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So far, my simple numerical analysis program has differentiation, integration, first-order ODEs and Taylor Series. What else can I add to this?

```
from math import*
"""Single-variable calculus."""
#DIFFERENTIATION
def gen_differentiation(f,x,h):
#finds the first derivative of any mathematical function
return (f(x+h)-f(x))/(h)
def special_differentiation(f,x):
#finds the first derivative of specific functions
if (f == sin(x)):
return cos(x)
elif (f == cos(x)):
return -sin(x)
elif (f == tan(x)):
return 1/(sin(x)*sin(x))
elif (f == exp(x)):
return exp(x)
elif (f == log(x,e)):
return 1/x
else:
return "Not a special function."
def second_derivative(f,x,h):
#Finds the second derivative of any mathematical function
return (f(x+h)-2*f(x)+f(x-h))/(h*h)
def third_derivative(f,x,h):
#Finds the third derivative of any mathematical function
numer = (h*h)*(gen_differentiation(f,x+h,h)-2*gen_differentiation(f,x,h)+\
gen_differentiation(f,x-h,h))
denom = h**4
return numer/denom
#INTEGRATION
def integration(f,a,b): #Using Simpson's Rule
return ((b-a)/6)*(f(a)+4*f(0.5*(a+b))+f(b))
def integration1(f,a,b): #Using Trapezoidal Rule (general)
return (b-a)*0.5*(f(a)+f(b))
def integration2(f,a,b,n): #Using Trapezoidal Rule (strips)
clist = [f(a+i*((b-a)/n)) for i in range(1,n)]
return ((b-a)/n)*(0.5*(f(a)+f(b))+ sum(clist))
#FIRST-ORDER DIFFERENTIAL EQUATIONS
def direct_method(dydx,x1,x2):
#Only works with single-variable
y = integration(dydx, x1, x2)
return y
def midpoint(f,x0,y0,h,limit): #The midpoint method
"""f is the value of dy/dx."""
while x0 <= limit:
k1 = h*f(x0,y0)
k2 = h*f(x0 + 0.5*h, y0 + 0.5*k1)
y0 = y0 + k2
x0 = x0 + h
return y0
def runge_kutta(f,x0,y0,h,limit): #The Runge-Kutta 4th-order Method
"""f is the value of dy/dx."""
while x0 <= limit:
k1 = h*f(x0,y0)
k2 = h*f(x0 + 0.5*h, y0 + 0.5*k1)
k3 = h*f(x0 + 0.5*h, y0 + 0.5*k2)
k4 = h*f(x0 + h, y0 + k3)
y0 = y0+(k1/6)+(k2/3)+(k3/3)+(k4/6)
x0 = x0 + h
return y0
#TAYLOR SERIES
def taylor_series(f,x,a): #Taylor series up to the fourth term
d = gen_differentiation(f,a,0.0000001)
sd = second_derivative(f,a,0.0000001)
td = third_derivative(f,a,0.0001)
second_term = d*(x-a)
third_term = (sd*(x-a)*(x-a))/factorial(2)
fourth_term = (td*(x-a)**3)/factorial(3)
series = [f(a), second_term, third_term, fourth_term]
return sum(series)
```