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

[robve]

(04-08-2021 04:27 PM)Albert Chan Wrote:  

(04-08-2021 01:32 PM)robve Wrote:  Using exp(t) in the inner loop instead of the strength-reduced expt recurrence should be the most accurate. Try it.

I did. It make no difference.

That's really nice. I was stating this more in "theory" that it would be most accurate, but practically makes no difference, which is really nice. On the SHARP Pocket Computers it makes a difference only because variables hold 10 digits whereas expressions are evaluated with 12 digit precision. So using EXP is a tiny bit better than repeated multiplication with a variable, e.g. we get the integral 2.0 exactly instead of 1.999999998.

(04-08-2021 04:27 PM)Albert Chan Wrote:  Same conclusion when I compared strength-reduced exp vs supposed more accurate expm1.
I even tried "improved u", when t get closer to 1. Again, no difference.

u = exp(1/t - t) = exp((1-t)*(1+1/t))

Think of this as Newton's method.
Once starting to converge, every iteration doubled accuracy.
In other words, correction term only required half precision. (guess had the other half)

Same with quadrature, required precision keep reducing.
Inaccuracy of strength-reduced expt does not contribute to the sum.

I didn't think about it that way. That is a nice way to confirm.

I would also like to confirm that removing if (r == 0 || r == 1) break; makes no difference with all integrals tested. Logically the loop terminates before these conditions are met. This check is redundant and can be removed as you observed correctly.

One thing I am still curious about is reusing or setting fp and fm to zero. I will look into the decaying factor idea. Perhaps there is also a difference between Inf and NaN, because with Inf the limit is clearly not finite and with NaN we don't necessarily know, but reusing in that case makes sense, I think. But I may be wrong if it makes no difference. I just don't know yet.

- Rob