In [2]:
import math
math.sqrt(3)
Out[2]:
1.7320508075688772
In [3]:
math.sqrt(8)
Out[3]:
2.8284271247461903
$\sqrt(8) = 2\sqrt(2)$, but it's hard to see that here
In [4]:
import sympy
sympy.sqrt(3)
Out[4]:
$\displaystyle \sqrt{3}$
SymPy can even simplify symbolic computations
In [5]:
sympy.sqrt(8)
Out[5]:
$\displaystyle 2 \sqrt{2}$
In [6]:
from sympy import symbols
x, y = symbols('x y')
expr = x + 2*y
expr
Out[6]:
$\displaystyle x + 2 y$
Note that simply adding two symbols creates an expression. Now let's play around with it.
In [7]:
expr + 1
Out[7]:
$\displaystyle x + 2 y + 1$
In [8]:
expr - x
Out[8]:
$\displaystyle 2 y$
Note that expr - x was not x + 2y -x
In [9]:
x*expr
Out[9]:
$\displaystyle x \left(x + 2 y\right)$
In [10]:
from sympy import expand, factor
expanded_expr = expand(x*expr)
expanded_expr
Out[10]:
$\displaystyle x^{2} + 2 x y$
In [11]:
factor(expanded_expr)
Out[11]:
$\displaystyle x \left(x + 2 y\right)$
In [12]:
from sympy import diff, sin, exp
diff(sin(x)*exp(x), x)
Out[12]:
$\displaystyle e^{x} \sin{\left(x \right)} + e^{x} \cos{\left(x \right)}$
In [13]:
from sympy import limit
limit(sin(x)/x, x, 0)
Out[13]:
$\displaystyle 1$
Exercise¶
Solve $x^2 - 2 = 0$ using sympy.solve
In [13]:
# Type solution here
from sympy import solve
Pretty printing¶
In [14]:
from sympy import init_printing, Integral, sqrt
init_printing(use_latex='mathjax')
In [15]:
Integral(sqrt(1/x), x)
Out[15]:
$\displaystyle \int \sqrt{\frac{1}{x}}\, dx$
In [16]:
from sympy import latex
latex(Integral(sqrt(1/x), x))
Out[16]:
'\\int \\sqrt{\\frac{1}{x}}\\, dx'
More symbols. Exercise: fix the following piece of code
In [18]:
# NBVAL_SKIP
# The following piece of code is supposed to fail as it is
# The exercise is to fix the code
expr2 = x + 2*y +3*z
--------------------------------------------------------------------------- NameError Traceback (most recent call last) <ipython-input-18-88f4dc46acc5> in <module> ----> 1 expr2 = x + 2*y +3*z NameError: name 'z' is not defined
Exercise¶
Solve $x + 2*y + 3*z$ for $x$
In [19]:
# Solution here
from sympy import solve
Difference between symbol name and python variable name
In [20]:
x, y = symbols("y z")
In [21]:
x
Out[21]:
$\displaystyle y$
In [22]:
y
Out[22]:
$\displaystyle z$
In [23]:
# NBVAL_SKIP
# The following code will error until the code in cell 16 above is
# fixed
z
--------------------------------------------------------------------------- NameError Traceback (most recent call last) <ipython-input-23-3a710d2a84f8> in <module> ----> 1 z NameError: name 'z' is not defined
Symbol names can be more than one character long
In [24]:
crazy = symbols('unrelated')
crazy + 1
Out[24]:
$\displaystyle unrelated + 1$
In [25]:
x = symbols("x")
expr = x + 1
x = 2
What happens when I print expr now? Does it print 3?
In [26]:
print(expr)
x + 1
How do we get 3?
In [27]:
x = symbols("x")
expr = x + 1
expr.subs(x, 2)
Out[27]:
$\displaystyle 3$
Equalities¶
In [28]:
x + 1 == 4
Out[28]:
False
In [29]:
from sympy import Eq
Eq(x + 1, 4)
Out[29]:
$\displaystyle x + 1 = 4$
Suppose we want to ask whether $(x + 1)^2 = x^2 + 2x + 1$
In [30]:
(x + 1)**2 == x**2 + 2*x + 1
Out[30]:
False
In [31]:
from sympy import simplify
a = (x + 1)**2
b = x**2 + 2*x + 1
simplify(a-b)
Out[31]:
$\displaystyle 0$
Exercise¶
Write a function that takes two expressions as input, and returns a tuple of two booleans. The first if they are equal symbolically, and the second if they are equal mathematically.
More operations¶
In [32]:
z = symbols("z")
expr = x**3 + 4*x*y - z
expr.subs([(x, 2), (y, 4), (z, 0)])
Out[32]:
$\displaystyle 36$
In [33]:
from sympy import sympify
str_expr = "x**2 + 3*x - 1/2"
expr = sympify(str_expr)
expr
Out[33]:
$\displaystyle x^{2} + 3 x - \frac{1}{2}$
In [34]:
expr.subs(x, 2)
Out[34]:
$\displaystyle \frac{19}{2}$
In [35]:
expr = sqrt(8)
In [36]:
expr
Out[36]:
$\displaystyle 2 \sqrt{2}$
In [37]:
expr.evalf()
Out[37]:
$\displaystyle 2.82842712474619$
In [38]:
from sympy import pi
pi.evalf(100)
Out[38]:
$\displaystyle 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068$
In [39]:
from sympy import cos
expr = cos(2*x)
expr.evalf(subs={x: 2.4})
Out[39]:
$\displaystyle 0.0874989834394464$
Exercise¶
In [41]:
from IPython.core.display import Image
Image(filename='figures/comic.png')
Out[41]:
Write a function that takes a symbolic expression (like pi), and determines the first place where 789 appears. Tip: Use the string representation of the number. Python starts counting at 0, but the decimal point offsets this
Solving an ODE¶
In [42]:
from sympy import Function
f, g = symbols('f g', cls=Function)
f(x)
Out[42]:
$\displaystyle f{\left(x \right)}$
In [43]:
f(x).diff()
Out[43]:
$\displaystyle \frac{d}{d x} f{\left(x \right)}$
In [44]:
diffeq = Eq(f(x).diff(x, x) - 2*f(x).diff(x) + f(x), sin(x))
diffeq
Out[44]:
$\displaystyle f{\left(x \right)} - 2 \frac{d}{d x} f{\left(x \right)} + \frac{d^{2}}{d x^{2}} f{\left(x \right)} = \sin{\left(x \right)}$
In [45]:
from sympy import dsolve
dsolve(diffeq, f(x))
Out[45]:
$\displaystyle f{\left(x \right)} = \left(C_{1} + C_{2} x\right) e^{x} + \frac{\cos{\left(x \right)}}{2}$
Finite Differences¶
In [48]:
f = Function('f')
dfdx = f(x).diff(x)
dfdx.as_finite_difference()
Out[48]:
$\displaystyle - f{\left(x - \frac{1}{2} \right)} + f{\left(x + \frac{1}{2} \right)}$
In [49]:
from sympy import Symbol
d2fdx2 = f(x).diff(x, 2)
h = Symbol('h')
d2fdx2.as_finite_difference(h)
Out[49]:
$\displaystyle - \frac{2 f{\left(x \right)}}{h^{2}} + \frac{f{\left(- h + x \right)}}{h^{2}} + \frac{f{\left(h + x \right)}}{h^{2}}$
Now that we have seen some relevant features of vanilla SymPy, let's move on to Devito, which could be seen as SymPy finite differences on steroids!