Symbolic Computing¶

Symbolic computing allows you to use a computer to do all the algebra you might otherwise do by hand.

SymPy is a popular package for symbolic computing.

conda install sympy

import sympy as sym
import math
import numpy as np

A good place to start is the SymPy tutorial

Let’s compare how sympy evaluates sqrt vs. math

print(math.sqrt(8))
print(np.sqrt(8))

2.8284271247461903
2.8284271247461903

sym.sqrt(8)

\[\displaystyle 2 \sqrt{2}\]

Note that jupyter notebooks render the output nicely.

the math package gives a floating point approximation to \(\sqrt{8}\), whereas sympy simplifies the expression by removing any divisors that are perfect squares.

a = sym.sqrt(8)
print(a)
type(a)

2*sqrt(2)

sympy.core.mul.Mul

Algebraic Expressions¶

Symbolic computing is particularly useful for manipulating algebraic expressions.

from sympy import symbols, expand, factor

You can build expressions using symbols:

x, y = symbols('x y')
expr = 2*x + y
expr

\[\displaystyle 2 x + y\]

Let’s say we forgot how to mulitply monomials

a, b, c, d = symbols('a b c d')
x, y = symbols('x, y')
ex1 = a*x + b
ex2 = c*x + d
ex1 * ex2

\[\displaystyle \left(a x + b\right) \left(c x + d\right)\]

we can expand an expression by multiplying everything out and combining like terms:

ex = expand(ex1 * ex2)
ex

\[\displaystyle a c x^{2} + a d x + b c x + b d\]

We can also factor an expression that has been expanded:

factor(ex)

\[\displaystyle \left(a x + b\right) \left(c x + d\right)\]

You can substitute values (or other symbols) for symbols:

ex.subs([(a, 1), (b, 2), (c, 3), (d, y)])

\[\displaystyle 3 x^{2} + x y + 6 x + 2 y\]

ex_sub = ex.subs([(a, 1), (b, 2), (c, 2), (d, 3)])
ex_sub

\[\displaystyle 2 x^{2} + 7 x + 6\]

You can evaluate expressions numerically:

ex_sub.evalf(subs={x: 0.1})

\[\displaystyle 6.72\]

SymPy uses arbitrary precision arithmetic. The default precision is 10 decimal digits (between single and double precision), but you can increase the precision in evalf

sym.pi.evalf()

\[\displaystyle 3.14159265358979\]

sym.pi.evalf(100)

\[\displaystyle 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068\]

Symbolic Mathematics¶

Sympy supports just about everything you might have learned in courses on algebra and calculus (and more)

import sympy as sym

Differentiation¶

expr = sym.sin(x)
print(expr)
print(sym.diff(expr)) # symbolic differentiation w.r.t x

sin(x)
cos(x)

expr = x * y
sym.diff(expr, x) # differentiate w.r.t. x

\[\displaystyle y\]

Integration¶

expr = sym.sin(x)
print(expr)
print(sym.integrate(expr, x)) # indefinite integral w.r.t x

sin(x)
-cos(x)

the sympy variable oo (two o symbols) is used to represent \(\infty\)

from sympy import oo
expr = sym.exp(-(x**2))
sym.integrate(expr, (x, -oo, oo)) # integration with limits

\[\displaystyle \sqrt{\pi}\]

sym.integrate(expr, (x, -1, 1))

\[\displaystyle \sqrt{\pi} \operatorname{erf}{\left(1 \right)}\]

Limits¶

You can compute limits. For example: \begin{equation} \lim_{x\to 0} \frac{\sin(x)}{x} \end{equation}

sym.limit(sym.sin(x)/x, x, 0) # limit as x \to 0

\[\displaystyle 1\]

Roots¶

You can compute roots of functions using solve (i.e. we solve the system f(x) = 0 for x)

sym.solve(x**2 - 3, x)

[-sqrt(3), sqrt(3)]

Symbolic to Numerical Functions¶

You can turn sympy functions into lambda functions that are compatible with numpy using lambdify:

f = sym.sin(x)
g = sym.lambdify(x, f, 'numpy')
g(np.array(np.linspace(0,1,10)))

array([0.        , 0.11088263, 0.22039774, 0.3271947 , 0.42995636,
       0.52741539, 0.6183698 , 0.70169788, 0.77637192, 0.84147098])

Plotting Functions¶

You can plot a symbolic function using the sympy plot function

f = sym.sin(x)
sym.plot(f)

<sympy.plotting.plot.Plot at 0x7fe8e85f9c70>

you can change the domain using a tuple with the variable and lower and upper bounds

sym.plot(f, (x, -5,5))

<sympy.plotting.plot.Plot at 0x7fe8cfac6fa0>

There are a variety of keyword arguments you can use for formatting as well. See help(sym.plot) for more information.

Exercises¶

Use sympy to compute solutions to answer the following questions

What is \(\int_{1}^{\infty} x^{-2}\, dx\)?
Find a value of x where sin(x) = cos(x)

sym.integrate((x)**(-2), (x, 1, oo))