def test_fps__inverse():
f1, f2, f3 = fps(sin(x)), fps(exp(x)), fps(cos(x))
raises(ValueError, lambda: f1.inverse(x))
finv = f2.inverse(x)
assert isinstance(finv, FormalPowerSeriesInverse)
assert isinstance(finv.ffps, FormalPowerSeries)
raises(ValueError, lambda: finv.gfps)
assert finv.f == exp(x)
assert finv.function == exp(-x)
assert finv._eval_terms(5) == 1 - x + x**2/2 - x**3/6 + x**4/24
assert finv.truncate() == 1 - x + x**2/2 - x**3/6 + x**4/24 - x**5/120 + O(x**6)
assert finv.truncate(5) == 1 - x + x**2/2 - x**3/6 + x**4/24 + O(x**5)
raises(NotImplementedError, lambda: finv._eval_term(5))
raises(ValueError, lambda: finv.g)
raises(NotImplementedError, lambda: finv.infinite)
raises(NotImplementedError, lambda: finv._eval_derivative(x))
raises(NotImplementedError, lambda: finv.integrate(x))
assert f2.inverse(x).truncate(8) == \
1 - x + x**2/2 - x**3/6 + x**4/24 - x**5/120 + x**6/720 - x**7/5040 + O(x**8)
assert f3.inverse(x).truncate() == 1 + x**2/2 + 5*x**4/24 + O(x**6)
assert f3.inverse(x).truncate(8) == 1 + x**2/2 + 5*x**4/24 + 61*x**6/720 + O(x**8)
def test_fps():
assert fps(1) == 1
assert fps(2, x) == 2
assert fps(2, x, dir='+') == 2
assert fps(2, x, dir='-') == 2
assert fps(x**2 + x + 1) == x**2 + x + 1
assert fps(1 / x + 1 / x**2) == 1 / x + 1 / x**2
assert fps(log(1 + x), hyper=False, rational=False) == log(1 + x)
f = fps(log(1 + x))
assert isinstance(f, FormalPowerSeries)
assert f.function == log(1 + x)
assert f.subs(x, y) == f
assert f[:5] == [0, x, -x**2 / 2, x**3 / 3, -x**4 / 4]
assert f.as_leading_term(x) == x
assert f.polynomial(6) == x - x**2 / 2 + x**3 / 3 - x**4 / 4 + x**5 / 5
k = f.ak.variables[0]
assert f.infinite == Sum((-(-1)**(-k) * x**k) / k, (k, 1, oo))
ft, s = f.truncate(n=None), f[:5]
for i, t in enumerate(ft):
if i == 5:
break
assert s[i] == t
f = sin(x).fps(x)
assert isinstance(f, FormalPowerSeries)
assert f.truncate() == x - x**3 / 6 + x**5 / 120 + O(x**6)
raises(NotImplementedError, lambda: fps(y * x))
raises(ValueError, lambda: fps(x, dir=0))
def test_fps():
assert fps(1) == 1
assert fps(2, x) == 2
assert fps(2, x, dir="+") == 2
assert fps(2, x, dir="-") == 2
assert fps(x ** 2 + x + 1) == x ** 2 + x + 1
assert fps(1 / x + 1 / x ** 2) == 1 / x + 1 / x ** 2
assert fps(log(1 + x), hyper=False, rational=False) == log(1 + x)
f = fps(log(1 + x))
assert isinstance(f, FormalPowerSeries)
assert f.function == log(1 + x)
assert f.subs(x, y) == f
assert f[:5] == [0, x, -x ** 2 / 2, x ** 3 / 3, -x ** 4 / 4]
assert f.as_leading_term(x) == x
assert f.polynomial(6) == x - x ** 2 / 2 + x ** 3 / 3 - x ** 4 / 4 + x ** 5 / 5
k = f.ak.variables[0]
assert f.infinite == Sum((-(-1) ** (-k) * x ** k) / k, (k, 1, oo))
ft, s = f.truncate(n=None), f[:5]
for i, t in enumerate(ft):
if i == 5:
break
assert s[i] == t
f = sin(x).fps(x)
assert isinstance(f, FormalPowerSeries)
assert f.truncate() == x - x ** 3 / 6 + x ** 5 / 120 + O(x ** 6)
raises(NotImplementedError, lambda: fps(y * x))
raises(ValueError, lambda: fps(x, dir=0))
def test_fps__fractional():
f = sin(sqrt(x)) / x
assert fps(f, x).truncate() == \
(1/sqrt(x) - sqrt(x)/6 + x**Rational(3, 2)/120 -
x**Rational(5, 2)/5040 + x**Rational(7, 2)/362880 -
x**Rational(9, 2)/39916800 + x**Rational(11, 2)/6227020800 + O(x**6))
f = sin(sqrt(x)) * x
assert fps(f, x).truncate() == \
(x**Rational(3, 2) - x**Rational(5, 2)/6 + x**Rational(7, 2)/120 -
x**Rational(9, 2)/5040 + x**Rational(11, 2)/362880 + O(x**6))
f = atan(sqrt(x)) / x**2
assert fps(f, x).truncate() == \
(x**Rational(-3, 2) - x**Rational(-1, 2)/3 + x**Rational(1, 2)/5 -
x**Rational(3, 2)/7 + x**Rational(5, 2)/9 - x**Rational(7, 2)/11 +
x**Rational(9, 2)/13 - x**Rational(11, 2)/15 + O(x**6))
f = exp(sqrt(x))
assert fps(f, x).truncate().expand() == \
(1 + x/2 + x**2/24 + x**3/720 + x**4/40320 + x**5/3628800 + sqrt(x) +
x**Rational(3, 2)/6 + x**Rational(5, 2)/120 + x**Rational(7, 2)/5040 +
x**Rational(9, 2)/362880 + x**Rational(11, 2)/39916800 + O(x**6))
f = exp(sqrt(x))*x
assert fps(f, x).truncate().expand() == \
(x + x**2/2 + x**3/24 + x**4/720 + x**5/40320 + x**Rational(3, 2) +
