'''
    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.
    Analytical solution x = 2^(1/3)
    
    In this second version we format output a bit better
    & break the iteration when the increment of x drops to 0.
    (addition gives less than machine accuracy, i.e. all 
    digits of x stabilize)
'''
x = 1.4                 # staring from a guess
print("\n 1st guess x= ",x)
for i in range(1,99):
# compute the increment (del_x) of x in this iteration
# this is the r.h.s. of the iteration equation minus x.
    del_x = ((-x**4 +2*x +6)/3)**(1/3) - x
    x = x + del_x
    print('iter. %d   x = %18.16f     del_x = %8.3e' %(i,x,del_x))
    if (del_x==0.): break   # breaks out from the loop when del_x==0.
print("analytically obtained, exact res. is  = 2**(1/3) =")
print("          x = ",2**(1/3))
print('\n')                 # one new line (empty line) after results