For the Mathematicians.

[doc hawk]

roger reasoned,

> Yet, as I mentioned in the beginning, Graphing tools like Good Grapher on the Mac can do it. And, I wonder how?

My guess would be using either differentials or differences, and drawing from the current point for a small distance, with some watching for branch points.

There are any number of numeric methods to approximate the derivative.   

You could even iterate after each point until “close enough”.

E.g., if your calculated x,y(x) is 4.2, 7.4, you would plug these into the original function and see if it indeed is zero.  

If so, good, go to the next point.  If not, try a smaller move, or plug in adjacent values of 7.3 and 7.5 for y, and see if either gets you closer to zero.

For my dissertation, I developed a third branch of dynamic programming, which built spaces around trial solutions, and then collapsing to successively finer grds.

In the process, I unwittingly reinvented cache memory . . . the search space was *far* to large to keep all potential moves, so they were dropped from the cache and recalculated when needed.

We never did calculate the dimensionality of the baseline problem we were working on—we know that it was *at least* sixty dimensional, but almost certainly past that, as the choice spaces were very much not compact . . .