Accuracy of Integral with epsilon

10102021, 11:51 AM
(This post was last modified: 10142021 12:30 PM by Albert Chan.)
Post: #20




RE: Accuracy of Integral with epsilon
(10092021 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? TanhSinh (and other DE variants) does not see a constant function ... it see a bellshaped curve. CAS> 'int(1,x,1,1)' (x = tanh(sinh(t))) \(\int _{\infty }^{+\infty }\mathrm{cosh}\left(t\right)\cdot (1\left(\mathrm{tanh}\left(\mathrm{sinh}\left(t\right)\right)\right)^{2})\, dt\) (10092021 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 1E4, 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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