x**Rational(5, 2)/6 + x**Rational(7, 2)/120 + x**Rational(9, 2)/5040 +
x**Rational(11, 2)/362880 + O(x**6))
def test_fps__fractional():
f = sin(sqrt(x)) / x
assert fps(f, x).truncate() == \
(1/sqrt(x) - sqrt(x)/6 + x**Rational(3, 2)/120 -
x**Rational(5, 2)/5040 + x**Rational(7, 2)/362880 -
x**Rational(9, 2)/39916800 + x**Rational(11, 2)/6227020800 + O(x**6))
f = sin(sqrt(x)) * x
assert fps(f, x).truncate() == \
(x**Rational(3, 2) - x**Rational(5, 2)/6 + x**Rational(7, 2)/120 -
x**Rational(9, 2)/5040 + x**Rational(11, 2)/362880 + O(x**6))
f = atan(sqrt(x)) / x**2
assert fps(f, x).truncate() == \
(x**Rational(-3, 2) - x**Rational(-1, 2)/3 + x**Rational(1, 2)/5 -
x**Rational(3, 2)/7 + x**Rational(5, 2)/9 - x**Rational(7, 2)/11 +
x**Rational(9, 2)/13 - x**Rational(11, 2)/15 + O(x**6))
f = exp(sqrt(x))
assert fps(f, x).truncate().expand() == \
(1 + x/2 + x**2/24 + x**3/720 + x**4/40320 + x**5/3628800 + sqrt(x) +
x**Rational(3, 2)/6 + x**Rational(5, 2)/120 + x**Rational(7, 2)/5040 +
x**Rational(9, 2)/362880 + x**Rational(11, 2)/39916800 + O(x**6))
f = exp(sqrt(x)) * x
assert fps(f, x).truncate().expand() == \
(x + x**2/2 + x**3/24 + x**4/720 + x**5/40320 + x**Rational(3, 2) +
x**Rational(5, 2)/6 + x**Rational(7, 2)/120 + x**Rational(9, 2)/5040 +
x**Rational(11, 2)/362880 + O(x**6))
def test_fps__logarithmic_singularity():
f = log(1 + 1 / x)
assert fps(
f,
x) != -log(x) + x - x**2 / 2 + x**3 / 3 - x**4 / 4 + x**5 / 5 + O(x**6)
assert fps(
f, x, rational=False
) != -log(x) + x - x**2 / 2 + x**3 / 3 - x**4 / 4 + x**5 / 5 + O(x**6)
def test_fps__Add_expr():
f = x * atan(x) - log(1 + x ** 2) / 2
assert fps(f, x).truncate() == x ** 2 / 2 - x ** 4 / 12 + O(x ** 6)
f = sin(x) + cos(x) - exp(x) + log(1 + x)
assert fps(f, x).truncate() == x - 3 * x ** 2 / 2 - x ** 4 / 4 + x ** 5 / 5 + O(x ** 6)
f = 1 / x + sin(x)
assert fps(f, x).truncate() == 1 / x + x - x ** 3 / 6 + x ** 5 / 120 + O(x ** 6)
f = sin(x) - cos(x) + 1 / (x - 1)
assert fps(f, x).truncate() == -2 - x ** 2 / 2 - 7 * x ** 3 / 6 - 25 * x ** 4 / 24 - 119 * x ** 5 / 120 + O(x ** 6)
def test_fps__Add_expr():
f = x*atan(x) - log(1 + x**2) / 2
assert fps(f, x).truncate() == x**2/2 - x**4/12 + O(x**6)
f = sin(x) + cos(x) - exp(x) + log(1 + x)
assert fps(f, x).truncate() == x - 3*x**2/2 - x**4/4 + x**5/5 + O(x**6)
f = 1/x + sin(x)
assert fps(f, x).truncate() == 1/x + x - x**3/6 + x**5/120 + O(x**6)
f = sin(x) - cos(x) + 1/(x - 1)
assert fps(f, x).truncate() == \
-2 - x**2/2 - 7*x**3/6 - 25*x**4/24 - 119*x**5/120 + O(x**6)
def test_fps_shift():
f = x ** -5 * sin(x)
assert fps(f, x).truncate() == 1 / x ** 4 - 1 / (6 * x ** 2) + S.One / 120 - x ** 2 / 5040 + x ** 4 / 362880 + O(
x ** 6
)
f = x ** 2 * atan(x)
assert fps(f, x, rational=False).truncate() == x ** 3 - x ** 5 / 3 + O(x ** 6)
f = cos(sqrt(x)) * x
assert fps(f, x).truncate() == x - x ** 2 / 2 + x ** 3 / 24 - x ** 4 / 720 + x ** 5 / 40320 + O(x ** 6)
f = x ** 2 * cos(sqrt(x))
assert fps(f, x).truncate() == x ** 2 - x ** 3 / 2 + x ** 4 / 24 - x ** 5 / 720 + O(x ** 6)
def test_fps__inverse():
f1, f2, f3 = fps(sin(x)), fps(exp(x)), fps(cos(x))
raises(ValueError, lambda: f1.inverse(x))
raises(ValueError, lambda: f1.inverse(x, n=8))
assert f2.inverse(
x) == 1 - x + x**2 / 2 - x**3 / 6 + x**4 / 24 - x**5 / 120 + O(x**6)
assert f2.inverse(x, n=8) == \
1 - x + x**2/2 - x**3/6 + x**4/24 - x**5/120 + x**6/720 - x**7/5040 + O(x**8)
assert f3.inverse(x) == 1 + x**2 / 2 + 5 * x**4 / 24 + O(x**6)
assert f3.inverse(
x, n=8) == 1 + x**2 / 2 + 5 * x**4 / 24 + 61 * x**6 / 720 + O(x**8)
def test_fps_shift():
f = x**-5 * sin(x)
assert fps(f, x).truncate() == 1 / x**4 - 1 / (6 * x**2) + Rational(
1, 120) - x**2 / 5040 + x**4 / 362880 + O(x**6)
f = x**2 * atan(x)
assert fps(f, x, rational=False).truncate() == x**3 - x**5 / 3 + O(x**6)
f = cos(sqrt(x)) * x
assert fps(f, x).truncate(
) == x - x**2 / 2 + x**3 / 24 - x**4 / 720 + x**5 / 40320 + O(x**6)
f = x**2 * cos(sqrt(x))
assert fps(
f, x).truncate() == x**2 - x**3 / 2 + x**4 / 24 - x**5 / 720 + O(x**6)
def plotAM(I):
xs = []
ys = []
x = Symbol('x')
for j in np.arange(-15, 15, 0.1):
