Newtons Method for Cube Root.ipynb

Kernel: Python 3 (system-wide)

Newton's Method to approximate cube roots

Approximating $\sqrt[3]{4}$

def newton(c, dc, x, n, err):
    i=0
    while abs(c(x)) > err and i <=n:
        x=x-c(x)/dc(x)
        i+=1
    if i > n:
        return False
    else:
        return x

def cubic(x):
    return x**3-4

def cubicDerivative(x):
    return 3*x**2

print("cube root of 4 = 4^(1/3) =", 4**(1/3), '\n')
print("Using Newton's Method:", '\n')
print("1st guess = 1:", '\n', newton(cubic, cubicDerivative, 1, 4, 0.0001), '\n')
print("2nd guess = 1.5:", '\n', newton(cubic, cubicDerivative, 1.4, 5, 0.0001),'\n')
print("3rd guess = 1.5 (decreasing err to 0.0000000001):", '\n', newton(cubic, cubicDerivative, 1.5, 4, 0.0000000001),"<-- this is the most accurate")

cube root of 4 = 4^(1/3) = 1.5874010519681994 

Using Newton's Method: 

1st guess = 1: 
 1.5874096961416333 

2nd guess = 1.5: 
 1.587401164777749 

3rd guess = 1.5 (decreasing err to 0.0000000001): 
 1.5874010519681996 <-- this is the most accurate