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.
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...
"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).
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
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.
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.
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.
> 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.
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.
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.
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.
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.
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.
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.
I first cam across the term "Chebyshev polynomials" when working on length parametrization of Bézier 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".
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
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.
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...
"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/
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
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.
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.
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.
I wonder if the “programmers” could be much sloppier back then. “Find the eigenvalues? Which ones? You know, the ones we always want!”
Prompt engineers, basically.
Are we talking about the book "Souls in the Great Machine" or real history?!
> 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-...
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.
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.
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.
Anything that can't be solved analytically, I suppose. That's a pretty huge range of problems!
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.
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.
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.
I second that. I bet such a documentary may even yield insights how to make current scientific breakthroughs
You might enjoy the book "The Language of Mathematics: Making the Invisible Visible"
I first cam across the term "Chebyshev polynomials" when working on length parametrization of Bézier 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".
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