To be clear I don't mean, provided the last two numbers in the sequence provide the next one:
(2, 3, -> 5)
But rather given any index provide the Fibonacci number:
(0 -> 1) or (7 -> 21) or (11 -> 144)
Adding two numbers is a very simple task for any machine learning structure, and by extension counting by ones, twos or any fixed number is a simple addition rule. Recursive calculations however...
To my understanding, most learning networks rely on forwards only evaluation, whereas most programming languages have loops, jumps, or circular flow patterns (all of which are usually ASM jumps of some kind), thus allowing recursion.
Sure some networks aren't forwards only; But can processing weights using the hyperbolic tangent or sigmoid function enter any computationally complete state?
i.e. conditional statements, conditional jumps, forced jumps, simple loops, complex loops with multiple conditions, providing sort order, actual reordering of elements, assignments, allocating extra registers, etc?
It would seem that even a non-forwards only network would only find a polynomial of best fit, reducing errors across the expanse of the training set and no further.
Am I missing something obvious, or did most of Machine Learning just look at recursion and pretend like those problems don't exist?
Update
Technically any programming language can be considered the DNA of a genetic algorithm, where the compiler (and possibly console out measurement) would be the fitness function.
The issue is that programming (so far) cannot be expressed in a hill climbing way - literally, the fitness is 0, until the fitness is 1. Things don't half work in programming, and if they do, there is no way of measuring how 'working' a program is for unknown situations. Even an off by one error could appear to be a totally different and chaotic system with no output. This is exactly the reason learning to code in the first place is so difficult, the learning curve is almost vertical.
Some might argue that you just need to provide stronger foundation rules for the system to exploit - but that just leads to attempting to generalize all programming problems, which circles right back to designing a programming language and loses all notion of some learning machine at all. Following this road brings you to a close variant of LISP with mutate-able code and virtually meaningless fitness functions that brute force the 'nice' and 'simple' looking code-space in attempt to follow human coding best practices.
Others might argue that we simply aren't using enough population or momentum to gain footing on the error surface, or make a meaningful step towards a solution. But as your population approaches the number of DNA permutations, you are really just brute forcing (and very inefficiently at that). Brute forcing code permutations is nothing new, and definitely not machine learning - it's actually quite common in regex golf, I think there's even an xkcd about it...
The real problem isn't finding a solution that works for some specific recursive function, but finding a solution space that can encompass the recursive domain in some useful way.
So other than Neural Networks trained using Backpropagation hypothetically finding the closed form of a recursive function (if a closed form even exists, and they don't in most real cases where recursion is useful), or a non-forwards only network acting like a pseudo-programming language with awful fitness prospects in the best case scenario, plus the virtually impossible task of tuning exit constraints to prevent infinite recursion... That's really it so far for machine learning and recursion?
According to Kolmogorov et al's On the representation of continuous functions of many variables by superposition of continuous functions of one variable and addition, a three layer neural network can model arbitrary function with the linear and logistic functions, including f(n) = ((1+sqrt(5))^n - (1-sqrt(5))^n) / (2^n * sqrt(5)), which is the close form solution of Fibonacci sequence.
If you would like to treat the problem as a recursive sequence without a closed-form solution, I would view it as a special sliding window approach (I called it special because your window size seems fixed as 2). There are more general studies on the proper window size for your interest. See these two posts:
Time Series Prediction via Neural Networks
Proper way of using recurrent neural network for time series analysis
Ok, where to start...
Firstly, you talk about 'machine learning' and 'perfectly emulate'. This is not generally the purpose of machine learning algorithms. They make informed guesses given some evidence and some general notions about structures that exist in the world. That typically means an approximate answer is better than an 'exact' one that is wrong. So, no, most existing machine learning approaches aren't the right tools to answer your question.
Second, you talk of 'recursive structures' as some sort of magic bullet. Yet they are merely convenient ways to represent functions, somewhat analogous to higher order differential equations. Because of the feedbacks they tend to introduce, the functions tend to be non-linear. Some machine learning approaches will have trouble with this, but many (neural networks for example) should be able to approximate you function quite well, given sufficient evidence.
As an aside, having or not having closed form solutions is somewhat irrelevant here. What matters is how well the function at hand fits with the assumptions embodied in the machine learning algorithm. That relationship may be complex (eg: try approximating fibbonacci with a support vector machine), but that's the essence.
Now, if you want a machine learning algorithm tailored to the search for exact representations of recursive structures, you could set up some assumptions and have your algorithm produce the most likely 'exact' recursive structure that fits your data. There are probably real world problems in which such a thing would be useful. Indeed the field of optimisation approaches similar problems.
The genetic algorithms mentioned in other answers could be an example of this, especially if you provided a 'genome' that matches the sort of recursive function you think you may be dealing with. Closed form primitives could form part of that space too, if you believe they are more likely to be 'exact' than more complex genetically generated algorithms.
Regarding your assertion that programming cannot be expressed in a hill climbing way, that doesn't prevent a learning algorithm from scoring possible solutions by how many much of your evidence it's able to reproduce and how complex they are. In many cases (most? though counting cases here isn't really possible) such an approach will find a correct answer. Sure, you can come up with pathological cases, but with those, there's little hope anyway.
