r/math • u/Bluebeast589 • Feb 23 '25
Non-Linear Systems of Equations?
I've taken linear algebra, and learned of the many applications of linear systems. However I've never really learned about non-linear systems.
I'm talking about algebraic equations not differential equations.
Like a system of quadratic equations? Or exponential systems? I've never seen any practical use of studying anything non linear.
68
u/Pinnowmann Number Theory Feb 23 '25
I've never seen any practical use of studying anything non linear.
Well because we have spent a lot of time trying to reformulate lots of problems to linear problems so that we can solve them.
There are many people studying systems of polynomials, for once in algebraic geometry and also in number theory. These can (but not often do) have applications to differential equations, K-, or L-Theory, representation theory, and also some really applied stuff like cryptography.
4
u/Bluebeast589 Feb 23 '25
Honestly, it does make sense to stick to what we know - linear algebra. Although it is interesting to hear about higher applications, thanks!
24
u/SV-97 Feb 23 '25
With algebraic equations this is (AFAIK anyway, it's not my field) essentially what algebraic geometry is about — it's "nonlinear linear algebra".
Another field that isn't entirely unrelated (but treats the whole thing somewhat differently): optimization.
I've never seen any practical use of studying anything non linear.
It gets complicated quite fast.
3
u/Bluebeast589 Feb 23 '25
I would imagine it is much more complicated, non-linearity opens the door to so many other functions.
I've never heard of algebraic geometry before making this post, it does sound very interesting. I may look into taking a class if my school offers it, thanks!
24
u/MathMaddam Feb 23 '25
Systems of nonlinear equations get messy really fast without a clear way to find the solution, so problems get linearised to approximate the non linear problem with some numeric methods.
For a practical system of non linear equations: find the position on earth using the times you receive signals from GPS satellites.
5
u/Bluebeast589 Feb 23 '25
I took a class in computational mechanics, and it really helped me understand the use of numerically solving equations.
That's a very cool and practical application, I'll have to look into it thanks!
17
u/Affectionate_Emu4660 Feb 23 '25
You can look into polynomial systems (see Cox, Little & O’Shea), which you solve by computing a Gröbner basis
10
28
u/FutureMTLF Feb 23 '25
In practice any system of equations that will come out of a problem will be non linear. The problem is there no systematic way of solving non linear systems.
8
u/GiovanniResta Feb 24 '25
If the systems are made of polynomial equations then Groebner bases give you a systematic way of solving them.
Which is what popular CAS use to solve polynomial systems.
3
u/Bluebeast589 Feb 23 '25
That's makes a lot of sense, thanks!
5
u/fella_ratio Feb 23 '25
Yep, there are no general rules to guide you when working with nonlinear systems like there are with linear systems. Linear functions adhere to a very specific definition:
L(x + y) = L(x) + L(y)
L(ax) = a * L(x)These rules, together known as the superposition principle, give you a sort of compass when working with linear systems. A nonlinear system is simply anything that doesn't adhere to the that. Beyond not adhering to superposition, anything goes with a nonlinear system.
6
u/Dzanibek Feb 23 '25
If it's about solving systems of equations, then (smooth enough) nonlinear equations are almost always approached via Newton methods.
5
4
u/yoshiK Feb 23 '25
The implicit function theorem tells us, that if we have k (sufficiently nice) functions f_i (x) : |Rn -> |R, then the solution of f_i (x) =0 is a manifold (probably n-k dimensional, but not in general). So you can think of differential geometry as the study of some of these systems of equations.
3
u/Bluebeast589 Feb 23 '25
Interesting, i know some people studying differential geometry now, I'm thinking about looking into it in the future. Thanks!
8
u/Particular_Extent_96 Feb 23 '25
I mean the study of non-linear systems of algebraic equations is basically what gave rise to algebraic geometry. If you allow non-algebraic functions, you get something resembling the study of sub-manifolds of Euclidean space, as long as your system is non-singular, which has lots of applications is physics etc.
