Tutorial Brief¶
SymPy is symbolic mathematics library written completely in Python and doesn't require any dependencies.
Finding Help:
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NumPyBase N-dimensional array package |
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SciPyFundamental library for scientific computing |
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MatplotlibComprehensive 2D Plotting |
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IPythonEnhanced Interactive Console |
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SymPySymbolic mathematics |
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PandasData structures & analysis |
SymPy is a Python library for symbolic mathematics. It aims to become a full-featured computer algebra system (CAS) while keeping the code as simple as possible in order to be comprehensible and easily extensible. SymPy is written entirely in Python and does not require any external libraries.¶
Import SymPy¶
In [1]:
from sympy import *
In [2]:
3 + math.sqrt(3)
Out[2]:
4.732050807568877
In [3]:
expr = 3 * sqrt(3)
expr
Out[3]:
3*sqrt(3)
In [4]:
init_printing(use_latex='mathjax')
In [5]:
expr
Out[5]:
$$3 \sqrt{3}$$
In [6]:
expr = sqrt(8)
expr
Out[6]:
$$2 \sqrt{2}$$
symbols() & Symbol()¶
In [7]:
x, y = symbols("x y")
In [8]:
expr = x**2 + y**2
expr
Out[8]:
$$x^{2} + y^{2}$$
In [9]:
expr = (x+y)**3
expr
Out[9]:
$$\left(x + y\right)^{3}$$
Assumptions for symbols¶
In [10]:
a = Symbol("a")
In [11]:
a.is_imaginary
In [12]:
b = Symbol("b", integer=True)
In [13]:
b.is_imaginary
Out[13]:
False
In [14]:
c = Symbol("c", positive=True)
In [15]:
c.is_positive
Out[15]:
True
In [16]:
c.is_imaginary
Out[16]:
False
Imaginary Numbers¶
In [17]:
I
Out[17]:
$$i$$
In [18]:
I ** 2
Out[18]:
$$-1$$
Rational()¶
In [19]:
Rational(1,3)
Out[19]:
$$\frac{1}{3}$$
In [20]:
Rational(1,3) + Rational(1,2)
Out[20]:
$$\frac{5}{6}$$
Numerical evaluation¶
In [21]:
expr = Rational(1,3) + Rational(1,2)
N(expr)
Out[21]:
$$0.833333333333333$$
In [22]:
N(pi, 100)
Out[22]:
$$3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068$$
In [23]:
pi.evalf(100)
Out[23]:
$$3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068$$
subs()¶
In [24]:
expr = x**2 + 2*x + 1
expr
Out[24]:
$$x^{2} + 2 x + 1$$
In [25]:
expr.subs(x, 1)
Out[25]:
$$4$$
In [26]:
expr = pi * x**2
expr
Out[26]:
$$\pi x^{2}$$
In [27]:
expr.subs(x, 3)
Out[27]:
$$9 \pi$$
In [28]:
N(_)
Out[28]:
$$28.2743338823081$$
factor() and expand()¶
In [29]:
expr = (x + y) ** 2
expr
Out[29]:
$$\left(x + y\right)^{2}$$
In [30]:
expand(expr)
Out[30]:
$$x^{2} + 2 x y + y^{2}$$
In [31]:
factor(_)
Out[31]:
$$\left(x + y\right)^{2}$$
simplify()¶
In [32]:
expr = (2*x + Rational(1,3)*x + 4) / x
expr
Out[32]:
$$\frac{1}{x} \left(\frac{7 x}{3} + 4\right)$$
In [33]:
simplify(expr)
Out[33]:
$$\frac{7}{3} + \frac{4}{x}$$
In [34]:
expr = "(2*x + 1/3*x + 4)/x"
simplify(expr)
Out[34]:
$$\frac{7}{3} + \frac{4}{x}$$
