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Free:
- rutgers-pxk-416 chapter "Memory management: lecture notes"
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SymPy Updated 2025-07-16
It also has serious applications obviously. www.sympy.org/scipy-2017-codegen-tutorial/ mentions code generation capabilities, which sounds super cool!
Let's start with some basics. fractions:outputs:Note that this is an exact value, it does not get converted to floating-point numbers where precision could be lost!
from sympy import *
sympify(2)/3 + sympify(1)/2
7/6
We can also do everything with symbols:outputs:We can now evaluate that expression object at any time:outputs:
from sympy import *
x, y = symbols('x y')
expr = x/3 + y/2
print(expr)
x/3 + y/2
expr.subs({x: 1, y: 2})
4/3
How about a square root?outputs:so we understand that the value was kept without simplification. And of course:outputs outputs:gives:
x = sqrt(2)
print(x)
sqrt(2)
sqrt(2)**2
2
. Also:sqrt(-1)
I
I
is the imaginary unit. We can use that symbol directly as well, e.g.:I*I
-1
Let's do some trigonometry:gives:and:gives:The exponential also works:gives;
cos(pi)
-1
cos(pi/4)
sqrt(2)/2
exp(I*pi)
-1
Now for some calculus. To find the derivative of the natural logarithm:outputs:Just read that. One over x. Beauty. And now for some integration:outputs:OK.
from sympy import *
x = symbols('x')
print(diff(ln(x), x))
1/x
print(integrate(1/x, x))
log(x)
Let's do some more. Let's solve a simple differential equation:Doing:outputs: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:it just blows up:Sad.
y''(t) - 2y'(t) + y(t) = sin(t)
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)))
Eq(f(x), (C1 + C2*x)*exp(x) + cos(x)/2)
diffeq = Eq(f(x).diff(x, x)**2 + f(x), 0)
NotImplementedError: solve: Cannot solve f(x) + Derivative(f(x), (x, 2))**2
Let's try some polynomial equations:which outputs:which is a not amazingly nice version of the quadratic formula. Let's evaluate with some specific constants after the fact:which outputsLet's see if it handles the quartic equation:Something comes out. It takes up the entire terminal. Naughty. And now let's try to mess with it:and this time it spits out something more magic:Oh well.
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)
FiniteSet(-b/(2*a) - sqrt(-4*a*c + b**2)/(2*a), -b/(2*a) + sqrt(-4*a*c + b**2)/(2*a))
sol.subs({a: 1, b: 2, c: 3})
FiniteSet(-1 + sqrt(2)*I, -1 - sqrt(2)*I)
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)
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)
ConditionSet(x, Eq(a + b*x + c*x**2 + d*x**3 + e*x**4 + f*x**5, 0), Complexes)
Let's try some linear algebra.Let's invert it:outputs:
m = Matrix([[1, 2], [3, 4]])
m**-1
Matrix([
[ -2, 1],
[3/2, -1/2]])
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