Symbolic Algebra with sympy

In [1]:
from sympy import *
In [2]:
from sympy import init_session
init_session()
IPython console for SymPy 0.7.6.1 (Python 3.5.1-64-bit) (ground types: python)

These commands were executed:
>>> from __future__ import division
>>> from sympy import *
>>> x, y, z, t = symbols('x y z t')
>>> k, m, n = symbols('k m n', integer=True)
>>> f, g, h = symbols('f g h', cls=Function)
>>> init_printing()

Documentation can be found at http://www.sympy.org

Basics

In [3]:
from sympy.stats import *
E(Die('X', 6))
Out[3]:
\[\frac{7}{2}\]
In [4]:
sqrt(8)
Out[4]:
\[2 \sqrt{2}\]
In [5]:
expr = x + 2*y
In [6]:
expr2 = x*expr
In [7]:
expr2
Out[7]:
\[x \left(x + 2 y\right)\]
In [8]:
expand(expr2)
Out[8]:
\[x^{2} + 2 x y\]
In [9]:
factor(expand(expr2))
Out[9]:
\[x \left(x + 2 y\right)\]
In [10]:
diff(sin(x) * exp(x), x)
Out[10]:
\[e^{x} \sin{\left (x \right )} + e^{x} \cos{\left (x \right )}\]
In [11]:
integrate(exp(x)*sin(x) + exp(x)*cos(x), x)
Out[11]:
\[e^{x} \sin{\left (x \right )}\]
In [12]:
integrate(sin(x**2), (x, -oo, oo))
Out[12]:
\[\frac{\sqrt{2} \sqrt{\pi}}{2}\]
In [13]:
dsolve(Eq(f(t).diff(t, t) - f(t), exp(t)), f(t))
Out[13]:
\[f{\left (t \right )} = C_{2} e^{- t} + \left(C_{1} + \frac{t}{2}\right) e^{t}\]
In [14]:
Matrix([[1,2],[2,2]]).eigenvals()
Out[14]:
\[\left \{ \frac{3}{2} + \frac{\sqrt{17}}{2} : 1, \quad - \frac{\sqrt{17}}{2} + \frac{3}{2} : 1\right \}\]
In [15]:
nu = symbols('nu')
besselj(nu, z).rewrite(jn)
Out[15]:
\[\frac{\sqrt{2} \sqrt{z}}{\sqrt{\pi}} j_{\nu - \frac{1}{2}}\left(z\right)\]
In [16]:
latex(Integral(cos(x)**2, (x, 0, pi)))
Out[16]:
'\\int_{0}^{\\pi} \\cos^{2}{\\left (x \\right )}\\, dx'
In [17]:
expr = cos(x) + 1
In [18]:
expr.subs(x, y)
Out[18]:
\[\cos{\left (y \right )} + 1\]
In [19]:
expr = x**y
expr = expr.subs(y, x**y)
expr = expr.subs(y, x**y)
expr = expr.subs(x, x**x)
In [20]:
expr
Out[20]:
\[\left(x^{x}\right)^{\left(x^{x}\right)^{\left(x^{x}\right)^{y}}}\]
In [21]:
expr = sin(2*x) + cos(2*x)
expand_trig(expr)
Out[21]:
\[2 \sin{\left (x \right )} \cos{\left (x \right )} + 2 \cos^{2}{\left (x \right )} - 1\]
In [22]:
expr.subs(sin(2*x), 2*sin(x)*cos(x))
Out[22]:
\[2 \sin{\left (x \right )} \cos{\left (x \right )} + \cos{\left (2 x \right )}\]
In [23]:
expr = x**4 - 4*x**3 + 4*x**2 - 2*x +3
In [24]:
replacements = [(x**i, y**i) for i in range(5) if i%2 == 0]
expr.subs(replacements)
Out[24]:
\[- 4 x^{3} - 2 x + y^{4} + 4 y^{2} + 3\]
In [25]:
sexpr = "x**4 - 4*x**3 + 4*x**2 - 2*x +3"
sympify(sexpr)
Out[25]:
\[x^{4} - 4 x^{3} + 4 x^{2} - 2 x + 3\]
In [26]:
sqrt(8).evalf()
Out[26]:
\[2.82842712474619\]
In [27]:
pi.evalf(100)
Out[27]:
\[3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068\]
In [28]:
expr = cos(2*x)
expr.evalf(subs = {x: 2.4})
Out[28]:
\[0.0874989834394464\]
In [29]:
expr = cos(x)**2 + sin(x)**2
expr.evalf(subs = {x: 1}, chop=True)
Out[29]:
\[1.0\]
In [30]:
a = range(10)
expr = sin(x)
f = lambdify(x, expr, "numpy")
f(a)
Out[30]:
array([ 0.        ,  0.84147098,  0.90929743,  0.14112001, -0.7568025 ,
       -0.95892427, -0.2794155 ,  0.6569866 ,  0.98935825,  0.41211849])

