def __mul__(self, other):
if isinstance(other, RayTransferMatrix):
return RayTransferMatrix(Matrix.__mul__(self, other))
elif isinstance(other, GeometricRay):
return GeometricRay(Matrix.__mul__(self, other))
elif isinstance(other, BeamParameter):
temp = self * Matrix(((other.q, ), (1, )))
q = (temp[0] / temp[1]).expand(complex=True)
return BeamParameter(other.wavelen,
together(re(q)),
z_r=together(im(q)))
else:
return Matrix.__mul__(self, other)
def _rational_case(cls, poly, func):
"""Handle the rational function case. """
roots = symbols('r:%d' % poly.degree())
var, expr = func.variables[0], func.expr
f = sum(expr.subs(var, r) for r in roots)
p, q = together(f).as_numer_denom()
domain = QQ[roots]
p = p.expand()
q = q.expand()
try:
p = Poly(p, domain=domain, expand=False)
except GeneratorsNeeded:
p, p_coeff = None, (p, )
else:
p_monom, p_coeff = zip(*p.terms())
try:
q = Poly(q, domain=domain, expand=False)
except GeneratorsNeeded:
q, q_coeff = None, (q, )
else:
q_monom, q_coeff = zip(*q.terms())
coeffs, mapping = symmetrize(p_coeff + q_coeff, formal=True)
formulas, values = viete(poly, roots), []
for (sym, _), (_, val) in zip(mapping, formulas):
values.append((sym, val))
for i, (coeff, _) in enumerate(coeffs):
coeffs[i] = coeff.subs(values)
n = len(p_coeff)
p_coeff = coeffs[:n]
q_coeff = coeffs[n:]
if p is not None:
p = Poly(dict(zip(p_monom, p_coeff)), *p.gens).as_expr()
else:
(p, ) = p_coeff
if q is not None:
q = Poly(dict(zip(q_monom, q_coeff)), *q.gens).as_expr()
else:
(q, ) = q_coeff
return factor(p / q)
def _rational_case(cls, poly, func):
"""Handle the rational function case. """
roots = symbols('r:%d' % poly.degree())
var, expr = func.variables[0], func.expr
f = sum(expr.subs(var, r) for r in roots)
p, q = together(f).as_numer_denom()
domain = QQ[roots]
p = p.expand()
q = q.expand()
try:
p = Poly(p, domain=domain, expand=False)
except GeneratorsNeeded:
p, p_coeff = None, (p,)
else:
p_monom, p_coeff = zip(*p.terms())
try:
q = Poly(q, domain=domain, expand=False)
except GeneratorsNeeded:
q, q_coeff = None, (q,)
else:
q_monom, q_coeff = zip(*q.terms())
coeffs, mapping = symmetrize(p_coeff + q_coeff, formal=True)
formulas, values = viete(poly, roots), []
for (sym, _), (_, val) in zip(mapping, formulas):
values.append((sym, val))
for i, (coeff, _) in enumerate(coeffs):
coeffs[i] = coeff.subs(values)
n = len(p_coeff)
p_coeff = coeffs[:n]
q_coeff = coeffs[n:]
if p is not None:
p = Poly(dict(zip(p_monom, p_coeff)), *p.gens).as_expr()
else:
(p,) = p_coeff
if q is not None:
q = Poly(dict(zip(q_monom, q_coeff)), *q.gens).as_expr()
else:
(q,) = q_coeff
return factor(p/q)
def test_apart_extension():
f = 2 / (x**2 + 1)
g = I / (x + I) - I / (x - I)
assert apart(f, extension=I) == g
assert apart(f, gaussian=True) == g
f = x / ((x - 2) * (x + I))
assert factor(together(apart(f)).expand()) == f
f, g = _make_extension_example()
# XXX: Only works with dotprodsimp. See test_apart_extension_xfail below
from sympy.matrices import dotprodsimp
with dotprodsimp(True):
assert apart(f, x, extension={sqrt(2)}) == g
def _quintic_simplify(expr):
expr = powsimp(expr)
expr = cancel(expr)
return together(expr)
def test_together():
assert together(0) == 0
assert together(1) == 1
assert together(x * y * z) == x * y * z
assert together(x + y) == x + y
assert together(1 / x) == 1 / x
assert together(1 / x + 1) == (x + 1) / x
assert together(1 / x + 3) == (3 * x + 1) / x
assert together(1 / x + x) == (x ** 2 + 1) / x
assert together(1 / x + S.Half) == (x + 2) / (2 * x)
assert together(S.Half + x / 2) == Mul(S.Half, x + 1, evaluate=False)
assert together(1 / x + 2 / y) == (2 * x + y) / (y * x)
assert together(1 / (1 + 1 / x)) == x / (1 + x)
