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

[robve]

(04-17-2021 12:03 AM)Albert Chan Wrote:  The plot showed a nice bell-shaped curve, easy to get area, simply by summing squares.

Rather than doing that, I think we can do better to quickly estimate an optimal d using a geometric sequence of points. I ran a few tests with this change to 'quad' for Exp-Sinh:

Code:

  else if (isfinite(a)) {
    int i;
    mode = 1; // Exp-Sinh
    for (i = 1; i <= 12; ++i) {
      v = 1 << i;
      printf("%d diff=%g\n", 1 << i, f(a + d/v)/v - f(a + v*d)*v);
    }
    c = a;
    v = a+d;
  }

Note what happens to 'diff' for these integrands:

exp(-.2*x) on [0,inf) d~45 is good
exp(-.1*x) on [0,inf) d~100 is good
exp(-.01*x) on [0,inf) d~700 is good
exp(-.001*x) on [0,inf) d~9000 is good
1/(x*x) on [1,inf) d~1 is good
1/(1+x*x) on [0,inf) d~1 is good
log(1+(x*x))/(x*x) on [0,inf] d~1 is good
(x*x*x)/(exp(x)-1) on [0,inf) d~7 is best d~8 is good
x/sinh(x) on [0,inf) d~9 is best d~8 is good
(x*x)/sqrt(exp(x)-1) on [0,inf) d~17 is best d~16 is good
pow(x,7)*exp(-x) on [0,inf) d~25 is best and d~32 is good
exp(-sqrt(x)) on [0,inf) d~81 is best and d~64 is good
exp(-x)*(x*x*x) on [0,inf) d~28 is good
exp(-x)*pow(x,5) on [0,inf) d~50 is good

Note the \( 2^i \) value when the diff sign changes, if it changes.

Periodic functions switch signs multiple times, these are not great to integrate with Exp-Sinh anyway e.g. sin(x)/(x*x) on [0,inf).

The idea of using geometric sequences to determine d aligns with the Exp-Sinh rule. This method is not perfect yet and certainly needs some tweaking.

- Rob

Edit: I checked 208 Exp-Sinh integrals from the benchmarks and they all show that this conceptually works. I've added some more interesting cases to the list.