Simplify

In [31]:
simplify(sin(x)**2 + cos(x)**2)
Out[31]:
\[1\]
In [32]:
simplify(gamma(x) / gamma(x-2))
Out[32]:
\[\left(x - 2\right) \left(x - 1\right)\]
In [33]:
expand((x + y)**3)
Out[33]:
\[x^{3} + 3 x^{2} y + 3 x y^{2} + y^{3}\]
In [34]:
factor(x**3 - x**2 + x - 1)
Out[34]:
\[\left(x - 1\right) \left(x^{2} + 1\right)\]
In [35]:
factor_list(x**3 - x**2 + x - 1)
Out[35]:
\[\left ( 1, \quad \left [ \left ( x - 1, \quad 1\right ), \quad \left ( x^{2} + 1, \quad 1\right )\right ]\right )\]
In [36]:
expand((cos(x) + sin(x))**2)
Out[36]:
\[\sin^{2}{\left (x \right )} + 2 \sin{\left (x \right )} \cos{\left (x \right )} + \cos^{2}{\left (x \right )}\]
In [37]:
expr = x*y + x - 3 + 2*x**2 - z*x**2 + x**3
cexpr = collect(expr, x)
cexpr
Out[37]:
\[x^{3} + x^{2} \left(- z + 2\right) + x \left(y + 1\right) - 3\]
In [38]:
cexpr.coeff(x**2)
Out[38]:
\[- z + 2\]
In [39]:
cancel((x**2 + 2*x + 1) / (x**2 + x))
Out[39]:
\[\frac{1}{x} \left(x + 1\right)\]
In [40]:
expr = (x*y**2 - 2*x*y*z + x*z**2 + y**2 - 2*y*z + z**2)/(x**2 - 1)
In [41]:
cancel(expr)
Out[41]:
\[\frac{1}{x - 1} \left(y^{2} - 2 y z + z^{2}\right)\]
In [42]:
factor(expr)
Out[42]:
\[\frac{\left(y - z\right)^{2}}{x - 1}\]
In [43]:
expr = (4*x**3 + 21*x**2 + 10*x + 12)/(x**4 + 5*x**3 + 5*x**2 + 4*x)
In [44]:
apart(expr)
Out[44]:
\[\frac{2 x - 1}{x^{2} + x + 1} - \frac{1}{x + 4} + \frac{3}{x}\]
In [45]:
trigsimp(sin(x)*tan(x)/sec(x))
Out[45]:
\[\sin^{2}{\left (x \right )}\]
In [46]:
trigsimp(cosh(x)**2 + sinh(x)**2)
Out[46]:
\[\cosh{\left (2 x \right )}\]
In [47]:
expand_trig(sin(x + y))
Out[47]:
\[\sin{\left (x \right )} \cos{\left (y \right )} + \sin{\left (y \right )} \cos{\left (x \right )}\]
In [48]:
x, y = symbols('x y', positive=True)
a, b = symbols('a b', real=True)
z, t, c = symbols('z t c')
In [49]:
powsimp(x**a*x**b)
Out[49]:
\[x^{a + b}\]
In [50]:
powsimp(x**a*y**a)
Out[50]:
\[\left(x y\right)^{a}\]
In [51]:
powsimp(t**c*z**c)
Out[51]:
\[t^{c} z^{c}\]
In [52]:
powsimp(t**c*z**c, force=True)
Out[52]:
\[\left(t z\right)^{c}\]
In [53]:
expand_power_exp(x**(a + b))
Out[53]:
\[x^{a} x^{b}\]
In [54]:
expand_power_base((x*y)**a)
Out[54]:
\[x^{a} y^{a}\]
In [55]:
powdenest((x**a)**b)
Out[55]:
\[x^{a b}\]
In [56]:
