'''
    1. Compute real, positive, root 4th order polynomial. 
        x^4 + 3x^3 -2x -6 = 0 
    by iteration(s). Try the following iterations:
        x^4  (next) = -3x^3 +2x +6
        3x^3 (next) = -x^4  +2x +6
        2x   (next) =  x^4  +3x^3 -6
    From someone who plotted this polynomial 
    we know that there is a single positive root 
    somewhere between x=1.2 and x=1.6.
    
    2. An unverified report says the result should be 
    x = 2**(1/3). Check it analytically and numerically.
'''

def f(x):           # right hand side of iteration eq.
    return ((-x**4 +2.*x +6.)/3.)**(1./3.) 


# main program 
x = 1.4                 # staring from a guess
print("\n    start with x = ",x)
for i in range(1,44):
    incr = f(x) - x
    x = f(x) 
    print("iteration",i,"   x = ",x,"   increment",incr)

if (x == 2.**(1/3) ): 
    print("iteration converged to 2^(1/3) exactly, i.e. to machine precision! \n")
else:
    print("iteration differs from 2^(1/3) by",x-2.**(1/3))