Accuracy of Integral with epsilon
10-10-2021, 11:51 AM (This post was last modified: 10-14-2021 12:30 PM by Albert Chan.)
Post: #20
 Albert Chan Senior Member Posts: 2,705 Joined: Jul 2018
RE: Accuracy of Integral with epsilon
(10-09-2021 08:37 PM)robve Wrote:  There is never a guarantee that a given function is constant by probing a few points. That should be obvious, no?

Tanh-Sinh (and other DE variants) does not see a constant function ... it see a bell-shaped curve.

CAS> 'int(1,x,-1,1)' (x = tanh(sinh(t)))

$$\int _{-\infty }^{+\infty }\mathrm{cosh}\left(t\right)\cdot (1-\left(\mathrm{tanh}\left(\mathrm{sinh}\left(t\right)\right)\right)^{2})\, dt$$

(10-09-2021 09:50 PM)robve Wrote:  You do realize that it is a fallacy to assume exp(x) and cosh(x)+sinh(x) equate numerically, in addition to algebraically?

It is interesting that my MAPM implementation of raw_exp() actually use above "identity"
Of course, |x| is limited to tiny value, below 1E-4, so catastrophic cancellation is not an issue.

sinh(x) converge with half as many terms, compared with exp(x)

exp(x) = x + x^2/2! + x^3/3! + ...
sinh(x) = x + x^3/3! + x^5/5! + ...
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 Messages In This Thread Accuracy of Integral with epsilon - lrdheat - 10-08-2021, 09:57 PM RE: Accuracy of Integral with epsilon - toml_12953 - 10-08-2021, 10:17 PM RE: Accuracy of Integral with epsilon - Thomas Okken - 10-08-2021, 10:18 PM RE: Accuracy of Integral with epsilon - toml_12953 - 10-08-2021, 10:28 PM RE: Accuracy of Integral with epsilon - lrdheat - 10-08-2021, 11:28 PM RE: Accuracy of Integral with epsilon - toml_12953 - 10-09-2021, 12:47 AM RE: Accuracy of Integral with epsilon - robve - 10-09-2021, 01:41 AM RE: Accuracy of Integral with epsilon - Valentin Albillo - 10-09-2021, 02:22 AM RE: Accuracy of Integral with epsilon - Paul Dale - 10-09-2021, 02:27 AM RE: Accuracy of Integral with epsilon - Thomas Okken - 10-09-2021, 08:47 AM RE: Accuracy of Integral with epsilon - John Keith - 10-09-2021, 05:40 PM RE: Accuracy of Integral with epsilon - robve - 10-09-2021, 08:37 PM RE: Accuracy of Integral with epsilon - Claudio L. - 10-12-2021, 04:47 PM RE: Accuracy of Integral with epsilon - robve - 10-14-2021, 02:43 AM RE: Accuracy of Integral with epsilon - robve - 10-09-2021, 09:50 PM RE: Accuracy of Integral with epsilon - Albert Chan - 10-10-2021 11:51 AM RE: Accuracy of Integral with epsilon - robve - 10-10-2021, 02:07 PM RE: Accuracy of Integral with epsilon - robve - 10-09-2021, 02:01 PM RE: Accuracy of Integral with epsilon - lrdheat - 10-09-2021, 03:13 AM RE: Accuracy of Integral with epsilon - toml_12953 - 10-09-2021, 05:44 AM RE: Accuracy of Integral with epsilon - toml_12953 - 10-09-2021, 11:26 AM RE: Accuracy of Integral with epsilon - lrdheat - 10-09-2021, 05:16 AM RE: Accuracy of Integral with epsilon - lrdheat - 10-09-2021, 03:25 PM

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