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sympy quick tutorial & example

SymPy notebook¶

This is a note about learning SymPy library¶

http://docs.sympy.org/latest/index.html¶

import sympy and other library¶

In [1]:

import sympy
import numpy as np
import matplotlib.pyplot as plt
sympy.plotting
%matplotlib inline

pretty printing (as latex) setting¶

In [2]:

sympy.init_printing()

define variables¶

In [3]:

x, y, k, t = sympy.symbols('x y k t')

symbolic expression¶

sqrt example¶

In [4]:

sympy.sqrt(9)

Out[4]:

$$3$$

In [5]:

sympy.sqrt(8)

Out[5]:

$$2 \sqrt{2}$$

derivative¶

In [6]:

expr = sympy.sin(k*x)
expr

Out[6]:

$$\sin{\left (k x \right )}$$

In [7]:

sympy.diff(expr, x)

Out[7]:

$$k \cos{\left (k x \right )}$$

In [8]:

sympy.diff(expr, x, x)

Out[8]:

$$- k^{2} \sin{\left (k x \right )}$$

integrate¶

In [9]:

expr = sympy.exp(x)*sympy.sin(x) + sympy.exp(x)*sympy.cos(x)
expr

Out[9]:

$$e^{x} \sin{\left (x \right )} + e^{x} \cos{\left (x \right )}$$

In [10]:

sympy.integrate(expr, x)

Out[10]:

$$e^{x} \sin{\left (x \right )}$$

In [11]:

expr = sympy.sin(sympy.pi*x)
expr

Out[11]:

$$\sin{\left (\pi x \right )}$$

In [12]:

sympy.integrate(expr, (x, 0, 1))

Out[12]:

$$\frac{2}{\pi}$$

limit¶

In [13]:

sympy.limit(sympy.sin(2*x)/x, x, 0)

Out[13]:

$$2$$

slove equation¶

In [14]:

sympy.solve(x**2 - 2, x)

Out[14]:

$$\left [ - \sqrt{2}, \quad \sqrt{2}\right ]$$

ODE¶

In [15]:

sympy.Eq(y(t).diff(t, t) - y(t))

Out[15]:

$$- y{\left (t \right )} + \frac{d^{2}}{d t^{2}} y{\left (t \right )} = 0$$

In [16]:

sympy.dsolve(sympy.Eq(y(t).diff(t, t) - y(t), sympy.exp(t)), y(t))

Out[16]:

$$y{\left (t \right )} = C_{2} e^{- t} + \left(C_{1} + \frac{t}{2}\right) e^{t}$$

In [17]:

eq = sympy.Eq(y(t).diff(t) - y(t))

In [18]:

sympy.dsolve(eq, y(t))

Out[18]:

$$y{\left (t \right )} = C_{1} e^{t}$$

print latex forma¶

In [19]:

sympy.latex(sympy.Integral(sympy.cos(x)**2, (x, 0, sympy.pi)))

Out[19]:

'\\int_{0}^{\\pi} \\cos^{2}{\\left (x \\right )}\\, dx'

In [20]:

%%latex
$\int_{0}^{\pi} \cos^{2}{\left (x \right )}\, dx$

$\int_{0}^{\pi} \cos^{2}{\left (x \right )}\, dx$

In [21]:

pwd

Out[21]:

'/Users/ap9035/Documents/ScientificComputing/sympy'

substitute symbol with value¶

In [22]:

expr = x+y**2
expr

Out[22]:

$$x + y^{2}$$

method 1¶

In [23]:

expr.subs({x:2, y:3})

Out[23]:

$$11$$

method 2¶

In [24]:

expr.subs(x, 2)

Out[24]:

$$y^{2} + 2$$

method 3¶

In [25]:

expr.subs([(x, 2), (y, 3)])

Out[25]:

$$11$$

Symbolic Equation¶

using sympy.Eq to create equation¶

In [26]:

eq1 = sympy.Eq(x**2, 1)
eq1

Out[26]:

$$x^{2} = 1$$

In [27]:

sympy.solve(eq1, x)

Out[27]:

$$\left [ -1, \quad 1\right ]$$

Equals signs¶

In [28]:

eq1 = (x+y)**2
eq1

Out[28]:

$$\left(x + y\right)^{2}$$

In [29]:

eq2 = x**2+2*x*y+y**2
eq2

Out[29]:

$$x^{2} + 2 x y + y^{2}$$

In [30]:

sympy.simplify(eq1-eq2)

