'''
    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.
    An unverified report says
    
'''
x = 1.4                 # staring from a guess
print("\n    start with x = ",x)
for i in range(1,44):
    # x = (x**4  + 3 * x**3 -6)/2 
    # x = (-3*x**3 +2*x +6)**0.25 
    x = ((-x**4 +2*x +6)/3)**(1./3.) 
    print("iteration",i,"   x = ",x)
print("next increment of x =",((-x**4 +2*x +6)/3)**(1./3.) -x)