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The generalized form of Newton's method uses something called the relaxation parameter to increase or decrease the step size in order to improve the convergence rate. This value should be close to one and tuned for the function you're trying to solve since random values are liable to make the method slower rather than faster. If you use significantly larger values (positive or negative), Newton's method completely fails to converge but you can still end up with something interesting.

The black areas are where iteration stopped due to the denominator being zero. For every other pixel iteration stops due to the method producing NaNs; these happen when numerator and denominator are both infinity.

Click on the links for live versions that you can play with.