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

[Albert Chan]

(04-29-2021 07:33 PM)robve Wrote:  I noticed that when a sign change is detected, peak() also does overall very well, except for one strange case that costs a lot more, 180 points more for log(1+9*(x*x))/(1+16*(x*x)) where d=10.9477 which for a small d<1 is better, meaning the movement of d is in the wrong direction?

Q.peak() got the direction wrong. (with points at 1/4, 1/2, 1. left side intercept ≈ 0.158)

Q.peak() always go for the x-intercept of the right.
In other words, it tried to push bulk of the area to \(\int_a^{a+d}\)
Sometimes that is not appropriate, when we really need more points close to a.

Without knowing the shape of curve, peak for d is the safe choice.

Quote:Albert, you have the following "magic numbers" hardcoded without explanation:
while L/C > 2.353 do
while R/C > 0.425 do
Are these empirically established?

With quadratic fit, last two points cannot be too flat, otherwise tangent line would be way off.
I had based on \(\int_0^∞ e^{-mx}\,dx\), with optimal d ≈ 6/m to 7/m.

Let f(x) = exp(-m*x), g(x) = f(x)*x

With g(2x)/g(x) ≈ 85%, we considered it just passes the peak.

f(2x)/f(x) ≈ 0.85/2 = 0.425, or reverse direction: 1/0.425 = 2.353

With 85% setting, x-intercept roughly hit the sweet spot.

lua> Q = require 'quad'
lua> F = function(m) return function(x) return exp(-m*x) end end
lua> for m=1,9 do p,x=Q.peak(F(m)); print(m, m*x) end
1       5.734205371025561
2       5.734205371025561
3       4.41325021172599
4       5.734205371025561
5       4.509031895026867
6       4.41325021172599
7       8.628727480200146
8       5.734205371025561
9       4.841791367031226