xs.append(x)
series = fps(myfx(x, I)).truncate(3)
print(series)
def test_fps__product():
f1, f2, f3 = fps(sin(x)), fps(exp(x)), fps(cos(x))
raises(ValueError, lambda: f1.product(exp(x), x))
raises(ValueError, lambda: f1.product(fps(exp(x), dir=-1), x, 4))
raises(ValueError, lambda: f1.product(fps(exp(x), x0=1), x, 4))
raises(ValueError, lambda: f1.product(fps(exp(y)), x, 4))
fprod = f1.product(f2, x)
assert isinstance(fprod, FormalPowerSeriesProduct)
assert isinstance(fprod.ffps, FormalPowerSeries)
assert isinstance(fprod.gfps, FormalPowerSeries)
assert fprod.f == sin(x)
assert fprod.g == exp(x)
assert fprod.function == sin(x) * exp(x)
assert fprod._eval_terms(4) == x + x**2 + x**3/3
assert fprod.truncate(4) == x + x**2 + x**3/3 + O(x**4)
assert fprod.polynomial(4) == x + x**2 + x**3/3
raises(NotImplementedError, lambda: fprod._eval_term(5))
raises(NotImplementedError, lambda: fprod.infinite)
raises(NotImplementedError, lambda: fprod._eval_derivative(x))
raises(NotImplementedError, lambda: fprod.integrate(x))
assert f1.product(f3, x)._eval_terms(4) == x - 2*x**3/3
assert f1.product(f3, x).truncate(4) == x - 2*x**3/3 + O(x**4)
def test_fps__symbolic():
f = x**n*sin(x**2)
assert fps(f, x).truncate(8) == x**2*x**n - x**6*x**n/6 + O(x**(n + 8), x)
f = x**(n - 2)*cos(x)
assert fps(f, x).truncate() == \
(x**n*(-S(1)/2 + x**(-2)) + x**2*x**n/24 - x**4*x**n/720 +
O(x**(n + 6), x))
f = x**n*log(1 + x)
fp = fps(f, x)
k = fp.ak.variables[0]
assert fp.infinite == \
Sum((-(-1)**(-k)*x**k*x**n)/k, (k, 1, oo))
f = x**(n - 2)*sin(x) + x**n*exp(x)
assert fps(f, x).truncate() == \
(x**n*(1 + 1/x) + 5*x*x**n/6 + x**2*x**n/2 + 7*x**3*x**n/40 +
x**4*x**n/24 + 41*x**5*x**n/5040 + O(x**(n + 6), x))
f = (x - 2)**n*log(1 + x)
assert fps(f, x, 2).truncate() == \
((x - 2)**n*log(3) - (x - 2)**2*(x - 2)**n/18 +
(x - 2)**3*(x - 2)**n/81 - (x - 2)**4*(x - 2)**n/324 +
(x - 2)**5*(x - 2)**n/1215 + (x/3 - S(2)/3)*(x - 2)**n +
O((x - 2)**(n + 6), (x, 2)))
f = x**n*atan(x)
assert fps(f, x, oo).truncate() == \
(-x**n/(5*x**5) + x**n/(3*x**3) + x**n*(pi/2 - 1/x) +
O(x**(n - 6), (x, oo)))
def test_fps__symbolic():
f = x**n * sin(x**2)
assert fps(
f, x).truncate(8) == x**2 * x**n - x**6 * x**n / 6 + O(x**(n + 8), x)
f = x**(n - 2) * cos(x)
assert fps(f, x).truncate() == \
(x**n*(-S(1)/2 + x**(-2)) + x**2*x**n/24 - x**4*x**n/720 +
O(x**(n + 6), x))
f = x**n * log(1 + x)
fp = fps(f, x)
k = fp.ak.variables[0]
assert fp.infinite == \
Sum((-(-1)**(-k)*x**k*x**n)/k, (k, 1, oo))
f = x**(n - 2) * sin(x) + x**n * exp(x)
assert fps(f, x).truncate() == \
(x**n*(1 + 1/x) + 5*x*x**n/6 + x**2*x**n/2 + 7*x**3*x**n/40 +
x**4*x**n/24 + 41*x**5*x**n/5040 + O(x**(n + 6), x))
f = (x - 2)**n * log(1 + x)
assert fps(f, x, 2).truncate() == \
((x - 2)**n*log(3) - (x - 2)**2*(x - 2)**n/18 +
(x - 2)**3*(x - 2)**n/81 - (x - 2)**4*(x - 2)**n/324 +
(x - 2)**5*(x - 2)**n/1215 + (x/3 - S(2)/3)*(x - 2)**n +
O((x - 2)**(n + 6), (x, 2)))
f = x**n * atan(x)
assert fps(f, x, oo).truncate() == \
(-x**n/(5*x**5) + x**n/(3*x**3) + x**n*(pi/2 - 1/x) +
O(x**(n - 6), (x, oo)))
def test_fps__convolution():
f1, f2, f3 = fps(sin(x)), fps(exp(x)), fps(cos(x))
raises(ValueError, lambda: f1.product(exp(x), x))
raises(ValueError, lambda: f1.product(fps(exp(x), dir=-1), x, 4))
raises(ValueError, lambda: f1.product(fps(exp(x), x0=1), x, 4))
raises(ValueError, lambda: f1.product(fps(exp(y)), x, 4))
assert f1.product(f2, x, 3) == x + x**2 + O(x**3)
assert f1.product(f2, x, 4) == x + x**2 + x**3 / 3 + O(x**4)
assert f1.product(f3, x, 4) == x - 2 * x**3 / 3 + O(x**4)
def test_fps_symbolic():
f = x**n * sin(x**2)
assert fps(
f, x).truncate(8) == x**(n + 2) - x**(n + 6) / 6 + O(x**(n + 8), x)
f = x**n * log(1 + x)
fp = fps(f, x)
k = fp.ak.variables[0]
assert fp.infinite == Sum((-((-1)**(-k)) * x**(k + n)) / k, (k, 1, oo))
f = (x - 2)**n * log(1 + x)
assert fps(
f, x,
2).truncate() == ((x - 2)**n * log(3) + (x - 2)**(n + 1) / 3 -
(x - 2)**(n + 2) / 18 + (x - 2)**(n + 3) / 81 -
(x - 2)**(n + 4) / 324 + (x - 2)**(n + 5) / 1215 + O(
(x - 2)**(n + 6), (x, 2)))
f = x**(n - 2) * cos(x)