n = symbols('n', real=True)
In [57]:
expand_log(log(x*y))
Out[57]:
\[\log{\left (x \right )} + \log{\left (y \right )}\]
In [58]:
expand_log(log(x**n))
Out[58]:
\[n \log{\left (x \right )}\]
In [59]:
logcombine(n*log(x))
Out[59]:
\[\log{\left (x^{n} \right )}\]
In [60]:
x, y, z = symbols('x y z')
k, m, n = symbols('k m n')
In [61]:
factorial(n)
Out[61]:
\[n!\]
In [62]:
binomial(n, k)
Out[62]:
\[{\binom{n}{k}}\]
In [63]:
hyper([1,2], [3], z)
Out[63]:
\[\begin{split}{{}_{2}F_{1}\left(\begin{matrix} 1, 2 \\ 3 \end{matrix}\middle| {z} \right)}\end{split}\]
In [64]:
factorial(x).rewrite(gamma)
Out[64]:
\[\Gamma{\left(x + 1 \right)}\]
In [65]:
tan(x).rewrite(sin)
Out[65]:
\[\frac{2 \sin^{2}{\left (x \right )}}{\sin{\left (2 x \right )}}\]
In [66]:
expand_func(gamma(x + 3))
Out[66]:
\[x \left(x + 1\right) \left(x + 2\right) \Gamma{\left(x \right)}\]
In [67]:
hyperexpand(hyper([1, 1], [2], z))
Out[67]:
\[- \frac{1}{z} \log{\left (- z + 1 \right )}\]
In [68]:
expr =  meijerg([[1],[1]], [[1],[]], -z)
In [69]:
expr
Out[69]:
\[\begin{split}{G_{2, 1}^{1, 1}\left(\begin{matrix} 1 & 1 \\1 & \end{matrix} \middle| {- z} \right)}\end{split}\]
In [70]:
hyperexpand(expr)
Out[70]:
\[e^{\frac{1}{z}}\]
In [71]:
combsimp(binomial(n+1, k+1)/binomial(n, k))
Out[71]:
\[\frac{n + 1}{k + 1}\]
In [72]:
def list_to_frac(l):
    expr = Integer(0)
    for i in reversed(l[1:]):
        expr += i
        expr = 1/expr
    return l[0] + expr
In [73]:
list_to_frac([1,2,3,4])
Out[73]:
\[\frac{43}{30}\]
In [74]:
syms = symbols('a0:5')
In [75]:
syms
Out[75]:
\[\left ( a_{0}, \quad a_{1}, \quad a_{2}, \quad a_{3}, \quad a_{4}\right )\]
In [76]:
frac = list_to_frac(syms)
In [77]:
frac
Out[77]:
\[a_{0} + \frac{1}{a_{1} + \frac{1}{a_{2} + \frac{1}{a_{3} + \frac{1}{a_{4}}}}}\]
In [78]:
frac = cancel(frac)
In [79]:
frac
Out[79]:
\[\frac{a_{0} a_{1} a_{2} a_{3} a_{4} + a_{0} a_{1} a_{2} + a_{0} a_{1} a_{4} + a_{0} a_{3} a_{4} + a_{0} + a_{2} a_{3} a_{4} + a_{2} + a_{4}}{a_{1} a_{2} a_{3} a_{4} + a_{1} a_{2} + a_{1} a_{4} + a_{3} a_{4} + 1}\]
In [80]:
from sympy.printing import print_ccode
print_ccode(frac)
(a0*a1*a2*a3*a4 + a0*a1*a2 + a0*a1*a4 + a0*a3*a4 + a0 + a2*a3*a4 + a2 + a4)/(a1*a2*a3*a4 + a1*a2 + a1*a4 + a3*a4 + 1)

