[Reliable_computing] sinc function for intervals
Ralph Baker Kearfott
C00255736 at louisiana.edu
Sun Jul 7 07:22:08 CDT 2019
Alan,
I find your questions interesting.
Yes, I think a quality interval implementation of the sinc function
for intervals would be somewhat lengthy, but using straightforward
ideas.
As you may know, using the Taylor series IS effective near x=0, for
rapidly converging series such as that of sinc(x).
Yes, you can leverage the points at which f(x) has its local extrema,
provided you can compute these accurately. You can combine that
with point evaluations of sinc(x) of known accuracy.
I would be willing to work with you on this; feel free to contact
me offline.
Best regards,
Baker (Ralph Baker Kearfott)
On 7/7/19 4:31 AM, Alan Eliasen wrote:
>
> I thought I would take a few minutes and implement the sinc function
> in my programming language Frink ( https://frinklang.org/ ) and then 2
> hours later realized it wasn't going to be that simple.
>
> As you all know, the sinc function is defined as:
>
> sin[x]/x
>
> where the value at x=0 is defined to be 1 because the limit converges
> to 1 at this point. That's the only special thing about this function.
>
> However, a version of this function over intervals has several issues:
>
> 1.) If the interval contains zero, the supremum of the result should
> be exactly 1. (Or should it? See 3.)
>
> 2.) The function is non-monotonic, so we can't evaluate it at just
> its endpoints. For example, if we evaluated the endpoints at 2 and 7,
> we would miss a local minimum.
>
> 3.) This function is the embodiment of the "dependence problem" or
> the "overestimation problem": that is, if a variable is used multiple
> times in an expression, as it is in sin[x]/x, then its interval bound
> may be larger than if each bound were computed naively. This has
> implications on your interpretations of intervals. These are:
>
> 3.a.) An interval contains its result somewhere between its bounds,
> but we're not sure where.
>
> 3.b) An interval contains *all* of the values between its bounds
> simultaneously.
>
> 4.) One could write the Taylor series expansion of sin[x] as
>
> sin[x] = x - x^3/3! + x^5/5! + x^7/7! ...
>
> and the expansion of sinc[x] = sin[x]/x as:
>
> sinc[x] = 1 - x^2/3! + x^4/5! + x^6/7! ...
>
> which begs the question of the dependence problem by probably making
> it much worse in most cases.
>
>
> 5.) A cool thing about the sinc[x] function is that it has its local
> extrema at exactly the points that sinc[x] intersects cos[x]. Can you
> leverage this with respect to the sinc function? Can your mathematical
> system solve this exactly?
>
> 6.) When evaluating the bounds of sinc[x], and its endpoints, where
> do you round down and round up?
>
>
> THE BIG QUESTION: Has anyone done analysis of the sinc function with
> respect to real intervals and provided opinion on the way it should be
> treated? Because any implementation of this function is dependent on
> your interpretation of 3.) and the "dependency problem."
>
> Solving sinc[x] naively gives wider bounds than we may want in any
> other case.
>
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