assert fps(f, x).truncate() == (x**(n - 2) - x**n / 2 + x**(n + 2) / 24 -
x**(n + 4) / 720 + O(x**(n + 6), x))
f = x**(n - 2) * sin(x) + x**n * exp(x)
assert fps(f, x).truncate() == (x**(n - 1) + x**n + 5 * x**(n + 1) / 6 +
x**(n + 2) / 2 + 7 * x**(n + 3) / 40 +
x**(n + 4) / 24 + 41 * x**(n + 5) / 5040 +
O(x**(n + 6), x))
f = x**n * atan(x)
assert fps(f, x, oo).truncate() == (-(x**(n - 5)) / 5 + x**(n - 3) / 3 +
x**n * (pi / 2 - 1 / x) + O(
(1 / x)**(-n) / x**6, (x, oo)))
f = x**(n / 2) * cos(x)
assert fps(
f,
x).truncate() == x**(n /
2) - x**(n / 2 + 2) / 2 + x**(n / 2 + 4) / 24 + O(
x**(n / 2 + 6), x)
f = x**(n + m) * sin(x)
assert fps(f, x).truncate() == x**(
m + n + 1) - x**(m + n + 3) / 6 + x**(m + n + 5) / 120 + O(
x**(m + n + 6), x)
def test_fps__asymptotic():
f = exp(x)
assert fps(f, x, oo) == f
assert fps(f, x, -oo).truncate() == O(1 / x**6, (x, oo))
f = erf(x)
assert fps(f, x, oo).truncate() == 1 + O(1 / x**6, (x, oo))
assert fps(f, x, -oo).truncate() == -1 + O(1 / x**6, (x, oo))
f = atan(x)
assert fps(f, x, oo, full=True).truncate() == \
-1/(5*x**5) + 1/(3*x**3) - 1/x + pi/2 + O(1/x**6, (x, oo))
assert fps(f, x, -oo, full=True).truncate() == \
-1/(5*x**5) + 1/(3*x**3) - 1/x - pi/2 + O(1/x**6, (x, oo))
f = log(1 + x)
assert fps(f, x, oo) != \
(-1/(5*x**5) - 1/(4*x**4) + 1/(3*x**3) - 1/(2*x**2) + 1/x - log(1/x) +
O(1/x**6, (x, oo)))
assert fps(f, x, -oo) != \
(-1/(5*x**5) - 1/(4*x**4) + 1/(3*x**3) - 1/(2*x**2) + 1/x + I*pi -
log(-1/x) + O(1/x**6, (x, oo)))
def test_fps__asymptotic():
f = exp(x)
assert fps(f, x, oo) == f
assert fps(f, x, -oo).truncate() == O(1/x**6, (x, oo))
f = erf(x)
assert fps(f, x, oo).truncate() == 1 + O(1/x**6, (x, oo))
assert fps(f, x, -oo).truncate() == -1 + O(1/x**6, (x, oo))
f = atan(x)
assert fps(f, x, oo, full=True).truncate() == \
-1/(5*x**5) + 1/(3*x**3) - 1/x + pi/2 + O(1/x**6, (x, oo))
assert fps(f, x, -oo, full=True).truncate() == \
-1/(5*x**5) + 1/(3*x**3) - 1/x - pi/2 + O(1/x**6, (x, oo))
f = log(1 + x)
assert fps(f, x, oo) != \
(-1/(5*x**5) - 1/(4*x**4) + 1/(3*x**3) - 1/(2*x**2) + 1/x - log(1/x) +
O(1/x**6, (x, oo)))
assert fps(f, x, -oo) != \
(-1/(5*x**5) - 1/(4*x**4) + 1/(3*x**3) - 1/(2*x**2) + 1/x + I*pi -
log(-1/x) + O(1/x**6, (x, oo)))
def test_fps__composition():
f1, f2, f3 = fps(exp(x)), fps(sin(x)), fps(cos(x))
raises(ValueError, lambda: f1.compose(sin(x), x))
raises(ValueError, lambda: f1.compose(fps(sin(x), dir=-1), x, 4))
raises(ValueError, lambda: f1.compose(fps(sin(x), x0=1), x, 4))
raises(ValueError, lambda: f1.compose(fps(sin(y)), x, 4))
raises(ValueError, lambda: f1.compose(f3, x))
raises(ValueError, lambda: f2.compose(f3, x))
assert f1.compose(f2,
x) == 1 + x + x**2 / 2 - x**4 / 8 - x**5 / 15 + O(x**6)
assert f1.compose(f2, x, n=4) == 1 + x + x**2 / 2 + O(x**4)
assert f1.compose(f2, x, n=8) == \
1 + x + x**2/2 - x**4/8 - x**5/15 - x**6/240 + x**7/90 + O(x**8)
assert f2.compose(f2, x, n=4) == x - x**3 / 3 + O(x**4)
assert f2.compose(
f2, x, n=8) == x - x**3 / 3 + x**5 / 10 - 8 * x**7 / 315 + O(x**8)
def test_fps__compose():
f1, f2, f3 = fps(exp(x)), fps(sin(x)), fps(cos(x))
raises(ValueError, lambda: f1.compose(sin(x), x))
raises(ValueError, lambda: f1.compose(fps(sin(x), dir=-1), x, 4))
raises(ValueError, lambda: f1.compose(fps(sin(x), x0=1), x, 4))
raises(ValueError, lambda: f1.compose(fps(sin(y)), x, 4))
raises(ValueError, lambda: f1.compose(f3, x))
raises(ValueError, lambda: f2.compose(f3, x))
fcomp = f1.compose(f2, x)
assert isinstance(fcomp, FormalPowerSeriesCompose)
assert isinstance(fcomp.ffps, FormalPowerSeries)
assert isinstance(fcomp.gfps, FormalPowerSeries)
assert fcomp.f == exp(x)
assert fcomp.g == sin(x)
assert fcomp.function == exp(sin(x))
assert fcomp._eval_terms(6) == 1 + x + x**2 / 2 - x**4 / 8 - x**5 / 15
assert fcomp.truncate() == 1 + x + x**2 / 2 - x**4 / 8 - x**5 / 15 + O(x**
6)
assert fcomp.truncate(5) == 1 + x + x**2 / 2 - x**4 / 8 + O(x**5)
raises(NotImplementedError, lambda: fcomp._eval_term(5))
raises(NotImplementedError, lambda: fcomp.infinite)
raises(NotImplementedError, lambda: fcomp._eval_derivative(x))
