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The short answer is yes, your intuition is correct, I would also have argued if someone had said the same thing. The approximation of a non polytopal domain by a straight mesh/ triangulation is called a "variational crime", due to the fact that consistency/ Galerkin orthogonality properties are violated. Unfortunately your experience is very common ...


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If real numbers are really implemented on your computer, then this is most likely acceptable. In other words, you should use a computer library that supports interval arithmetic. There are many such libraries. In particular, if you compute $1/\sqrt{2\pi}$ as lying in an interval that does not include $0.4$, then you will have proved your statement.


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I have no references, but i'm actually coding in Julia these days, and the profiling tool allows me do dig the computational time up until functions Base.+() and Base.*(). For the computations i do (and this is not a generality), there are a lot more additions than multiplications in numbers, but a lot more multiplications than additions in computing time. ...


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A complex Inf has no need for a sign, but arithmetic should work similarly. Inf plus anything should be Inf, Inf multiplied by anything should be Inf unless multiplied by 0 in which case the result should be 0. I found a good resource for checking how various programming languages handle infinity here: https://rosettacode.org/wiki/...


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Regarding complex version of IEEE-754 It is true that no complex version of IEE754 exists, but the same design rules as for IEEE-754 can be applied for complex arithmetic. However, this is more subtle than it seems in the first place. For example the direct cartesian implementation of the complex multiplication (i.e. $(a+ib)(c+id)$ does not work with $inf$. ...


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For the cubic spline case, it appears the spline interpolation scheme is using the "not-a-knot" spline interpolation, which enforces the 3rd derivative continuity at the 2nd point and the next-to-last point. This is the reason why you did not see a 3rd derivative discontinuity at x=1 and x=9. For details about the "not-a-knot" spline, ...


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