Calculus

In [81]:
diff(cos(x), x)
Out[81]:
\[- \sin{\left (x \right )}\]
In [82]:
diff(x**4, x, 3)
Out[82]:
\[24 x\]
In [83]:
expr = exp(x*y*z)
diff(expr, x, y, 2, z, 4)
Out[83]:
\[x^{3} y^{2} \left(x^{3} y^{3} z^{3} + 14 x^{2} y^{2} z^{2} + 52 x y z + 48\right) e^{x y z}\]
In [84]:
deriv = Derivative(expr, x, y, 2, z, 4)
deriv
Out[84]:
\[\frac{\partial^{7}}{\partial x\partial y^{2}\partial z^{4}} e^{x y z}\]
In [85]:
deriv.doit()
Out[85]:
\[x^{3} y^{2} \left(x^{3} y^{3} z^{3} + 14 x^{2} y^{2} z^{2} + 52 x y z + 48\right) e^{x y z}\]
In [86]:
integrate(cos(x), x)
Out[86]:
\[\sin{\left (x \right )}\]
In [87]:
integrate(exp(-x), (x, 0, oo))
Out[87]:
\[1\]
In [88]:
integrate(exp(-x**2 - y**2), (x, -oo, oo), (y, -oo, oo))
Out[88]:
\[\pi\]
In [89]:
integrate(x**x, x)
Out[89]:
\[\int x^{x}\, dx\]
In [90]:
expr = Integral(log(x)**2, x)
expr
Out[90]:
\[\int \log^{2}{\left (x \right )}\, dx\]
In [91]:
expr.doit()
Out[91]:
\[x \log^{2}{\left (x \right )} - 2 x \log{\left (x \right )} + 2 x\]
In [92]:
integrate(sin(x**2), x)
Out[92]:
\[\frac{3 \sqrt{2} \sqrt{\pi} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{8 \Gamma{\left(\frac{7}{4} \right)}} \Gamma{\left(\frac{3}{4} \right)}\]
In [93]:
integrate(x**y*exp(-x), (x, 0, oo))
Out[93]:
\[\begin{split}\begin{cases} \Gamma{\left(y + 1 \right)} & \text{for}\: - \Re{y} < 1 \\\int_{0}^{\infty} x^{y} e^{- x}\, dx & \text{otherwise} \end{cases}\end{split}\]
In [94]:
limit(sin(x)/x, x, 0)
Out[94]:
\[1\]
In [95]:
expr = Limit((cos(x) - 1)/x, x, 0)
expr
Out[95]:
\[\lim_{x \to 0^+}\left(\frac{1}{x} \left(\cos{\left (x \right )} - 1\right)\right)\]
In [96]:
expr.doit()
Out[96]:
\[0\]
In [97]:
limit(1/x, x, 0, '-')
Out[97]:
\[-\infty\]
In [98]:
expr = exp(sin(x))
expr.series(x, 0, 4)
Out[98]:
\[1 + x + \frac{x^{2}}{2} + \mathcal{O}\left(x^{4}\right)\]
In [99]:
x + x**3 + x**6 + O(x**4)
Out[99]:
\[x + x^{3} + \mathcal{O}\left(x^{4}\right)\]
In [100]:
x * O(1)
Out[100]:
\[\mathcal{O}\left(x\right)\]
In [101]:
expr.series(x, 0, 4).removeO()
Out[101]:
\[\frac{x^{2}}{2} + x + 1\]
In [102]:
exp(x - 6).series(x, 6)
Out[102]:
\[-5 + \frac{1}{2} \left(x - 6\right)^{2} + \frac{1}{6} \left(x - 6\right)^{3} + \frac{1}{24} \left(x - 6\right)^{4} + \frac{1}{120} \left(x - 6\right)^{5} + x + \mathcal{O}\left(\left(x - 6\right)^{6}; x\rightarrow6\right)\]
In [103]:
exp(x - 6).series(x, 6).removeO().subs(x, x - 6)
Out[103]:
\[x + \frac{1}{120} \left(x - 12\right)^{5} + \frac{1}{24} \left(x - 12\right)^{4} + \frac{1}{6} \left(x - 12\right)^{3} + \frac{1}{2} \left(x - 12\right)^{2} - 11\]