raises(NotImplementedError, lambda: fcomp.integrate(x))
assert f1.compose(f2, x).truncate(4) == 1 + x + x**2 / 2 + O(x**4)
assert f1.compose(f2, x).truncate(
8
) == 1 + x + x**2 / 2 - x**4 / 8 - x**5 / 15 - x**6 / 240 + x**7 / 90 + O(
x**8)
assert f1.compose(
f2, x).truncate(6) == 1 + x + x**2 / 2 - x**4 / 8 - x**5 / 15 + O(x**6)
assert f2.compose(f2, x).truncate(4) == x - x**3 / 3 + O(x**4)
assert f2.compose(
f2,
x).truncate(8) == x - x**3 / 3 + x**5 / 10 - 8 * x**7 / 315 + O(x**8)
assert f2.compose(f2, x).truncate(6) == x - x**3 / 3 + x**5 / 10 + O(x**6)
def test_fps__slow():
f = x * exp(x) * sin(2 * x) # TODO: rsolve needs improvement
assert fps(f, x).truncate() == 2 * x ** 2 + 2 * x ** 3 - x ** 4 / 3 - x ** 5 + O(x ** 6)
def test_fps__logarithmic_singularity_fail():
f = asech(x) # Algorithms for computing limits probably needs improvemnts
assert fps(f, x) == log(2) - log(x) - x**2 / 4 - 3 * x**4 / 64 + O(x**6)
def solve(FLT_MIN, FLT_MAX):
"""Solving cos(x) <= -0.99, dx/dt=1, x(0) = 0
# Basic steps:
# 1. First compute the n terms for each ode
# 2. Next replace the guard with ode(t), so that it is only in t
# 3. Then compute the number of terms needed for g(t)
# 4. Finally, compute g(t) = 0 and g(t)-2g(0) = 0
# 5. Note that computing number of terms "n" in taylor essentially
# guarantees that tᵣ - t ≤ floating point error only, specified by the
# polynomial solver.
"""
# XXX: This is the theta
def test_multivariate():
# LTI is easy to solve
# Xdiff = S.sympify('(5*x(t) + 2*y(t) + 1)')
# Time varying, takes more time in general,
# with increasing power for t^n
# Xdiff = S.sympify('(5*x(t) + 2*y(t) + t**3)')
# Non linear with periodic functions
# Xdiff = S.sympify('sin(sqrt(x(t)+1))')
# import math
# FLT_MIN = 0
# FLT_MAX = 2*math.pi
# More complex ode
# Xdiff = S.sympify('sin(sin(x(t)+1))')
# The angles can only be between 0 and 2π
# import math
# FLT_MIN = -2*math.pi
# FLT_MAX = 2*math.pi
# A sqrt
# Xdiff = S.sympify('sqrt(x(t)+1)')
# The ones below need to have a reduced search space bound for
# continous variables.
# Another sqrt, does not seem to converge
# Xdiff = S.sympify('x(t)*t') # Does not work
# Now multiplication, seems to not coverge ever.
Xdiff = S.sympify('exp(2*x(t))') # Does not work either
# Using scaling factor, to reduce the bounds of the maximisation
# problem.
FLT_MIN = -1e1
FLT_MAX = 1e1
return FLT_MIN, FLT_MAX, Xdiff
FLT_MIN, FLT_MAX, tomaximize = test_multivariate()
xt = S.sympify('x(t)')
x = S.abc.x
yt = S.sympify('y(t)')
y = S.abc.y
# Coupled ode example
(tokens, nx) = getN({xt.diff(t): ([tomaximize],
{yt.diff(t): (xt,
# args always in
# same order for
# everyone
[x, y, t])},
# Always list all the replacements
{xt: x, yt: y},
[x, y, t])},
FLT_MIN=FLT_MIN, FLT_MAX=FLT_MAX, epsilon=1e-6)
# print(tokens)
print('required terms for θ satisfying Lipschitz constant:', nx)
# Now make the taylor polynomial
taylorxcoeffs = [5*S.pi/2, 1] + [0]*(nx-2)
# These are the smooth tokens
taylorxpoly = sum([t**i*v for i, v in zip(range(nx), taylorxcoeffs)])
# The theta' taylor polynomial
print('θ(t) = ', taylorxpoly)
# The guard function that needs the lipschitz constant
def guard():
gt = (S.cos(taylorxpoly)+0.99)
return gt.diff(t)
gt = S.sympify('g(t)')
tokens, n = getN({gt.diff(t): ([guard()], dict(), dict(), [t])})
# print(tokens)
print('Number of terms for cos(%s)+0.99: %s' % (taylorxpoly, n))
# Now we do the example of the ode with taylor polynomial
cosseries1 = S.fps(S.cos(taylorxpoly)+0.99, x0=0).polynomial(n=n)
print('Guard taylor polynomial:', cosseries1, '\n')
# print(S.simplify(cosseries1))
root = None
try:
root1 = S.nsolve(cosseries1, t, 0, dict=True)[0][t]
root = root1
except ValueError:
print('No root for g(t)=0')
# Now the second one, this one fails
# g(t) - 2*g(0) = 0
cosseries2 = S.fps(S.cos((5*S.pi/2) + t)-1.98, x0=0).polynomial(n=n)
# print(S.simplify(cosseries2))
try:
root2 = S.nsolve(cosseries2, t, 0, dict=True)[0][t]
root = min(root, root2)
except ValueError:
print('No root for g(t)-2*g(0) = 0')
print('guard Δt:', root)
def test_fps__hyper():
f = sin(x)
