration if r and sympy.simplify(r[0].subs(dummy, 1)) == sympy.simplify(dv): du = u.diff(symbol) v_step = integral_steps(sympy.simplify(dv), symbol) v = _manualintegrate(v_step) return u, dv, v, du, v_step