assert together(x / (1 + 1 / x)) == x ** 2 / (1 + x)
assert together(1 / x + 1 / y + 1 / z) == (x * y + x * z + y * z) / (x * y * z)
assert together(1 / (1 + x + 1 / y + 1 / z)) == y * z / (y + z + y * z + x * y * z)
assert together(1 / (x * y) + 1 / (x * y) ** 2) == y ** (-2) * x ** (-2) * (
1 + x * y
)
assert together(1 / (x * y) + 1 / (x * y) ** 4) == y ** (-4) * x ** (-4) * (
1 + x ** 3 * y ** 3
)
assert together(1 / (x ** 7 * y) + 1 / (x * y) ** 4) == y ** (-4) * x ** (-7) * (
x ** 3 + y ** 3
)
assert together(5 / (2 + 6 / (3 + 7 / (4 + 8 / (5 + 9 / x))))) == Rational(5, 2) * (
(171 + 119 * x) / (279 + 203 * x)
)
assert together(1 + 1 / (x + 1) ** 2) == (1 + (x + 1) ** 2) / (x + 1) ** 2
assert together(1 + 1 / (x * (1 + x))) == (1 + x * (1 + x)) / (x * (1 + x))
assert together(1 / (x * (x + 1)) + 1 / (x * (x + 2))) == (3 + 2 * x) / (
x * (1 + x) * (2 + x)
)
assert together(1 + 1 / (2 * x + 2) ** 2) == (4 * (x + 1) ** 2 + 1) / (
4 * (x + 1) ** 2
)
assert together(sin(1 / x + 1 / y)) == sin(1 / x + 1 / y)
assert together(sin(1 / x + 1 / y), deep=True) == sin((x + y) / (x * y))
assert together(1 / exp(x) + 1 / (x * exp(x))) == (1 + x) / (x * exp(x))
assert together(1 / exp(2 * x) + 1 / (x * exp(3 * x))) == (1 + exp(x) * x) / (
x * exp(3 * x)
)
assert together(Integral(1 / x + 1 / y, x)) == Integral((x + y) / (x * y), x)
assert together(Eq(1 / x + 1 / y, 1 + 1 / z)) == Eq((x + y) / (x * y), (z + 1) / z)
assert together((A * B) ** -1 + (B * A) ** -1) == (A * B) ** -1 + (B * A) ** -1
def test_together():
assert together(0) == 0
assert together(1) == 1
assert together(x*y*z) == x*y*z
assert together(x + y) == x + y
assert together(1/x) == 1/x
assert together(1/x + 1) == (x + 1)/x
assert together(1/x + 3) == (3*x + 1)/x
assert together(1/x + x) == (x**2 + 1)/x
assert together(1/x + Rational(1, 2)) == (x + 2)/(2*x)
assert together(Rational(1, 2) + x/2) == Mul(S.Half, x + 1, evaluate=False)
assert together(1/x + 2/y) == (2*x + y)/(y*x)
assert together(1/(1 + 1/x)) == x/(1 + x)
assert together(x/(1 + 1/x)) == x**2/(1 + x)
assert together(1/x + 1/y + 1/z) == (x*y + x*z + y*z)/(x*y*z)
assert together(1/(1 + x + 1/y + 1/z)) == y*z/(y + z + y*z + x*y*z)
assert together(1/(x*y) + 1/(x*y)**2) == y**(-2)*x**(-2)*(1 + x*y)
assert together(1/(x*y) + 1/(x*y)**4) == y**(-4)*x**(-4)*(1 + x**3*y**3)
assert together(1/(x**7*y) + 1/(x*y)**4) == y**(-4)*x**(-7)*(x**3 + y**3)
assert together(5/(2 + 6/(3 + 7/(4 + 8/(5 + 9/x))))) == \
(S(5)/2)*((171 + 119*x)/(279 + 203*x))
assert together(1 + 1/(x + 1)**2) == (1 + (x + 1)**2)/(x + 1)**2
assert together(1 + 1/(x*(1 + x))) == (1 + x*(1 + x))/(x*(1 + x))
assert together(1/(x*(x + 1)) + 1/(x*(x + 2))) == (3 + 2*x)/(x*(1 + x)*(2 + x))
assert together(1 + 1/(2*x + 2)**2) == (4*(x + 1)**2 + 1)/(4*(x + 1)**2)
assert together(sin(1/x + 1/y)) == sin(1/x + 1/y)
assert together(sin(1/x + 1/y), deep=True) == sin((x + y)/(x*y))
assert together(1/exp(x) + 1/(x*exp(x))) == (1+x)/(x*exp(x))
assert together(1/exp(2*x) + 1/(x*exp(3*x))) == (1+exp(x)*x)/(x*exp(3*x))
assert together(Integral(1/x + 1/y, x)) == Integral((x + y)/(x*y), x)
assert together(Eq(1/x + 1/y, 1 + 1/z)) == Eq((x + y)/(x*y), (z + 1)/z)
assert together((A*B)**-1 + (B*A)**-1) == (A*B)**-1 + (B*A)**-1
def continuous_domain(f, symbol, domain):
"""
Returns the intervals in the given domain for which the function
is continuous.
This method is limited by the ability to determine the various
singularities and discontinuities of the given function.
Parameters
==========
f : Expr
The concerned function.
symbol : Symbol
The variable for which the intervals are to be determined.
domain : Interval
The domain over which the continuity of the symbol has to be checked.