Out[30]:

$$0$$

In [31]:

eq1.equals(eq2)

Out[31]:

True

Cover string to SymPy expressions¶

In [32]:

# a and b must define first
a, b = sympy.symbols('a b')
express_string = "a**2+b**2+5"
expr = sympy.simplify(express_string)

In [33]:

expr

Out[33]:

$$a^{2} + b^{2} + 5$$

In [34]:

expr.subs(a, 2)

Out[34]:

$$b^{2} + 9$$

Evaluate SymPy expressions into float number¶

In [35]:

expr.subs([(a, 2), (b, 3)]).evalf()

Out[35]:

$$18.0$$

In [36]:

sympy.pi.evalf(100)

Out[36]:

$$3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068$$

lambdify¶

In [37]:

expr = a**2+b
expr

Out[37]:

$$a^{2} + b$$

In [38]:

f = sympy.lambdify([a, b], expr, "numpy")

In [39]:

a = np.arange(5)
b = 1
print(a)
print(b)
f(a, b)

[0 1 2 3 4]
1

Out[39]:

array([ 1,  2,  5, 10, 17])

plotting function¶

http://docs.sympy.org/dev/modules/plotting.html

In [40]:

p1 = sympy.plotting.plot(x**3)
p2 = sympy.plotting.plot(sympy.sin(x))
p3 = sympy.plotting.plot(sympy.sin(x), sympy.cos(x))
p3[0].line_color = 'green'
p3[1].line_color = 'red'
p3.xlabel='x'
p3.ylabel='y'
p3.show()

random variable¶

http://docs.sympy.org/latest/modules/stats.html

sympy.stats.Die(name, items)¶

Create a dice which have [items] face¶

In [41]:

from sympy.stats import Die, density, Normal, Uniform
from sympy import symbols, Symbol

In [42]:

D6 = Die('D6', 6)
density(D6).dict

Out[42]:

$$\left \{ 1 : \frac{1}{6}, \quad 2 : \frac{1}{6}, \quad 3 : \frac{1}{6}, \quad 4 : \frac{1}{6}, \quad 5 : \frac{1}{6}, \quad 6 : \frac{1}{6}\right \}$$

sympy.stats.Normal(name, mean, std)¶

normal distrubution¶

In [43]:

z, mu = symbols('z mu')
sigma = Symbol("sigma", positive=True)
X = Normal("X", mu, sigma)
expr = density(X)(z)
expr

Out[43]:

$$\frac{\sqrt{2}}{2 \sqrt{\pi} \sigma} e^{- \frac{\left(- \mu + z\right)^{2}}{2 \sigma^{2}}}$$

In [44]:

ee = expr.subs([(mu, 0), (sigma, 5)])
sympy.plot(ee, (z, -30, 30))
ee.subs(z, 5)

Out[44]:

$$\frac{\sqrt{2}}{10 \sqrt{\pi} e^{\frac{1}{2}}}$$

In [45]:

sympy.integrate(expr, (z, -sympy.oo, mu))

Out[45]:

$$\frac{1}{2}$$

Uniform distrubution¶

In [46]:

a = Symbol("a", negative=True)
b = Symbol("b", positive=True)
X = Uniform("x", a, b)
expr = density(X)(x)
expr

Out[46]:

$$\begin{cases} \frac{1}{- a + b} & \text{for}\: a \leq x \wedge x \leq b \\0 & \text{otherwise} \end{cases}$$

In [47]:

sympy.integrate(expr, (x, a, b))

Out[47]:

$$- \frac{a}{- a + b} + \frac{b}{- a + b}$$

In [48]:

sympy.simplify(sympy.integrate(expr, (x, a, b)))

Out[48]:

$$1$$

In [49]:

expr2 = expr.subs([(a, 0), (b, 10)])
expr2

Out[49]:

$$\begin{cases} \frac{1}{10} & \text{for}\: 0 \leq x \wedge x \leq 10 \\0 & \text{otherwise} \end{cases}$$

In [50]:

sympy.simplify(expr2)
print(type(expr2))

Piecewise