'''
    Compute real, positive, root of the 4th order polynomial:
        x^4 + 3x^3 -2x -6 = 0 
    by iteration(s). Try the following iterations, obtained by 
    leaving only one term containing x on the left-hand side:
      x^4  = -3x^3 +2x +6  => next x <-- [-3x^3 +2x +6]^(1/4)
      3x^3 = -x^4 +2x +6   => next x <-- [(-x^4  +2x +6)/3]^(1/3)
      2x   =  x^4 +3x^3 -6 => next x <-- [(x^4 +3x^3 -6)/2]^(1/3)
    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 that the polynomial 
    is actually (x^3-2)(x+3), so our real positive 
    root should be x= 2^(1/3) ~ 1.26
    Let's check! Our method will then be applicable to polynomials 
    that don't have known factorization/solutions.
'''
x = 1.4                 # starting from this guess
i = 0                   # couter of iterations
while (i < 45) :        # do 45-1 iterations
    i = i + 1           # update counter 
    #  x = (-3*x**3 +2*x +6)**0.25  <-approached two *complex* roots
    x = ( (-x**4 +2*x +6) / 3) **(1/3)  # converges to +-machine prec.
    print("in iteration",i,"   x = ",x)
print("exact sol. x = 2^(1/3)=",2**(1/3) )
    
# now see if the 3rd iteration scheme works!