Examples
========
>>> from sympy import Symbol, S, tan, log, pi, sqrt
>>> from sympy.sets import Interval
>>> from sympy.calculus.util import continuous_domain
>>> x = Symbol('x')
>>> continuous_domain(1/x, x, S.Reals)
Union(Interval.open(-oo, 0), Interval.open(0, oo))
>>> continuous_domain(tan(x), x, Interval(0, pi))
Union(Interval.Ropen(0, pi/2), Interval.Lopen(pi/2, pi))
>>> continuous_domain(sqrt(x - 2), x, Interval(-5, 5))
Interval(2, 5)
>>> continuous_domain(log(2*x - 1), x, S.Reals)
Interval.open(1/2, oo)
Returns
=======
Interval
Union of all intervals where the function is continuous.
Raises
======
NotImplementedError
If the method to determine continuity of such a function
has not yet been developed.
"""
from sympy.solvers.inequalities import solve_univariate_inequality
from sympy.solvers.solveset import solveset, _has_rational_power
if domain.is_subset(S.Reals):
constrained_interval = domain
for atom in f.atoms(Pow):
predicate, denomin = _has_rational_power(atom, symbol)
constraint = S.EmptySet
if predicate and denomin == 2:
constraint = solve_univariate_inequality(atom.base >= 0,
symbol).as_set()
constrained_interval = Intersection(constraint,
constrained_interval)
for atom in f.atoms(log):
constraint = solve_univariate_inequality(atom.args[0] > 0,
symbol).as_set()
constrained_interval = Intersection(constraint,
constrained_interval)
domain = constrained_interval
try:
sings = S.EmptySet
if f.has(Abs):
sings = solveset(1/f, symbol, domain) + \
solveset(denom(together(f)), symbol, domain)
else:
for atom in f.atoms(Pow):
predicate, denomin = _has_rational_power(atom, symbol)
if predicate and denomin == 2:
sings = solveset(1/f, symbol, domain) +\
solveset(denom(together(f)), symbol, domain)
break
else:
sings = Intersection(solveset(1/f, symbol), domain) + \
solveset(denom(together(f)), symbol, domain)
except NotImplementedError:
import sys
raise (NotImplementedError("Methods for determining the continuous domains"
" of this function have not been developed."),
None,
sys.exc_info()[2])
return domain - sings
def _quintic_simplify(expr):
from sympy.simplify.simplify import powsimp
expr = powsimp(expr)
expr = cancel(expr)
return together(expr)
def continuous_domain(f, symbol, domain):
"""
Returns the intervals in the given domain for which the function
is continuous.
This method is limited by the ability to determine the various
singularities and discontinuities of the given function.
Examples
========
>>> from sympy import Symbol, S, tan, log, pi, sqrt
>>> from sympy.sets import Interval
>>> from sympy.calculus.util import continuous_domain
>>> x = Symbol('x')
>>> continuous_domain(1/x, x, S.Reals)
Union(Interval.open(-oo, 0), Interval.open(0, oo))
>>> continuous_domain(tan(x), x, Interval(0, pi))
Union(Interval.Ropen(0, pi/2), Interval.Lopen(pi/2, pi))
>>> continuous_domain(sqrt(x - 2), x, Interval(-5, 5))
Interval(2, 5)
>>> continuous_domain(log(2*x - 1), x, S.Reals)
Interval.open(1/2, oo)
"""
from sympy.solvers.inequalities import solve_univariate_inequality
from sympy.solvers.solveset import solveset, _has_rational_power
if domain.is_subset(S.Reals):
constrained_interval = domain
for atom in f.atoms(Pow):
predicate, denomin = _has_rational_power(atom, symbol)
constraint = S.EmptySet
if predicate and denomin == 2:
constraint = solve_univariate_inequality(
atom.base >= 0, symbol).as_set()
constrained_interval = Intersection(constraint,
constrained_interval)
for atom in f.atoms(log):
constraint = solve_univariate_inequality(atom.args[0] > 0,
symbol).as_set()
constrained_interval = Intersection(constraint,
constrained_interval)
domain = constrained_interval
try:
sings = S.EmptySet
if f.has(Abs):
sings = solveset(1/f, symbol, domain) + \
solveset(denom(together(f)), symbol, domain)
else:
for atom in f.atoms(Pow):
predicate, denomin = _has_rational_power(atom, symbol)
if predicate and denomin == 2:
sings = solveset(1/f, symbol, domain) +\
solveset(denom(together(f)), symbol, domain)
break
else:
sings = Intersection(solveset(1/f, symbol), domain) + \
solveset(denom(together(f)), symbol, domain)
except NotImplementedError:
raise NotImplementedError(
"Methods for determining the continuous domains"
" of this function have not been developed.")
return domain - sings
def test_Limits_simple_3a():
a = Symbol('a')
#issue 3513
assert together(limit((x**2 - (a + 1)*x + a)/(x**3 - a**3), x, a)) == \
(a - 1)/(3*a**2) # 196