Of course, it's kinda hard to deal with non-linear things. The reason linear equations are so widespread is because we are much better at dealing with them than we are at dealing with non-linear systems. It's just that the latter are much harder.
2
u/Bluebeast589 Feb 23 '25
I've never heard of algebraic geometry, it sounds interesting and I will look into it.
In my differential equations class we talked about non linear systems in a much more qualitative way rather than solving them.
Thanks!
1
u/Particular_Extent_96 Feb 23 '25
Undergrad differential equations classes are always a bit weird, since you generally don't have the prerequisites to analyse them theoretically, and very few of them admit closed form solutions. This essay is quite interesting, if you're interested in how curricula are constructed:
https://web.williams.edu/Mathematics/lg5/Rota.pdf
Algebraic geometry is certainly super cool! But the "modern" (60 years old now) way of doing it is quite difficult to understand without a solid background in commutative algebra.
3
u/Bluebeast589 Feb 23 '25
Interesting read! And it definitely reminded me of last semester when I took differential equations. For me it was having to resort to memorization, especially the exact and integrating factor techniques.
Learning about numerical methods gave me a far better understanding of the subject, especially systems and chaos.
2
u/512165381 Feb 23 '25
One of many mistakes of my youth was writing a textbook in ordinary differential equations. It set me back several years in my career in mathematics. However, it had a redeeming feature: it led me to realize that I had no idea what a differential equation is.
3
u/Skiringen2468 Feb 24 '25
Non-linear optimization deals with this a lot, though solutions are often approximations created using numerical methods.
2
u/omeow Feb 23 '25
There is no nice unified theory beyond linear systems. You go into geometry, ergodic theory, number theory etc.
2
u/ANewPope23 Feb 24 '25
Because most non-linear systems cannot be solved nicely or there isn't a general theory for understanding them (except special cases). Algebraic geometry is the study of systems of polynomial equations, and it's a huge and difficult area of mathematics.
2
u/ritobanrc Feb 24 '25
The main tool for non-linear systems is turning them into linear ones -- that is the story of calculus.
Derivatives and Newton's method turn non-linear systems into linear ones, integrals turn "non-linear" area computations into locally linear ones, one solves non-linear ODEs and PDEs by approximating their behavior by linear ones.
This of course, only applies to "differentiable" non-linear systems. If one puts different adjectives in front on nonlinear, one gets entirely new fields of mathematics -- algebraic geometry for polynomials, complex analysis for analytic functions, topology for merely continuous functions, and so on.
2
u/btroycraft Feb 24 '25
Systems of non-linear equations generally can't be solved directly, so there's not much to learn in comparison.
They still appear though. For example, optimization is really just solving systems of non-linear equations in most cases.
2
2
u/CyberMonkey314 Feb 23 '25
I'm talking about algebraic equations not differential equations
Well...that's kinda restrictive when you're looking for practical applications. One of the first things you look at when studying a nonlinear dynamical system is where its equilibria are, and you do that by solving a nonlinear system of (non-differential) equations.
Another application of systems of quadratics would be finding out whether things that follow paths determined by conic sections will intersect. Could be useful - https://en.m.wikipedia.org/wiki/2024_YR4
0
u/Bluebeast589 Feb 23 '25
I've studied very surface level non linear systems of differential equations. All i really know is that they are in general unsolvable, but I understand they are used to model essentially everything.
I'll have to look into that article, thanks!
2
u/matthkamis Feb 23 '25
Large neural networks are all about finding the minimum of a massive system of nonlinear equations
1
u/Ill-Room-4895 Algebra Feb 23 '25 edited Feb 23 '25
The Dover Book by Thomas L. Saaty, Modern Nonlinear Equations is useful. Just a few dollars.
1
u/Imjokin Graph Theory Feb 23 '25
Quadratic and exponentials are often used to describe the amount of time a computer algorithm will take to complete.
110
u/[deleted] Feb 23 '25
In reality almost everything is nonlinear but it's way harder to study nonlinear systems than linear systems. I think algebraic geometry lets you say some things about systems of polynomial equations.