How much precision can you squeeze out of a table?
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How much precision can you squeeze out of a table?
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Re: How much precision can you squeeze out of a table?
#2When I was in high school I thought book this was so much fun
http://182.160.97.198:8080/xmlui/bitstream/handle/123456789/...
which is all about the kind of numerical analysis you would do by hand and introduces a lot of really cool math like the calculus of differences
Re: How much precision can you squeeze out of a table?
#3Nitpicking:
> You’re never going to get error less than 10E−15 since that’s the error in the tabulated values,
If you have like 100 (or 400) values in the table, you can squeeze one more digit.
In the simple case, imagine you have the constant pi, with 15 digits, repeated 100 times. If the rounding was done in a random direction like
floor(pi * 1E15 + random() - 1/2 ) / 1E15
Nobody builds a table for a constant and uses a uniform offset for rounding. Anyway, with some smooth function that has no special values in the points where it's sample, the exact value of the cut decimal is quite "random" and you get a similar randomization.
Re: How much precision can you squeeze out of a table?
#4When I was in high school I thought book this was so much fun http://182.160.97.198:8080/xmlui/bitstream/handle/123456789/... which is all about the kind of numerical analysis you would do by hand and introduces a lot of really cool math like the calculus of differences https://en.wikipedia.org/wiki/Finite_difference
i've been missing this knowledge! thank you for the recommendation.
Re: How much precision can you squeeze out of a table?
#5Nitpicking: > You’re never going to get error less than 10E−15 since that’s the error in the tabulated values, If you have like 100 (or 400) values in the table, you can squeeze one more digit. In the simple case, imagine you have the constant pi, with 15 digits, repeated 100 times. If the rounding was done in a random direction like floor(pi * 1E15 + random() - 1/2 ) / 1E15 then you can use statistic to get an avera…
Also nobody in his right mind uses lookup tables where the table value is actually the float approximation of the true f(x) - you choose the support values to minimize an error (e.g. mse) of a dense sampling of your interpolated value over x (or, in the limit, the integral of the chosen error function between the true curve and the interpolation of your supports). If you want to e.g. approximate a convex function using linear interpolation, all the tabulated values f'(x) would be <= the true f(x).
Re: How much precision can you squeeze out of a table?
#6When I was in high school I thought book this was so much fun http://182.160.97.198:8080/xmlui/bitstream/handle/123456789/... which is all about the kind of numerical analysis you would do by hand and introduces a lot of really cool math like the calculus of differences https://en.wikipedia.org/wiki/Finite_difference
The typography and design inside the book is beautiful but the cover looks like an old bargain-store flyer. There's an expression about that, something about judging a book by its cover? I can't remember exactly.
Re: How much precision can you squeeze out of a table?
#7Nitpicking: > You’re never going to get error less than 10E−15 since that’s the error in the tabulated values, If you have like 100 (or 400) values in the table, you can squeeze one more digit. In the simple case, imagine you have the constant pi, with 15 digits, repeated 100 times. If the rounding was done in a random direction like floor(pi * 1E15 + random() - 1/2 ) / 1E15 then you can use statistic to get an avera…
Also nobody in his right mind uses lookup tables where the table value is actually the float approximation of the true f(x) - you choose the support values to minimize an error (e.g. mse) of a dense sampling of your interpolated value over x (or, in the limit, the integral of the chosen error function between the true curve and the interpolation of your supports). If you want to e.g. approximate a convex function usi…
The true value is far more useful in a lot of cases. If you're building a table indexed by the upper mantissa bits of the float, for example, it's difficult to distribute the error properly across all intervals.
Re: How much precision can you squeeze out of a table?
#8When I was in high school I thought book this was so much fun http://182.160.97.198:8080/xmlui/bitstream/handle/123456789/... which is all about the kind of numerical analysis you would do by hand and introduces a lot of really cool math like the calculus of differences https://en.wikipedia.org/wiki/Finite_difference
In order to mitigate link rot, the book is Numerical Analysis Second Edition, from Francis Scheid (Schaum's Outlines)
Re: How much precision can you squeeze out of a table?
#9When I was in high school I thought book this was so much fun http://182.160.97.198:8080/xmlui/bitstream/handle/123456789/... which is all about the kind of numerical analysis you would do by hand and introduces a lot of really cool math like the calculus of differences https://en.wikipedia.org/wiki/Finite_difference
Ah, very cool. My personal favourite is this one:
https://archive.org/details/engineeringformu00giec_0/page/n5...
My father owned a copy of it which I still have to this day. It had a lot of different formulas than the ones we were taught in school, and I liked the range of topics that it covered.
Re: How much precision can you squeeze out of a table?
#10When I was in high school I thought book this was so much fun http://182.160.97.198:8080/xmlui/bitstream/handle/123456789/... which is all about the kind of numerical analysis you would do by hand and introduces a lot of really cool math like the calculus of differences https://en.wikipedia.org/wiki/Finite_difference
I just tried a portion of the url & it took me to Bangladesh university - http://182.160.97.198:8080/xmlui/bitstream/handle/ . Intersecting. When I go to root of this url, the error messages are listing the softwares this site is powered by. Generally this is not considered a secure way of protecting a site.