Working wiht matrices

In [104]:
M = Matrix([[1,2,3],[3,2,1]])
P = Matrix([0,1,1])
M*P
Out[104]:
\[\begin{split}\left[\begin{matrix}5\\3\end{matrix}\right]\end{split}\]
In [105]:
M
Out[105]:
\[\begin{split}\left[\begin{matrix}1 & 2 & 3\\3 & 2 & 1\end{matrix}\right]\end{split}\]
In [106]:
M.shape
Out[106]:
\[\left ( 2, \quad 3\right )\]
In [107]:
M.col(-1)
Out[107]:
\[\begin{split}\left[\begin{matrix}3\\1\end{matrix}\right]\end{split}\]
In [108]:
M = Matrix([[1,3], [-2,3]])
M**-1
Out[108]:
\[\begin{split}\left[\begin{matrix}\frac{1}{3} & - \frac{1}{3}\\\frac{2}{9} & \frac{1}{9}\end{matrix}\right]\end{split}\]
In [109]:
M.T
Out[109]:
\[\begin{split}\left[\begin{matrix}1 & -2\\3 & 3\end{matrix}\right]\end{split}\]
In [110]:
eye(3)
Out[110]:
\[\begin{split}\left[\begin{matrix}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{matrix}\right]\end{split}\]
In [111]:
diag(1,2,3)
Out[111]:
\[\begin{split}\left[\begin{matrix}1 & 0 & 0\\0 & 2 & 0\\0 & 0 & 3\end{matrix}\right]\end{split}\]
In [112]:
M.det()
Out[112]:
\[9\]
In [113]:
M= Matrix([[1,0,1,3],[2,3,4,7],[-1,-3,-3,-4]])
M
Out[113]:
\[\begin{split}\left[\begin{matrix}1 & 0 & 1 & 3\\2 & 3 & 4 & 7\\-1 & -3 & -3 & -4\end{matrix}\right]\end{split}\]
In [114]:
M.rref()
Out[114]:
\[\begin{split}\left ( \left[\begin{matrix}1 & 0 & 1 & 3\\0 & 1 & \frac{2}{3} & \frac{1}{3}\\0 & 0 & 0 & 0\end{matrix}\right], \quad \left [ 0, \quad 1\right ]\right )\end{split}\]
In [115]:
M = Matrix([[1,2,3,0,0],[4,10,0,0,1]])
M
Out[115]:
\[\begin{split}\left[\begin{matrix}1 & 2 & 3 & 0 & 0\\4 & 10 & 0 & 0 & 1\end{matrix}\right]\end{split}\]
In [116]:
M.nullspace()
Out[116]:
\[\begin{split}\left [ \left[\begin{matrix}-15\\6\\1\\0\\0\end{matrix}\right], \quad \left[\begin{matrix}0\\0\\0\\1\\0\end{matrix}\right], \quad \left[\begin{matrix}1\\- \frac{1}{2}\\0\\0\\1\end{matrix}\right]\right ]\end{split}\]
In [117]:
M = Matrix([[3, -2, 4, -2], [5,3,-3,-2], [5,-2,2,-2], [5,-2,-3,3]])
M
Out[117]:
\[\begin{split}\left[\begin{matrix}3 & -2 & 4 & -2\\5 & 3 & -3 & -2\\5 & -2 & 2 & -2\\5 & -2 & -3 & 3\end{matrix}\right]\end{split}\]
In [118]:
M.eigenvals()
Out[118]:
\[\left \{ -2 : 1, \quad 3 : 1, \quad 5 : 2\right \}\]
In [119]:
M.eigenvects()
Out[119]:
\[\begin{split}\left [ \left ( -2, \quad 1, \quad \left [ \left[\begin{matrix}0\\1\\1\\1\end{matrix}\right]\right ]\right ), \quad \left ( 3, \quad 1, \quad \left [ \left[\begin{matrix}1\\1\\1\\1\end{matrix}\right]\right ]\right ), \quad \left ( 5, \quad 2, \quad \left [ \left[\begin{matrix}1\\1\\1\\0\end{matrix}\right], \quad \left[\begin{matrix}0\\-1\\0\\1\end{matrix}\right]\right ]\right )\right ]\end{split}\]
In [120]:
P, D = M.diagonalize()
In [121]:
P
Out[121]:
\[\begin{split}\left[\begin{matrix}0 & 1 & 1 & 0\\1 & 1 & 1 & -1\\1 & 1 & 1 & 0\\1 & 1 & 0 & 1\end{matrix}\right]\end{split}\]
In [122]:
D
Out[122]:
\[\begin{split}\left[\begin{matrix}-2 & 0 & 0 & 0\\0 & 3 & 0 & 0\\0 & 0 & 5 & 0\\0 & 0 & 0 & 5\end{matrix}\right]\end{split}\]
In [123]:
P*D*P**-1
Out[123]:
\[\begin{split}\left[\begin{matrix}3 & -2 & 4 & -2\\5 & 3 & -3 & -2\\5 & -2 & 2 & -2\\5 & -2 & -3 & 3\end{matrix}\right]\end{split}\]
In [124]:
lamda = symbols('lamda')
p = M.charpoly(lamda)
p
Out[124]:
\[\operatorname{PurePoly}{\left( \lambda^{4} - 11 \lambda^{3} + 29 \lambda^{2} + 35 \lambda - 150, \lambda, domain=\mathbb{Z} \right)}\]
In [125]:
factor(p)
Out[125]:
\[\left(\lambda - 5\right)^{2} \left(\lambda - 3\right) \left(\lambda + 2\right)\]

