Re: [math-fun] Quintic equation

Another technique effective if the parameters a,b,c vary systematically, is to get good starting values by extrapolation. E.g. suppose c is stepping up by constant increments in an inner loop, while a,b fixed, and the previous solutions were x_{i-1}, x_{i-2}, .... Then x_{i-1} and x_{i-1} + 2(x_{i-1} - x_{i-2}) are good starters for secant iteration; or x_{i-1} + (x_{i-1} - x_{i-2}) for Newton. Obviously once the loop is underway, the approach could be extended to use higher order methods: difference tables, Newton interpolation, Pad\'e approximation etc. Intelligently applied in a continuous mechanical simulation requiring only low accuracy, this approach often requires just a single iteration to refine the initial guess! Fred Lunnon