In [35]:
expr = sin(x)/cos(x)
expr
Out[35]:
$$\frac{\sin{\left (x \right )}}{\cos{\left (x \right )}}$$
In [36]:
simplify(expr)
Out[36]:
$$\tan{\left (x \right )}$$
apart() and together()¶
In [37]:
expr = 1/(x**2 + 2*x)
expr
Out[37]:
$$\frac{1}{x^{2} + 2 x}$$
In [38]:
apart(expr)
Out[38]:
$$- \frac{1}{2 x + 4} + \frac{1}{2 x}$$
In [39]:
together(_)
Out[39]:
$$\frac{1}{x \left(x + 2\right)}$$
Calculus¶
In [40]:
diff(sin(x), x)
Out[40]:
$$\cos{\left (x \right )}$$
In [41]:
diff(log(x**2 + 1) + 2*x, x)
Out[41]:
$$\frac{2 x}{x^{2} + 1} + 2$$
In [42]:
integrate(cos(x), x)
Out[42]:
$$\sin{\left (x \right )}$$
In [43]:
Integral(sin(x), (x,0,pi))
Out[43]:
$$\int_{0}^{\pi} \sin{\left (x \right )}\, dx$$
In [44]:
N(_)
Out[44]:
$$2.0$$
Sum()¶
In [45]:
expr = Sum(1/(x**2 + 2*x), (x, 1, 10))
expr
Out[45]:
$$\sum_{x=1}^{10} \frac{1}{x^{2} + 2 x}$$
In [46]:
expr.doit()
Out[46]:
$$\frac{175}{264}$$
Product()¶
In [47]:
expr = Product(1/(x**2 + 2*x), (x, 1, 10))
expr
Out[47]:
$$\prod_{x=1}^{10} \frac{1}{x^{2} + 2 x}$$
In [48]:
expr.doit()
Out[48]:
$$\frac{1}{869100503040000}$$
Solve()¶
In [49]:
expr = 2*x + 1
solve(expr)
Out[49]:
$$\begin{bmatrix}- \frac{1}{2}\end{bmatrix}$$
In [50]:
expr = x**2 - 1
solve(expr)
Out[50]:
$$\begin{bmatrix}-1, & 1\end{bmatrix}$$
In [51]:
expr_1 = 2*x + y + 3
expr_2 = 2*y - x
solve([expr_1, expr_2],(x,y))
Out[51]:
$$\begin{Bmatrix}x : - \frac{6}{5}, & y : - \frac{3}{5}\end{Bmatrix}$$
Units¶
In [52]:
from sympy.physics import units as u
In [53]:
5. * u.milligram
Out[53]:
$$5.0 \cdot 10^{-6} kg$$
In [54]:
1./2 * u.inch
Out[54]:
$$0.0127 m$$
In [55]:
1. * u.nano
Out[55]:
$$1.0 \cdot 10^{-9}$$
In [56]:
u.watt
Out[56]:
$$\frac{kg m^{2}}{s^{3}}$$
In [57]:
u.ohm
Out[57]:
$$\frac{kg m^{2}}{A^{2} s^{3}}$$
Converting from Kilometers/hours to Miles/hours¶
In [58]:
kmph = u.km / u.hour
mph = u.mile / u.hour
N(mph / kmph)
Out[58]:
$$1.609344$$
In [69]:
80 * N(mph / kmph)
Out[69]:
$$128.74752$$
Working with NumPy / Pandas and Matplotlib¶
In [59]:
def sympy_expr(x_val):
expr = x**2 + sqrt(3)*x - Rational(1,3)
return expr.subs(x, x_val)
In [60]:
sympy_expr(3)
Out[60]:
$$3 \sqrt{3} + \frac{26}{3}$$
In [61]:
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
In [62]:
list1 = np.arange(1,1000)
list2 = pd.Series(list1)
In [63]:
%timeit [sympy_expr(item) for item in list1]
%timeit [sympy_expr(item) for item in list2]
1 loops, best of 3: 103 ms per loop 10 loops, best of 3: 105 ms per loop
In [64]:
%timeit np.vectorize(sympy_expr)(list1)
%timeit list2.apply(sympy_expr)
10 loops, best of 3: 776 ms per loop 1 loops, best of 3: 1.09 s per loop
In [65]:
expr = x**2 + sqrt(3)*x - Rational(1,3)
lf = lambdify(x, expr)
In [66]:
%timeit lf(list1)
%timeit lf(list2)
1000 loops, best of 3: 242 µs per loop 100 loops, best of 3: 4.8 ms per loop
In [67]:
fig = plt.figure()
axes = fig.add_subplot(111)
x_vals = np.linspace(-5.,5.)
y_vals = lf(x_vals)
axes.grid()
axes.plot(x_vals, y_vals)
plt.show();
