Notes on Taylor and Maclaurin Series • Hacker News Board

Re: Notes on Taylor and Maclaurin Series

#31

Post by bee_rider » Tue, Jul 30, 2024, 5:37 PM UTC

sfpotter wrote: post #23

bee_rider wrote: post #6

Earlier quoted context omitted.

Chebyshev’s polynomials seem to have eclipsed his semi-iterative method for solving linear systems, which is too bad IMO.

Why is that bad? Are there any cases where Chebyshev iteration is the best option? It seems like acquiring the necessary information about the spectrum would be prohibitive in practice, although I have never tried it out.

I’ve only really played with it on a single-node machine, which isn’t where it ought to shine.

IMO it sits at a really interesting spot as a sort of “more robust” (hand-waves) iterative solver that doesn’t require inner products. You need to know something about the spectrum sure, but sneakily figuring out things about the spectrum is somewhere where people can show off their expertise I think.

Re: Notes on Taylor and Maclaurin Series

#32

Post by antognini » Tue, Jul 30, 2024, 5:40 PM UTC

mkarrmann wrote: post #19

antognini wrote: post #18

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…

Why do you think a series which absolutely converges but whose first few terms are a poor approximation is related to a series which diverges but whose first few terms match the empirically expected value? Not only is it a stretch to wonder about parallels between a basic Calc II concept and Quantum Field Theory, but is seems like the exponential function is the exact opposite of the example you provided.

I was being a bit loose in my language, but I was more specifically thinking of the connection to "resurgence theory" which aims at understanding more complicated kinds of apparently divergent infinite sums. See this article for a popular overview: https://www.quantamagazine.org/alien-calculus-could-save-par...

Re: Notes on Taylor and Maclaurin Series

#33

Post by tomsmeding » Tue, Jul 30, 2024, 5:58 PM UTC

Nitpick:

> Let's start with our Maclaurin series for cos(x):

> p(x) = 1 - x^2/2! + x^4/4! - x^6/6! + x^8/8! - ... = 1 + \sum_{n=1}^∞ ((-1)^n x^(2n)) / (2n)!

> Ignoring the constant term, we'll write out the ratio limit. [...]

No need to ignore the constant term: it fits the formula just fine! Evaluating ((-1)^n x^(2n)) / (2n)! at n=0 gives ((-1)^0 x^0) / 0! = (1 * 1) / 1 = 1, which is precisely the constant term.

Each of those three 1s is because the empty product, i.e. the product of an empty list of numbers, is 1. This is sensible because it means that (product of L1) * (product of L2) = product of (L1 append L2).

Re: Notes on Taylor and Maclaurin Series

#34

Post by sfpotter » Wed, Jul 31, 2024, 10:08 PM UTC

bee_rider wrote: post #31

sfpotter wrote: post #23

Earlier quoted context omitted.

Why is that bad? Are there any cases where Chebyshev iteration is the best option? It seems like acquiring the necessary information about the spectrum would be prohibitive in practice, although I have never tried it out.

I’ve only really played with it on a single-node machine, which isn’t where it ought to shine. IMO it sits at a really interesting spot as a sort of “more robust” (hand-waves) iterative solver that doesn’t require inner products. You need to know something about the spectrum sure, but sneakily figuring out things about the spectrum is somewhere where people can show off their expertise I think.

That seems plausible.

I actually spent a little time digging into this, and I'm not sure if this method is actually due to Chebyshev! This link has the most extensive references I found:

    https://encyclopediaofmath.org/wiki/Chebyshev_iteration_method

and from what I can tell it's actually due to Richardson.