Newton's method is also used in the fast reciprocal square root algorithm (computing half of one of the numbers can be done using integer arithmetic (addition o

Newton's method is also used in the fast reciprocal square root algorithm (computing half of one of the numbers can be done using integer arithmetic (addition or subtraction) if you make the assumption that the input number is not too small; I have done this in MMIX). Furthermore, some map projections (e.g. Eckert IV) can be solved using Newton's method (I have implemented this too, if I remember correctly). I wrote my own program to render Newton fractals. You can use the algorithm like: 1. Input the list of the roots. If the roots are called c0, c1, etc then you will have: (z-c0)(z-c1)(z-c2)(z-c3)(z-c4)=0 2. Make the equation expanded by multiplying them together. 3. Make the derivative (which can now be made easily because it is expanded). 4. For each point of the picture, make Newton's method as many times as it is specified. If it exceeds the number of maximum iterations then make it transparent, otherwise use the colour of the root point that it reaches closely enough.