# SymPy

This is the dream cheating software every student should know about.
It also has serious applications obviously. www.sympy.org/scipy-2017-codegen-tutorial/ mentions code generation capabilities, which sounds super cool!
The code in this section was tested on sympy==1.8 and Python 3.9.5.
from sympy import *
sympify(2)/3 + sympify(1)/2
outputs:
7/6
Note that this is an exact value, it does not get converted to floating-point numbers where precision could be lost!
We can also do everything with symbols:
from sympy import *
x, y = symbols('x y')
expr = x/3 + y/2
print(expr)
outputs:
x/3 + y/2
We can now evaluate that expression object at any time:
expr.subs({x: 1, y: 2})
outputs:
4/3
x = sqrt(2)
print(x)
outputs:
sqrt(2)
so we understand that the value was kept without simplification. And of course:
sqrt(2)**2
outputs 2. Also:
sqrt(-1)
outputs:
I
I is the imaginary unit. We can use that symbol directly as well, e.g.:
I*I
gives:
-1
Let's do some trigonometry:
cos(pi)
gives:
-1
and:
cos(pi/4)
gives:
sqrt(2)/2
The exponential also works:
exp(I*pi)
gives;
-1
Now for some calculus. To find the derivative of the natural logarithm:
from sympy import *
x = symbols('x')
diff(ln(x), x)
outputs:
1/x
Just read that. One over x. Beauty.
Let's do some more. Let's solve a simple differential equation:
y''(t) - 2y'(t) + y(t) = sin(t)
Doing:
from sympy import *
x = symbols('x')
f, g = symbols('f g', cls=Function)
diffeq = Eq(f(x).diff(x, x) - 2*f(x).diff(x) + f(x), sin(x)**4)
print(dsolve(diffeq, f(x)))
outputs:
Eq(f(x), (C1 + C2*x)*exp(x) + cos(x)/2)
which means:
To be fair though, it can't do anything crazy, it likely just goes over known patterns that it has solvers for, e.g. if we change it to:
diffeq = Eq(f(x).diff(x, x)**2 + f(x), 0)
it just blows up:
NotImplementedError: solve: Cannot solve f(x) + Derivative(f(x), (x, 2))**2
Let's try some polynomial equations:
from sympy import *
x, a, b, c = symbols('x a b c d e f')
eq = Eq(a*x**2 + b*x + c, 0)
sol = solveset(eq, x)
print(sol)
which outputs:
FiniteSet(-b/(2*a) - sqrt(-4*a*c + b**2)/(2*a), -b/(2*a) + sqrt(-4*a*c + b**2)/(2*a))
which is a not amazingly nice version of the quadratic formula. Let's evaluate with some specific constants after the fact:
sol.subs({a: 1, b: 2, c: 3})
which outputs
FiniteSet(-1 + sqrt(2)*I, -1 - sqrt(2)*I)
Let's see if it handles the quartic equation:
x, a, b, c, d, e, f = symbols('x a b c d e f')
eq = Eq(e*x**4 + d*x**3 + c*x**2 + b*x + a, 0)
solveset(eq, x)
Something comes out. It takes up the entire terminal. Naughty. And now let's try to mess with it:
x, a, b, c, d, e, f = symbols('x a b c d e f')
eq = Eq(f*x**5 + e*x**4 + d*x**3 + c*x**2 + b*x + a, 0)
solveset(eq, x)
and this time it spits out something more magic:
ConditionSet(x, Eq(a + b*x + c*x**2 + d*x**3 + e*x**4 + f*x**5, 0), Complexes)
Oh well.
Let's try some linear algebra.
m = Matrix([[1, 2], [3, 4]])
Let's invert it:
m**-1
outputs:
Matrix([
[ -2,    1],
[3/2, -1/2]])