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Dear Mehran On 22 Oct 2015, at 20:21, Mehran Mazandarani <me.mazandarani@xxxxxxxxx> wrote: > Dear John > Thank you for your reply. > Let me inherit, to some extent, obstinacy from you, and say that, it > can be concluded that the notion of differentiability in that page is > lost. Please explain why? For the theory you need to differentiate *real-valued* functions. Maybe someone has invented a concept of derivative of interval-valued functions, but here there's no need of it. I still think you are looking for some silver bullet that will come up with the "right" interval answer to a problem, by automatic computation without the need to think about what's going on. There is no such silver bullet. The example of the Interval Newton Method in my Würzburg talk was intended to show just that. Namely, to get the right interval computation you must study how the quantifiers "for all" and "exists" occur in a statement about *real-valued* functions. Attached is the writeup of that talk, for the conference Proceedings. It goes into more detail about that argument. Symbolic calculation sometimes gets the "right" answer mindlessly, but usually intelligence must be added. I suspect RDM is just the same (sorry, I hadn't heard of it before this discussion). But as Richard Fateman said, any well-trained interval analyst knows that you can get the exact range of 10*x-x^2 by completing the square to write it as 25 - (x-5)^2, and using ordinary interval evaluation on this. Regards John Pryce > On 10/21/15, John Pryce <PryceJD1@xxxxxxxxxxxxx> wrote: >> Hi Mehran >> >> On 21 Oct 2015, at 12:03, Mehran Mazandarani <me.mazandarani@xxxxxxxxx> >> wrote: >>> Now, let me ask you some questions. What was the definition of derivative >>> for interval-valued function in the page 7 of your presentation at 16th >>> GAMM-IMACS symposium on Scientific Computing, Computer Arithmetic and >>> Validated Numerics 21-26 September 2014, Wurzburg, Germany? >> Read it again. It isn't a "derivative for interval-valued function". >> It is "interval version of a derivative". This is explicitly said half way >> down page 7. >> f is a point function. >> We form the derivative f', another point function. >> Then we form some interval version (=extension) of f', "bold f'" which I'll >> write ~f'. It doesn't matter which extension, but there will be some >> expression for f' and we can take ~f' to be the result of >> interval-evaluating this expression. >> >>> What is the notion of differentiability of interval-valued function? >> I don't know! >> >>> Is it possible you can present a well-define of the notion? >> No. >> >> John P
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1788TalkWriteup.pdf
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