[HN Gopher] Notes on Taylor and Maclaurin Series
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Notes on Taylor and Maclaurin Series
Author : ibobev
Score : 65 points
Date : 2024-07-24 13:30 UTC (5 days ago)
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| zamadatix wrote:
| All 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 Pade approximant looks
| like another interesting candidate to read about.
| raphlinus wrote:
| 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.
| bee_rider wrote:
| Chebyshev's polynomials seem to have eclipsed his semi-
| iterative method for solving linear systems, which is too bad
| IMO.
| bee_rider wrote:
| 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."
| abdullahkhalids wrote:
| Weird. When I was doing boxing, having to run sprints every
| training session, doing dozens of pushups, situps and what not
| did not beat the interest I had in boxing :).
| borroka wrote:
| Not at all weird, because you were also boxing. But if your
| boxing sessions were only to be sprints, pushups, situps etc,
| and no boxing, you would surely lose interest in doing boxing
| at that place. Likewise, if your jiu jitsu training were only
| drills and no sparring, there would be no people doing jiu
| jitsu after a while.
| abdullahkhalids wrote:
| My boxing training sessions were about 50-50 exercises and
| the padding/sparring session (in which there were big
| breaks).
|
| In my undergrad, I was made to take 6 math courses, and
| about 18 physics courses which used that math [1]. Plus
| additional Engineering/CS electives. So over a 25-75 ratio
| of exercise to "real" stuff.
|
| Much better ratio than boxing.
|
| [1] If you decide that the math courses have nothing
| real/interesting in them. Which is not true.
| brianm wrote:
| A 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.
| curt15 wrote:
| Are you referring to generating functions?
| ziofill wrote:
| For 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.
| skzv wrote:
| One of the most useful things I learned in mathematics.
|
| Gauss-Newton algorithm, GNSS (GPS) solution solving, financial
| calculus (Ito calculus), and more.
| btilly wrote:
| In 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."
| curt15 wrote:
| Instead of thinking of real analysis as a zoo of weird
| exceptions, it's probably more accurate to think of complex
| analyic functions as the exceptions. For example, when viewed
| as a two dimensional mapping from the plane to itself, complex-
| analytic functions are conformal (angle-preserving) whereas
| most differentiable mappings from the plane to itself are not.
| dhosek wrote:
| On the other hand, complex analysis explains things that
| aren't obvious in real analysis, like why the radius of
| convergence for a continuous function on the real line might
| be 1 (turns out there are singularities off the real line on
| the complex plane).
| kaashif wrote:
| Complex analysis is far too nice. There are so many theorems
| where my first thought was "there's no way that's true".
|
| Picard's great theorem is totally insane. As is even its little
| brother, and even Liouville's theorem.
|
| The proofs aren't even that long. They just feel totally false.
| antognini wrote:
| One of the things that has always seemed rather magical to me is
| the Taylor series of the exponential function for very large
| negative values. We know that a number like e^-100 is a number
| extremely close to zero. Yet when you write out the Taylor series
| you see an alternating some of increasingly large, seemingly
| arbitrary numbers (at least for ~100 terms):
|
| e^-100 = 1 - 100 + 5000 - 166,666.6 + 4,166,666.6 - ...
|
| If you were just given this sum and knew nothing about Taylor
| series or the exponential function you'd assume that its value
| was some extremely large number. Yet everything manages to cancel
| out just right so that the resulting sum is almost exactly zero,
| but not quite.
|
| I can't help but wonder if there's some parallel to parts of
| quantum field theory. If you expand out the interactions between
| two particles you also end up with a series that is apparently
| divergent. Yet we know experimentally that the first few terms
| work quite well as an approximation. It feels a bit like you're
| looking at the series of e^-100 without knowing about the
| exponential function.
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