assert fps(f, x).truncate() == x - x**3 / 6 + x**5 / 120 + O(x**6)
f = cos(x)
assert fps(f, x).truncate() == 1 - x**2 / 2 + x**4 / 24 + O(x**6)
f = exp(x)
assert fps(f, x).truncate() == \
1 + x + x**2/2 + x**3/6 + x**4/24 + x**5/120 + O(x**6)
f = atan(x)
assert fps(f, x).truncate() == x - x**3 / 3 + x**5 / 5 + O(x**6)
f = exp(acos(x))
assert fps(f, x).truncate() == \
(exp(pi/2) - x*exp(pi/2) + x**2*exp(pi/2)/2 - x**3*exp(pi/2)/3 +
5*x**4*exp(pi/2)/24 - x**5*exp(pi/2)/6 + O(x**6))
f = exp(acosh(x))
assert fps(f,
x).truncate() == I + x - I * x**2 / 2 - I * x**4 / 8 + O(x**6)
f = atan(1 / x)
assert fps(f, x).truncate() == pi / 2 - x + x**3 / 3 - x**5 / 5 + O(x**6)
f = x * atan(x) - log(1 + x**2) / 2
assert fps(f, x,
rational=False).truncate() == x**2 / 2 - x**4 / 12 + O(x**6)
f = log(1 + x)
assert fps(f, x, rational=False).truncate() == \
x - x**2/2 + x**3/3 - x**4/4 + x**5/5 + O(x**6)
f = airyai(x**2)
assert fps(f, x).truncate() == \
(3**Rational(5, 6)*gamma(Rational(1, 3))/(6*pi) -
3**Rational(2, 3)*x**2/(3*gamma(Rational(1, 3))) + O(x**6))
f = exp(x) * sin(x)
assert fps(f, x).truncate() == x + x**2 + x**3 / 3 - x**5 / 30 + O(x**6)
f = exp(x) * sin(x) / x
assert fps(
f, x).truncate() == 1 + x + x**2 / 3 - x**4 / 30 - x**5 / 90 + O(x**6)
f = sin(x) * cos(x)
assert fps(f, x).truncate() == x - 2 * x**3 / 3 + 2 * x**5 / 15 + O(x**6)
def test_fps__rational():
assert fps(1 / x) == (1 / x)
assert fps((x**2 + x + 1) / x**3, dir=-1) == (x**2 + x + 1) / x**3
f = 1 / ((x - 1)**2 * (x - 2))
assert fps(f, x).truncate() == \
(-Rational(1, 2) - 5*x/4 - 17*x**2/8 - 49*x**3/16 - 129*x**4/32 -
321*x**5/64 + O(x**6))
f = (1 + x + x**2 + x**3) / ((x - 1) * (x - 2))
assert fps(f, x).truncate() == \
(Rational(1, 2) + 5*x/4 + 17*x**2/8 + 49*x**3/16 + 113*x**4/32 +
241*x**5/64 + O(x**6))
f = x / (1 - x - x**2)
assert fps(f, x, full=True).truncate() == \
x + x**2 + 2*x**3 + 3*x**4 + 5*x**5 + O(x**6)
f = 1 / (x**2 + 2 * x + 2)
assert fps(f, x, full=True).truncate() == \
Rational(1, 2) - x/2 + x**2/4 - x**4/8 + x**5/8 + O(x**6)
f = log(1 + x)
assert fps(f, x).truncate() == \
x - x**2/2 + x**3/3 - x**4/4 + x**5/5 + O(x**6)
assert fps(f, x, dir=1).truncate() == fps(f, x, dir=-1).truncate()
assert fps(f, x, 2).truncate() == \
(log(3) - Rational(2, 3) - (x - 2)**2/18 + (x - 2)**3/81 -
(x - 2)**4/324 + (x - 2)**5/1215 + x/3 + O((x - 2)**6, (x, 2)))
assert fps(f, x, 2, dir=-1).truncate() == \
(log(3) - Rational(2, 3) - (-x + 2)**2/18 - (-x + 2)**3/81 -
(-x + 2)**4/324 - (-x + 2)**5/1215 + x/3 + O((x - 2)**6, (x, 2)))
f = atan(x)
assert fps(f, x, full=True).truncate() == x - x**3 / 3 + x**5 / 5 + O(x**6)
assert fps(f, x, full=True, dir=1).truncate() == \
fps(f, x, full=True, dir=-1).truncate()
assert fps(f, x, 2, full=True).truncate() == \
(atan(2) - Rational(2, 5) - 2*(x - 2)**2/25 + 11*(x - 2)**3/375 -
6*(x - 2)**4/625 + 41*(x - 2)**5/15625 + x/5 + O((x - 2)**6, (x, 2)))
assert fps(f, x, 2, full=True, dir=-1).truncate() == \
(atan(2) - Rational(2, 5) - 2*(-x + 2)**2/25 - 11*(-x + 2)**3/375 -
6*(-x + 2)**4/625 - 41*(-x + 2)**5/15625 + x/5 + O((x - 2)**6, (x, 2)))
f = x * atan(x) - log(1 + x**2) / 2
assert fps(f, x, full=True).truncate() == x**2 / 2 - x**4 / 12 + O(x**6)
f = log((1 + x) / (1 - x)) / 2 - atan(x)
assert fps(
f, x,
full=True).truncate(n=10) == 2 * x**3 / 3 + 2 * x**7 / 7 + O(x**10)
def test_fps__rational():
assert fps(1 / x) == (1 / x)
assert fps((x ** 2 + x + 1) / x ** 3, dir=-1) == (x ** 2 + x + 1) / x ** 3
f = 1 / ((x - 1) ** 2 * (x - 2))
assert fps(f, x).truncate() == (
-Rational(1, 2)
- 5 * x / 4
- 17 * x ** 2 / 8
- 49 * x ** 3 / 16
- 129 * x ** 4 / 32
- 321 * x ** 5 / 64
+ O(x ** 6)
)
f = (1 + x + x ** 2 + x ** 3) / ((x - 1) * (x - 2))
assert fps(f, x).truncate() == (
Rational(1, 2)
+ 5 * x / 4
+ 17 * x ** 2 / 8
+ 49 * x ** 3 / 16
+ 113 * x ** 4 / 32
+ 241 * x ** 5 / 64
+ O(x ** 6)
)
f = x / (1 - x - x ** 2)
assert fps(f, x, full=True).truncate() == x + x ** 2 + 2 * x ** 3 + 3 * x ** 4 + 5 * x ** 5 + O(x ** 6)
f = 1 / (x ** 2 + 2 * x + 2)
assert fps(f, x, full=True).truncate() == Rational(1, 2) - x / 2 + x ** 2 / 4 - x ** 4 / 8 + x ** 5 / 8 + O(x ** 6)
