[HN Gopher] Chebyshev Polynomials in the 16th Century (2022)
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Chebyshev Polynomials in the 16th Century (2022)
Author : IdealeZahlen
Score : 99 points
Date : 2024-11-27 14:58 UTC (1 days ago)
(HTM) web link (arxiv.org)
(TXT) w3m dump (arxiv.org)
| bee_rider wrote:
| I will always be amazed at these guys who did numerical
| algorithms before computers were a type of machine.
|
| Something that has always confused me about these Russians,
| Chebyschev and Krylov, what use did they have for their iterative
| methods and subspaces? I guess they weren't solving big sparse
| linear systems on distributed computers in the year 1900.
| random3 wrote:
| Ballistics, celestial bodies, even a ton of competitions
| between mathematicians. You can trace down analysis concepts to
| Archimedes easily, but by Descartes (rolling tangent) and
| eventually Newton / Leibniz who formalized calculus there was a
| lot of stuff happening. E.g. Descartes was contemporary with
| Galileo. So applied math and the theoretical part was under
| natural philosophy that eventually became physics.
| IdealeZahlen wrote:
| "The History of Approximation Theory" by Karl-Georg Steffens is
| a great reference for historical contexts.
|
| For Chebyshev, who devoted his life to the construction of
| various 'mechanisms' [1][2], his motivation was to determine
| the parameters of mechanisms (that minimizes the maximal error
| of the approximation on the whole interval).
|
| [1] https://en.wikipedia.org/wiki/Mechanism_(engineering)
|
| [2] https://tcheb.ru/
| vector_spaces wrote:
| In particular he studied Watt's mechanism, which was an
| integral component of steam engines powering the industrial
| revolution in Western Europe. Its optimal configuration
| wasn't really well understood at the time which led to
| practical problems. Chebyshev traveled from Russia (which
| wouldn't really enjoy an industrial revolution till much
| later) to Western Europe and discussed with experts and
| people who operated these engines. He brought back to Russia
| with him notes and experimental data, and those informed the
| development of what would later be known as minimax theory,
| and Chebyshev polynomials which provide polynomial solutions
| to minimax problems.
|
| In the course of developing that theory he founded the modern
| field of approximation theory, and the St. Petersburg school
| of mathematics. I think his approach of using applied
| problems and techniques to inform the development of pure
| math deeply influenced the whole of Soviet and Slavic
| mathematics in the century that followed
|
| (and yes, the book by Karl-Georg Steffens is beautiful!)
|
| Edit: To answer the grandparent's question, aside from things
| directly invented by Chebyshev or his students, often things
| are called "Chebyshev" when there's either a Chebyshev
| polynomial or a minimax problem lurking in the background
| dhosek wrote:
| I was playing around with an idea that there might be some
| insight into Fermat primes by looking at products of complex
| solutions of polynomials of the form x^{2^n)+1=0, and Chebyshev
| polynomials came up as I was looking at the exact values of
| those roots.12 As I recall, looking at finding half angle sines
| and cosines of increasing fractions, I ended up seeing
| Chebyshev polynomials emerging from the results.
|
| [?]
|
| 1. I may have this confused with my similar investigations into
| Mersenne primes and x^p-1=0.
|
| 2. My hypothesis that I could find factors of a Fermat number
| with _n_ > 5 by multiplying the roots together and setting _x_
| = 2 failed on writing a program to actually check the result,
| but looking back on my memories of doing this, I may have made
| an error.
| SideQuark wrote:
| Chebyshev polynomials have as roots nth roots of unity, so of
| course these are going to show up. It's one way to define
| them.
|
| https://en.wikipedia.org/wiki/Chebyshev_nodes
|
| The nth roots of unity are incredibly well studied, and some
| of that stemmed in the 1700-1800s on trying to factor things.
| The entire field of analytic numbers theory has taken these
| ideas to incredible (think decades of study and research to
| be state of the art) depths.
| jacobolus wrote:
| More explicitly: Chebyshev polynomials are what you get
| when you take trigonometric polynomials for a periodic
| interval or Laurent polynomials on the unit complex circle
| and project onto a diameter of the circle.
| gmiller123456 wrote:
| I was reading a book from the early 1900's, and it referenced
| using computers to calculate some complex algorithms. Threw me
| for a loop, and I finally realized the author was talking about
| people. Apparently it was a thing to send long computations to
| a room/building full of people and get the answer back.
| bee_rider wrote:
| I wonder if the "programmers" could be much sloppier back
| then. "Find the eigenvalues? Which ones? You know, the ones
| we always want!"
| HPsquared wrote:
| Prompt engineers, basically.
| SideQuark wrote:
| The word "computer" to describe a person who does
| computations dates back into the 1600s, and is exactly where
| we got the current word.
|
| Up to around 1940, the vast majority of the world's computers
| were people, and there were legions of them across all areas
| of government and industry.
|
| There were around 250 total automated computers in 1955,
| around 20,000 in 1965, so I doubt human computers were
| outnumberd until the 1970s/1980s at best.
| srean wrote:
| > I finally realized the author was talking about people.
| Apparently it was a thing to send long computations to a
| room/building full of people and get the answer back.
|
| s/people/women/g
|
| https://www.smithsonianmag.com/science-nature/history-
| human-...
| gowld wrote:
| s/women/people/g
|
| No need to promote sex-based divisiveness.
|
| From your own link:
|
| "So the French mathematician Alexis-Claude Clairaut decided
| to break the work up--by dividing the calculations among
| several people. In 1757, he sat down with two friends, the
| young astronomer Jerome-Joseph Lalande and Nicole-Reine
| Lepaute, a clockmaker's wife with a penchant for numbers.
