On Convergence Rates of Root-Seeking Methods

[Claudio L.]

(03-14-2018 11:26 PM)Mike (Stgt) Wrote:  So I correct my statement: this Precise Calculator is not too bad, and fast. Sufficient for many problems.

There's one thing preccalc does differently from the rest. I struggled with this at first, then I read the docs and I realized what it does: It searches for the solution at dynamic precision using an iterative method, until the final result doesn't change for the selected precision. But it uses more than the requested precision for intermediate steps (as needed for convergence).
I say I struggled because I was trying to get the rounding error to match after several operations and this one always got me the exact result, no rounding error.
If you try asin(acos(atan(tan(cos(sin(9)))) with 8000 digits, for example (in Deg), you'll see a result with 40 or 50 digits correct, then several seconds later it goes to 9 exactly and displays the total time of computation only after it reaches convergence (on my system that's around 9 seconds). That explains the difference in speed with Calc. This one does the same operation many times, increasing precision each time, while Calc does each operation only once, at the requested precision. In this case, Calc would report some rounding error (different from any other calculator since it uses 'a/b' representation of numbers rather than 'a*2^b' or 'a*10^b', therefore rounding errors will be very different, but will be there), while preccalc won't stop searching until it reports the exact final result of 9.
PS: Try 9-asin(acos(atan(tan(cos(sin(9)))) at 3000 digits and watch it go nuts refining the solution. It seems to have a cutoff, reporting zero when the result is < 10^-(1.3*Prec).