f = log(1 + x)
assert fps(f, x).truncate() == x - x ** 2 / 2 + x ** 3 / 3 - x ** 4 / 4 + x ** 5 / 5 + O(x ** 6)
assert fps(f, x, dir=1).truncate() == fps(f, x, dir=-1).truncate()
assert fps(f, x, 2).truncate() == (
log(3)
- Rational(2, 3)
- (x - 2) ** 2 / 18
+ (x - 2) ** 3 / 81
- (x - 2) ** 4 / 324
+ (x - 2) ** 5 / 1215
+ x / 3
+ O((x - 2) ** 6, (x, 2))
)
assert fps(f, x, 2, dir=-1).truncate() == (
log(3)
- Rational(2, 3)
- (-x + 2) ** 2 / 18
- (-x + 2) ** 3 / 81
- (-x + 2) ** 4 / 324
- (-x + 2) ** 5 / 1215
+ x / 3
+ O((x - 2) ** 6, (x, 2))
)
f = atan(x)
assert fps(f, x, full=True).truncate() == x - x ** 3 / 3 + x ** 5 / 5 + O(x ** 6)
assert fps(f, x, full=True, dir=1).truncate() == fps(f, x, full=True, dir=-1).truncate()
assert fps(f, x, 2, full=True).truncate() == (
atan(2)
- Rational(2, 5)
- 2 * (x - 2) ** 2 / 25
+ 11 * (x - 2) ** 3 / 375
- 6 * (x - 2) ** 4 / 625
+ 41 * (x - 2) ** 5 / 15625
+ x / 5
+ O((x - 2) ** 6, (x, 2))
)
assert fps(f, x, 2, full=True, dir=-1).truncate() == (
atan(2)
- Rational(2, 5)
- 2 * (-x + 2) ** 2 / 25
- 11 * (-x + 2) ** 3 / 375
- 6 * (-x + 2) ** 4 / 625
- 41 * (-x + 2) ** 5 / 15625
+ x / 5
+ O((x - 2) ** 6, (x, 2))
)
f = x * atan(x) - log(1 + x ** 2) / 2
assert fps(f, x, full=True).truncate() == x ** 2 / 2 - x ** 4 / 12 + O(x ** 6)
f = log((1 + x) / (1 - x)) / 2 - atan(x)
assert fps(f, x, full=True).truncate(n=10) == 2 * x ** 3 / 3 + 2 * x ** 7 / 7 + O(x ** 10)
def test_fps__hyper():
f = sin(x)
assert fps(f, x).truncate() == x - x ** 3 / 6 + x ** 5 / 120 + O(x ** 6)
f = cos(x)
assert fps(f, x).truncate() == 1 - x ** 2 / 2 + x ** 4 / 24 + O(x ** 6)
f = exp(x)
assert fps(f, x).truncate() == 1 + x + x ** 2 / 2 + x ** 3 / 6 + x ** 4 / 24 + x ** 5 / 120 + O(x ** 6)
f = atan(x)
assert fps(f, x).truncate() == x - x ** 3 / 3 + x ** 5 / 5 + O(x ** 6)
f = exp(acos(x))
assert fps(f, x).truncate() == (
exp(pi / 2)
- x * exp(pi / 2)
+ x ** 2 * exp(pi / 2) / 2
- x ** 3 * exp(pi / 2) / 3
+ 5 * x ** 4 * exp(pi / 2) / 24
- x ** 5 * exp(pi / 2) / 6
+ O(x ** 6)
)
f = exp(acosh(x))
assert fps(f, x).truncate() == I + x - I * x ** 2 / 2 - I * x ** 4 / 8 + O(x ** 6)
f = atan(1 / x)
assert fps(f, x).truncate() == pi / 2 - x + x ** 3 / 3 - x ** 5 / 5 + O(x ** 6)
f = x * atan(x) - log(1 + x ** 2) / 2
assert fps(f, x, rational=False).truncate() == x ** 2 / 2 - x ** 4 / 12 + O(x ** 6)
f = log(1 + x)
assert fps(f, x, rational=False).truncate() == x - x ** 2 / 2 + x ** 3 / 3 - x ** 4 / 4 + x ** 5 / 5 + O(x ** 6)
f = airyai(x ** 2)
assert fps(f, x).truncate() == (
3 ** Rational(5, 6) * gamma(Rational(1, 3)) / (6 * pi)
- 3 ** Rational(2, 3) * x ** 2 / (3 * gamma(Rational(1, 3)))
+ O(x ** 6)
)
f = exp(x) * sin(x)
assert fps(f, x).truncate() == x + x ** 2 + x ** 3 / 3 - x ** 5 / 30 + O(x ** 6)
f = exp(x) * sin(x) / x
assert fps(f, x).truncate() == 1 + x + x ** 2 / 3 - x ** 4 / 30 - x ** 5 / 90 + O(x ** 6)
f = sin(x) * cos(x)
assert fps(f, x).truncate() == x - 2 * x ** 3 / 3 + 2 * x ** 5 / 15 + O(x ** 6)
def test_fps__logarithmic_singularity():
f = log(1 + 1 / x)
assert fps(f, x) != -log(x) + x - x ** 2 / 2 + x ** 3 / 3 - x ** 4 / 4 + x ** 5 / 5 + O(x ** 6)
assert fps(f, x, rational=False) != -log(x) + x - x ** 2 / 2 + x ** 3 / 3 - x ** 4 / 4 + x ** 5 / 5 + O(x ** 6)
def test_fps__logarithmic_singularity_fail():
f = asech(x) # Algorithms for computing limits probably needs improvemnts
assert fps(f, x) == log(2) - log(x) - x ** 2 / 4 - 3 * x ** 4 / 64 + O(x ** 6)
def test_fps__slow():
f = x * exp(x) * sin(2 * x) # TODO: rsolve needs improvement
assert fps(f,
x).truncate() == 2 * x**2 + 2 * x**3 - x**4 / 3 - x**5 + O(x**6)
def test_fps__operations():
f1, f2 = fps(sin(x)), fps(cos(x))
fsum = f1 + f2
assert fsum.function == sin(x) + cos(x)
assert fsum.truncate() == \
1 + x - x**2/2 - x**3/6 + x**4/24 + x**5/120 + O(x**6)
fsum = f1 + 1
assert fsum.function == sin(x) + 1
assert fsum.truncate() == 1 + x - x**3 / 6 + x**5 / 120 + O(x**6)
fsum = 1 + f2
assert fsum.function == cos(x) + 1
assert fsum.truncate() == 2 - x**2 / 2 + x**4 / 24 + O(x**6)
assert (f1 + x) == Add(f1, x)