| ... The age of human computers began."
|
| "By the 19th century, scientists and governments were
| beginning to collect reams of data that needed to be
| processed, particularly in astronomy, navigation and
| surveying. So they began breaking their calculations down
| into tiny basic math problems and hiring gangs of people to
| solve them. The work wasn't always hard, though it required
| precision and an ability to work for long hours. Mostly,
| the computers were young men."
|
| "But by the late 19th century, some scientists realized
| that hiring women could reduce the cost of computation. The
| growth of education and middle-class prosperity had
| produced a generation of young women trained in math. So
| when the Harvard Observatory decided to process years of
| astronomic data it had gathered using its telescope, it
| assembled one all-female team of computers."
| MonkeyClub wrote:
| In this case, I read the substitution as increasing
| visibility rather than divisiveness, but I get your
| point: women are first-class people too.
| srean wrote:
| Right. I was conflicted about whether I should draw
| attention to this or not. In the end I felt there exists
| a lack of knowledge/appreciation of the contribution made
| by women computers.
| tliltocatl wrote:
| My uneducated guess is that the notion of computers being
| mostly female comes from the WWII era.
| Kabootit wrote:
| Are we talking about the book "Souls in the Great Machine" or
| real history?!
| gmiller123456 wrote:
| It was a book on positional astronomy, I don't remember the
| title.
| adrian_b wrote:
| The first job of my father, after finishing his university
| studies, three quarters of century ago, when there were only
| a handful of electronic computers in the entire world, was as
| a "computer" at an astronomical observatory.
|
| With the revenue secured by that job, he decided that he can
| afford to marry my mother.
| HPsquared wrote:
| Anything that can't be solved analytically, I suppose. That's a
| pretty huge range of problems!
| bee_rider wrote:
| For sure it is!
|
| The odd thing with what I listed is that these methods
| (nowadays) are really mostly useful for massive sparse
| problems, which wouldn't really be practical without
| computing machines.
|
| I'm pretty sure the Chebyschev semi-iterative method for
| solving linear systems is just named after his polynomials
| (and you can use his polynomials by hand for other stuff),
| but I really am at a loss as to what Krylov was up to.
| Jun8 wrote:
| Not in this case but sometimes mathematicians of old _did_ have
| access to "computers", ie savants who can do large calculations
| in their heads easily. Gauss used the services of such a
| person: https://hsm.stackexchange.com/questions/15609/autistic-
| assis...
| bibanez wrote:
| Very interesting anecdote!
| dukeofdoom wrote:
| Would love to watch a videos on historical math breakthroughs. In
| the style of Indiana Jones, I mean just told as a big adventure.
| I used to watch connections and loved it.
| incognito124 wrote:
| I second that. I bet such a documentary may even yield insights
| how to make current scientific breakthroughs
| panarchy wrote:
| You might enjoy the book "The Language of Mathematics: Making
| the Invisible Visible"
| jansan wrote:
| I first cam across the term "Chebyshev polynomials" when working
| on length parametrization of Bezier curves. Although I still do
| not know what they really are, I fell in love with the term,
| because it sounds super smart and is easy to remember. Sometimes
| when I want to impress non-science people I say "I have to go
| back to work, those Chebyshev polynomials aren't gonna solve
| themselves".
| vector_spaces wrote:
| There are lots of fun characterizations of them. The most
| significant one is that they are the polynomials that minimize
| the uniform norm on a given set -- classically, this set was
| the closed interval [-1, 1] or sometimes [-2, 2]. This means
| they are the polynomials that attain the smallest maximum
| absolute value on the set or interval of interest. This minimax
| property is essential for their utility and ubiquity throughout
| pure and applied math
|
| The minimax property is one of the ways you can generalize to
| other kinds of sets, allowing you to talk about the Chebyshev
| polynomial of, say, an arc or a Jordan curve or a union of
| intervals, and many of the properties enjoyed by the classical
| Chebyshev polynomials end up carrying over as well, but faaaaar
| less is known about these generalizations. In many cases all
| you can hope for are asymptotics, and you need much more in the
| way of machinery and sophisticated tools to prove anything --
| even to compute explicit formulas for the coefficients.
|
| The classical case is nice though because they can be explored
| fruitfully without very much background
| alphanumeric0 wrote:
| Ah cool to see this on HN! I'm taking a numerical calc class
| right now and it's nice to get some historical context around
| something you're studying. I'd recommend checking out some cool
| graphs about Runge's phenomenon and Chebyshev polynomials.
| throw_m239339 wrote:
| I know that name because in Blender there is a Voronoi texture
| named that way.
| ykonstant wrote:
| Kind of off topic, but look at how beautiful the ar5iv link is:
|
| https://ar5iv.org/html/2203.10955
|
| I am getting more and more excited about converting TeX sources
| to HTML5 to be more accessible to students and researchers.
|
| I do think PDF is still king for final results and of course
| print, but the accessibility and searchability the web format
| provides is fantastic.
| JaDogg wrote:
| Nice I did not know about this servie. looks cool. Replace x
| with 5 and view the PDF rendered
| namibj wrote:
| It's not the PDF; it's directly from the TeX source.
| venusenvy47 wrote:
| I can't find that link on the ar5iv.org page for this paper.
| Where did you find it? Under "Access Paper", if I click on TeX
| it downloads a .tar file.
| ykonstant wrote:
| You edit the URL from arxiv to ar5iv. If it works, the paper
| is available; if not, it is not. I don't know of any other
| way to link!
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