Solving Algebraic and Differential Equations

In [126]:
solve(x**2 - 1, x)
Out[126]:
\[\left [ -1, \quad 1\right ]\]
In [127]:
solve((x - y + 2, x + y -3), (x, y))
Out[127]:
\[\left \{ x : \frac{1}{2}, \quad y : \frac{5}{2}\right \}\]
In [128]:
solve(x**3 - 6*x**2 + 9*x, x)
Out[128]:
\[\left [ 0, \quad 3\right ]\]
In [129]:
roots(x**3 - 6*x**2 + 9*x, x)
Out[129]:
\[\left \{ 0 : 1, \quad 3 : 2\right \}\]
In [130]:
f, g = symbols('f g', cls=Function)
In [131]:
f(x).diff(x)
Out[131]:
\[\frac{d}{d x} f{\left (x \right )}\]
In [132]:
diffeq = Eq(f(x).diff(x, 2) - 2*f(x).diff(x) + f(x), sin(x))
In [133]:
diffeq
Out[133]:
\[f{\left (x \right )} - 2 \frac{d}{d x} f{\left (x \right )} + \frac{d^{2}}{d x^{2}} f{\left (x \right )} = \sin{\left (x \right )}\]
In [134]:
dsolve(diffeq, f(x))
Out[134]:
\[f{\left (x \right )} = \left(C_{1} + C_{2} x\right) e^{x} + \frac{1}{2} \cos{\left (x \right )}\]
In [135]:
dsolve(f(x).diff(x)*(1 - sin(f(x))), f(x))
Out[135]:
\[f{\left (x \right )} + \cos{\left (f{\left (x \right )} \right )} = C_{1}\]
In [136]:
a, t = symbols('a t')
f(t).diff(t)
diffeq = Eq(f(t).diff(t), a*t)
In [137]:
diffeq
Out[137]:
\[\frac{d}{d t} f{\left (t \right )} = a t\]
In [138]:
dsolve(diffeq, f(t))
Out[138]:
\[f{\left (t \right )} = C_{1} + \frac{a t^{2}}{2}\]
In [139]:
x = symbols('x', cls=Function)
diffeq = Eq(x(t).diff(t), a*x(t))
diffeq
Out[139]:
\[\frac{d}{d t} x{\left (t \right )} = a x{\left (t \right )}\]
In [140]:
dsolve(diffeq, x(t))
Out[140]:
\[x{\left (t \right )} = C_{1} e^{a t}\]

