Notes on Taylor and Maclaurin Series
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Notes on Taylor and Maclaurin Series
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Re: Notes on Taylor and Maclaurin Series
#2All interests I had in Taylor/Macluarin series were promptly beat out of me when having to manually calculate hundreds of them in Calculus courses :).
The Remez algorithm linked at the end still inspires curiosity though. Any other "numerically more useful" approximation algorithms folks want to highlight? The Padé approximant looks like another interesting candidate to read about.
Re: Notes on Taylor and Maclaurin Series
#3All interests I had in Taylor/Macluarin series were promptly beat out of me when having to manually calculate hundreds of them in Calculus courses :). The Remez algorithm linked at the end still inspires curiosity though. Any other "numerically more useful" approximation algorithms folks want to highlight? The Padé approximant looks like another interesting candidate to read about.
Yes! The workhorse numerical technique is Chebyshev approximation. Remez exchange usually starts with it, for fine-tuning with respect to a "maximum error" norm, but it also works quite well by itself, and can be computed efficiently (even with a high degree) even from a function you can only evaluate numerically.
A really good place to read up on it is the documentation for Chebfun.
https://www.chebfun.org/docs/guide/guide04.html
Also: be on the lookout for a blog post on using Chebyshev polynomials to efficiently compute error metrics for curves.
Re: Notes on Taylor and Maclaurin Series
#4A coworker, once, had a cool idea to use Taylor Series to encode histograms for metrics collection. Basically a digest method akin to sketches or t-digests. We wound up using t-digests as Stripe (iirc) had a good OSS implementation at the time, but using Taylor Series has been lingering in my mind ever since.
Re: Notes on Taylor and Maclaurin Series
#5For some functions you can turn the Taylor series into a recurrence relation, which makes it blazing fast to calculate.
For instance for the family of functions f(x0,x1,...) = exp(poly(x0,x1,...)), where poly is a multivariate polynomial of order m, you can compute the Taylor coefficients with a recurrence relation of order m (that needs to look back m steps). This shows up in quantum optics, for example.
Re: Notes on Taylor and Maclaurin Series
#6All interests I had in Taylor/Macluarin series were promptly beat out of me when having to manually calculate hundreds of them in Calculus courses :). The Remez algorithm linked at the end still inspires curiosity though. Any other "numerically more useful" approximation algorithms folks want to highlight? The Padé approximant looks like another interesting candidate to read about.
Yes! The workhorse numerical technique is Chebyshev approximation. Remez exchange usually starts with it, for fine-tuning with respect to a "maximum error" norm, but it also works quite well by itself, and can be computed efficiently (even with a high degree) even from a function you can only evaluate numerically. A really good place to read up on it is the documentation for Chebfun. https://www.chebfun.org/docs/guid…
Chebyshev’s polynomials seem to have eclipsed his semi-iterative method for solving linear systems, which is too bad IMO.
Re: Notes on Taylor and Maclaurin Series
#7All interests I had in Taylor/Macluarin series were promptly beat out of me when having to manually calculate hundreds of them in Calculus courses :). The Remez algorithm linked at the end still inspires curiosity though. Any other "numerically more useful" approximation algorithms folks want to highlight? The Padé approximant looks like another interesting candidate to read about.
I had the opposite reaction.
Math department: “oh, look, we are very clever, see these pretty plots the differential equations generate and all the beautiful closed form solution we can make”
Me: “Hmm, very nice, I am not smart enough for this.”
Engineering department: “Complicated equation bad, smash with Taylor series, little part go bye-bye, big part is smooth.”
Me: “Yes this equation will match my brain nicely.”
Re: Notes on Taylor and Maclaurin Series
#8One of the most useful things I learned in mathematics.
Gauss-Newton algorithm, GNSS (GPS) solution solving, financial calculus (Ito calculus), and more.
Re: Notes on Taylor and Maclaurin Series
#9In complex analysis, all differentiable (in a region) functions are infinitely differentiable, and are the same as their Taylor series.
Real analysis is a zoo of weird exceptions. Including 1/e^(-1/x^2) away from 0, 0 at 0. Its Maclaurin series is just 0, which is clearly not the function we wrote down.
I can't explain why real analysis fit my brain and complex analysis doesn't. But to me complex analysis looks like, "We draw a path, then calculate this contour integral, and magic happens."
Re: Notes on Taylor and Maclaurin Series
#10A coworker, once, had a cool idea to use Taylor Series to encode histograms for metrics collection. Basically a digest method akin to sketches or t-digests. We wound up using t-digests as Stripe (iirc) had a good OSS implementation at the time, but using Taylor Series has been lingering in my mind ever since.
Are you referring to generating functions?