(PC-12xx~14xx) qthsh Tanh-Sinh quadrature

[Albert Chan]

(04-20-2021 02:19 AM)robve Wrote:  I have not found a single counter example yet that does not work with the predictor.
...
If the diff is zero, we are at a reasonably good or optimal r to adjust d. This can happen very early
for example 1/(1+(x*x)) and log((1+(x*x))/x)/(1+(x*x)) ...

Is these two already counter-examples ?
Both functions, two sides overlapped, f(v)*v = f(1/v)/v

XCas> f1(x) := 1/(1+x*x)
XCas> f2(x) := log((1+(x*x))/x)/(1+(x*x))
XCas> simplify([f1(v)*v-f1(1/v)/v, f2(v)*v-f2(1/v)/v])       → [0, 0]

In other words, f(e^t)*(e^t) is even function, with respect to t.

Technically, there were no sign changes.
But, with rounding errors, if lucky, we expected sign changes early on.
In other words, closed to whatever starting guess of d was.

It just happpened we started with d=1, and d=1 is close to optimal.

Now, try g1(x) = f1(x*10), g2(x) = f2(x*10)

Since this is simple scaling, we expected both have optimal d ≈ 1/10

I don't think you'd catch a sign change here (except for v=1, useless to predict d)