assert -f2.truncate() == -1 + x**2 / 2 - x**4 / 24 + O(x**6)
assert (f1 - f1) == S.Zero
fsub = f1 - f2
assert fsub.function == sin(x) - cos(x)
assert fsub.truncate() == \
-1 + x + x**2/2 - x**3/6 - x**4/24 + x**5/120 + O(x**6)
fsub = f1 - 1
assert fsub.function == sin(x) - 1
assert fsub.truncate() == -1 + x - x**3 / 6 + x**5 / 120 + O(x**6)
fsub = 1 - f2
assert fsub.function == -cos(x) + 1
assert fsub.truncate() == x**2 / 2 - x**4 / 24 + O(x**6)
raises(ValueError, lambda: f1 + fps(exp(x), dir=-1))
raises(ValueError, lambda: f1 + fps(exp(x), x0=1))
fm = f1 * 3
assert fm.function == 3 * sin(x)
assert fm.truncate() == 3 * x - x**3 / 2 + x**5 / 40 + O(x**6)
fm = 3 * f2
assert fm.function == 3 * cos(x)
assert fm.truncate() == 3 - 3 * x**2 / 2 + x**4 / 8 + O(x**6)
assert (f1 * f2) == Mul(f1, f2)
assert (f1 * x) == Mul(f1, x)
fd = f1.diff()
assert fd.function == cos(x)
assert fd.truncate() == 1 - x**2 / 2 + x**4 / 24 + O(x**6)
fd = f2.diff()
assert fd.function == -sin(x)
assert fd.truncate() == -x + x**3 / 6 - x**5 / 120 + O(x**6)
fd = f2.diff().diff()
assert fd.function == -cos(x)
assert fd.truncate() == -1 + x**2 / 2 - x**4 / 24 + O(x**6)
f3 = fps(exp(sqrt(x)))
fd = f3.diff()
assert fd.truncate().expand() == \
(1/(2*sqrt(x)) + S(1)/2 + x/12 + x**2/240 + x**3/10080 + x**4/725760 +
x**5/79833600 + sqrt(x)/4 + x**(S(3)/2)/48 + x**(S(5)/2)/1440 +
x**(S(7)/2)/80640 + x**(S(9)/2)/7257600 + x**(S(11)/2)/958003200 +
O(x**6))
assert f1.integrate((x, 0, 1)) == -cos(1) + 1
assert integrate(f1, (x, 0, 1)) == -cos(1) + 1
fi = integrate(f1, x)
assert fi.function == -cos(x)
assert fi.truncate() == -1 + x**2 / 2 - x**4 / 24 + O(x**6)
fi = f2.integrate(x)
assert fi.function == sin(x)
assert fi.truncate() == x - x**3 / 6 + x**5 / 120 + O(x**6)
def test_fps__operations():
f1, f2 = fps(sin(x)), fps(cos(x))
fsum = f1 + f2
assert fsum.function == sin(x) + cos(x)
assert fsum.truncate() == 1 + x - x ** 2 / 2 - x ** 3 / 6 + x ** 4 / 24 + x ** 5 / 120 + O(x ** 6)
fsum = f1 + 1
assert fsum.function == sin(x) + 1
assert fsum.truncate() == 1 + x - x ** 3 / 6 + x ** 5 / 120 + O(x ** 6)
fsum = 1 + f2
assert fsum.function == cos(x) + 1
assert fsum.truncate() == 2 - x ** 2 / 2 + x ** 4 / 24 + O(x ** 6)
assert (f1 + x) == Add(f1, x)
assert -f2.truncate() == -1 + x ** 2 / 2 - x ** 4 / 24 + O(x ** 6)
assert (f1 - f1) == S.Zero
fsub = f1 - f2
assert fsub.function == sin(x) - cos(x)
assert fsub.truncate() == -1 + x + x ** 2 / 2 - x ** 3 / 6 - x ** 4 / 24 + x ** 5 / 120 + O(x ** 6)
fsub = f1 - 1
assert fsub.function == sin(x) - 1
assert fsub.truncate() == -1 + x - x ** 3 / 6 + x ** 5 / 120 + O(x ** 6)
fsub = 1 - f2
assert fsub.function == -cos(x) + 1
assert fsub.truncate() == x ** 2 / 2 - x ** 4 / 24 + O(x ** 6)
raises(ValueError, lambda: f1 + fps(exp(x), dir=-1))
raises(ValueError, lambda: f1 + fps(exp(x), x0=1))
fm = f1 * 3
assert fm.function == 3 * sin(x)
assert fm.truncate() == 3 * x - x ** 3 / 2 + x ** 5 / 40 + O(x ** 6)
fm = 3 * f2
assert fm.function == 3 * cos(x)
assert fm.truncate() == 3 - 3 * x ** 2 / 2 + x ** 4 / 8 + O(x ** 6)
assert (f1 * f2) == Mul(f1, f2)
assert (f1 * x) == Mul(f1, x)
fd = f1.diff()
assert fd.function == cos(x)
assert fd.truncate() == 1 - x ** 2 / 2 + x ** 4 / 24 + O(x ** 6)
fd = f2.diff()
assert fd.function == -sin(x)
assert fd.truncate() == -x + x ** 3 / 6 - x ** 5 / 120 + O(x ** 6)
fd = f2.diff().diff()
assert fd.function == -cos(x)
assert fd.truncate() == -1 + x ** 2 / 2 - x ** 4 / 24 + O(x ** 6)
f3 = fps(exp(sqrt(x)))
fd = f3.diff()
assert fd.truncate().expand() == (
1 / (2 * sqrt(x))
+ S(1) / 2
+ x / 12
+ x ** 2 / 240
+ x ** 3 / 10080
+ x ** 4 / 725760
+ x ** 5 / 79833600
+ sqrt(x) / 4
+ x ** (S(3) / 2) / 48
+ x ** (S(5) / 2) / 1440
+ x ** (S(7) / 2) / 80640
+ x ** (S(9) / 2) / 7257600
+ x ** (S(11) / 2) / 958003200
+ O(x ** 6)
)
assert f1.integrate((x, 0, 1)) == -cos(1) + 1
fi = f1.integrate(x)
assert fi.function == -cos(x)
assert fi.truncate() == -1 + x ** 2 / 2 - x ** 4 / 24 + O(x ** 6)
fi = f2.integrate()
assert fi.function == sin(x)
assert fi.truncate() == x - x ** 3 / 6 + x ** 5 / 120 + O(x ** 6)