Numerics

In [141]:
N(pi, 10)
Out[141]:
\[3.141592654\]
In [142]:
x = symbols('x')
In [143]:
expr = Integral(sin(x)/(x**2), (x, 1, oo))
In [144]:
expr.evalf()
Out[144]:
\[0.5\]
In [145]:
expr.evalf(maxn=20)
Out[145]:
\[0.5\]
In [146]:
expr.evalf(quad='osc')
Out[146]:
\[0.504067061906928\]
In [147]:
expr.evalf(20, quad='osc')
Out[147]:
\[0.50406706190692837199\]
In [148]:
expr = Integral(sin(1/x), (x, 0, 1))
expr
Out[148]:
\[\int_{0}^{1} \sin{\left (\frac{1}{x} \right )}\, dx\]
In [149]:
expr.evalf()
Out[149]:
\[0.5\]
In [150]:
expr = expr.transform(x, 1/x)
expr
Out[150]:
\[\int_{1}^{\infty} \frac{1}{x^{2}} \sin{\left (x \right )}\, dx\]
In [151]:
expr.evalf(quad='osc')
Out[151]:
\[0.504067061906928\]
In [152]:
nsimplify(pi, tolerance=0.001)
Out[152]:
\[\frac{355}{113}\]
In [153]:
expr = sin(x)/x
In [154]:
%timeit expr.evalf(subs={x: 3.14})
1000 loops, best of 3: 397 µs per loop
In [155]:
f1 = lambdify(x, expr)
%timeit f1(3.14)
The slowest run took 13.44 times longer than the fastest. This could mean that an intermediate result is being cached
1000000 loops, best of 3: 308 ns per loop
In [156]:
f2 = lambdify(x, expr, 'numpy')
%timeit f2(3.14)
The slowest run took 16.41 times longer than the fastest. This could mean that an intermediate result is being cached
100000 loops, best of 3: 2.2 µs per loop
In [157]:
%timeit f2(np.linspace(1, 10, 10000))
1000 loops, best of 3: 306 µs per loop
In [158]:
%timeit [f1(x) for x in np.linspace(1, 10, 10000)]
100 loops, best of 3: 5.19 ms per loop
In [159]:
from mpmath import *
In [160]:
f = odefun(lambda x, y: [-y[1], y[0]], 0, [1, 0])
for x in [0, 1, 2.5, 10]:
    nprint(f(x), 15)
    nprint([cos(x), sin(x)], 15)
[1.0, 0.0]
[1.0, 0.0]
[0.54030230586814, 0.841470984807897]
[0.54030230586814, 0.841470984807897]
[-0.801143615546934, 0.598472144103957]
[-0.801143615546934, 0.598472144103957]
[-0.839071529076452, -0.54402111088937]
[-0.839071529076452, -0.54402111088937]
In [161]:
from sympy.plotting import plot
%matplotlib inline

plot(x*y**3 - y*x**3)
pass
_images/10_SymbolicAlgebra_167_0.png
In [162]:
from sympy.plotting import plot3d_parametric_surface
from sympy import sin, cos
u, v = symbols('u v')
plot3d_parametric_surface(cos(u + v), sin(u - v), u-v, (u, -5, 5), (v, -5, 5))
pass
_images/10_SymbolicAlgebra_168_0.png

Statistics

In [163]:
from sympy.stats import *
In [164]:
k = Symbol("k", positive=True)
theta = Symbol("theta", positive=True)
z = Symbol("z")
X = Gamma("x", k, theta)
In [165]:
D = density(X)(z)
D
Out[165]:
\[\frac{z^{k - 1} e^{- \frac{z}{\theta}}}{\theta^{k} \Gamma{\left(k \right)}}\]
In [166]:
C = cdf(X, meijerg=True)(z)
C
Out[166]:
\[\begin{split}\begin{cases} - \frac{k \gamma\left(k, 0\right)}{\Gamma{\left(k + 1 \right)}} + \frac{k \gamma\left(k, \frac{z}{\theta}\right)}{\Gamma{\left(k + 1 \right)}} & \text{for}\: z \geq 0 \\0 & \text{otherwise} \end{cases}\end{split}\]
In [167]:
E(X)
Out[167]:
\[\frac{\theta}{\Gamma{\left(k \right)}} \Gamma{\left(k + 1 \right)}\]
In [168]:
V = variance(X)
V
Out[168]:
\[\frac{\theta^{3} \theta^{k - 1}}{\theta^{k} \Gamma^{2}{\left(k \right)}} \Gamma^{2}{\left(k + 1 \right)} - \frac{2 \theta^{2}}{\Gamma^{2}{\left(k \right)}} \Gamma^{2}{\left(k + 1 \right)} + \frac{\theta \theta^{k + 1}}{\theta^{k} \Gamma{\left(k \right)}} \Gamma{\left(k + 2 \right)}\]
In [169]:
simplify(V)
Out[169]:
\[k \theta^{2}\]
In [170]:
N = Normal('Gaussian', 10, 5)
density(N)(z)
Out[170]:
\[\frac{\sqrt{2}}{10 \sqrt{\pi}} e^{- \frac{1}{50} \left(z - 10\right)^{2}}\]
In [171]:
density(N)(3).evalf()
Out[171]:
\[0.029945493127149\]
In [172]:
simplify(cdf(N)(z))
Out[172]:
\[\frac{1}{2} \operatorname{erf}{\left (\frac{\sqrt{2}}{10} \left(z - 10\right) \right )} + \frac{1}{2}\]
In [173]:
P(N > 10)
Out[173]:
\[\frac{1}{2}\]
In [174]:
sample(N)
Out[174]:
\[6.30254802079348\]