bug#58929: 29.0.50; Calc: finding roots utpn doesn't work as advertised

[Mattias Engdegård]

1 nov. 2022 kl. 23.00 skrev Matt Armstrong <matt@rfc20.org>:

> See attached patch for what I wish the docs had said all along
> (would have saved me a lot of time).

Thank you, I have not worked through your table to confirm all the details but 
it looks fine on the whole.

One detail: it's not just that the functions are "flat" (ie, all derivatives 
rapidly vanishing as we stray from the mean) but at least for the normal 
distribution, Newton actively diverges with a bad starting guess even if 
computed with infinite precision (draw the graph).

> I wonder if the implementation can be improved.
> https://www.gnu.org/software/emacs/manual/html_node/calc/Root-Finding.html
> suggests that there are some cases where calc will switch over to the
> bisection method.  Perhaps the heuristics can be tweaked?

From what I can tell, bisection is used if there is an interval bracketing the 
root and either no derivative could be computed or Newton fails to converge. 
Clearly Newton is preferable when it converges since it's much faster than 
bisection.

We could employ special tactics by pattern-matching the function expression 
(perhaps according to your table); not sure if it would be worth the trouble.

Of course an actual numerical analyst would know immediately what to do. Is 
there one in the audience?