Summing up, machine learning algorithms are not usually designed to tackle finding 'exact' solutions, so aren't the right tools as they stand. But, by embedding some prior assumptions that exact solutions are best, and perhaps the sort of exact solution you're after, you'll probably do pretty well with genetic algorithms, and likely also with algorithms like support vector machines.
I think you also sum things up nicely with this:
The real problem isn't finding a solution that works for some specific recursive function, but finding a solution space that can encompass the recursive domain in some useful way.
The other answers go a long way to telling you where the state of the art is. If you want more, a bright new research path lies ahead!
See this article:
Turing Machines are Recurrent Neural Networks
http://lipas.uwasa.fi/stes/step96/step96/hyotyniemi1/
The paper describes how a recurrent neural network can simulate a register machine, which is known to be a universal computational model equivalent to a Turing machine. The result is "academic" in the sense that the neurons have to be capable of computing with unbounded numbers. This works mathematically, but would have problems pragmatically.
Because the Fibonacci function is just one of many computable functions (in fact, it is primitive recursive), it could be computed by such a network.
Genetic algorithms should do be able to do the trick. The important this is (as always with GAs) the representation.
If you define the search space to be syntax trees representing arithmetic formulas and provide enough training data (as you would with any machine learning algorithm), it probably will converge to the closed-form solution for the Fibonacci numbers, which is:
Fib(n) = ( (1+srqt(5))^n - (1-sqrt(5))^n ) / ( 2^n * sqrt(5) )
[Source]
If you were asking for a machine learning algorithm to come up with the recursive formula to the Fibonacci numbers, then this should also be possible using the same method, but with individuals being syntax trees of a small program representing a function.
Of course, you also have to define good cross-over and mutation operators as well as a good evaluation function. And I have no idea how well it would converge, but it should at some point.
Edit: I'd also like to point out that in certain cases there is always a closed-form solution to a recursive function:
Like every sequence defined by a linear recurrence with constant coefficients, the Fibonacci numbers have a closed-form solution.
The Fibonacci sequence, where a specific index of the sequence must be returned, is often used as a benchmark problem in Genetic Programming research. In most cases recursive structures are generated, although my own research focused on imperative programs so used an iterative approach.
There's a brief review of other GP research that uses the Fibonacci problem in Section 3.4.2 of my PhD thesis, available here: http://kar.kent.ac.uk/34799/. The rest of the thesis also describes my own approach, which is covered a bit more succinctly in this paper: http://www.cs.kent.ac.uk/pubs/2012/3202/
Other notable research which used the Fibonacci problem is Simon Harding's work with Self-Modifying Cartesian GP (http://www.cartesiangp.co.uk/papers/eurogp2009-harding.pdf).
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For example, math logic, graph theory.
Everyone around tells me that math is necessary for programmer. I saw a lot of threads where people say that they used linear algebra and some other math, but no one described concrete cases when they used it.
I know that there are similar threads, but I couldn't see any description of such a case.
Computer graphics.
It's all matrix multiplication, vector spaces, affine spaces, projection, etc. Lots and lots of algebra.
For more information, here's the Wikipedia article on projection, along with the more specific case of 3D projection, with all of its various matrices. OpenGL, a common computer graphics library, is an example of applying affine matrix operations to transform and project objects onto a computer screen.
I think that a lot of programmers use more math than they think they do. It's just that it comes so intuitively to them that they don't even think about it. For instance, every time you write an if statement are you not using your Discrete Math knowledge?
In graphic world you need a lot of transformations.
In cryptography you need geometry and number theory.
In AI, you need algebra.
And statistics in financial environments.
Computer theory needs math theory: actually almost all the founders are from Maths.
Given a list of locations with latitudes and longitudes, sort the list in order from closest to farthest from a specific position.
All applications that deal with money need math.
I can't think of a single app that I have written that didn't require math at some point.
I wrote a parser compiler a few months back, and that's full of graph-theory. This was only designed to be slightly more powerful than regular expressions (in that multiple matches were allowed, and some other features were added), but even such a simple compiler requires loop detection, finite state automata, and tons more math.
Implementing the Advanced Encryption Standard (AES) algorithm required some basic understanding of finite field math. See act 4 of my blog post on it for details (code sample included).
I've used a lot of algebra when writing business apps.
Simple Examples
BMI = weight / (height * height);
compensation = 10 * hours * ((pratio * 2.3) + tratio);
A few years ago, I had a DSP project that had to compute a real radix-2 FFT of size N, in a given time. The vendor-supplied real radix-2 FFT wouldn't run in the allocated time, but their complex FFT of size N/2 would. It is easy to feed the real data into the complex FFT. Getting the answers out afterwards is not so easy: it is called post-weaving, or post-unweaving, or unweaving. Deriving the unweave equations from the FFT and complex number theory was not fun. Going from there to tightly-optimized DSP code was equally not fun.
Naturally, the signal I was measuring did not match the FFT sample size, which causes artifacts. The standard fix is to apply a Hanning window. This causes other artifacts. As part of understanding (and testing) that code, I had to understand the artifacts caused by the Hanning window, so I could interpret the results and decide whether the code was working or not.
I've used tons of math in various projects, including:
Graph theory for dealing with dependencies in large systems (e.g. a Makefile is a kind of directed graph)
Statistics and linear regression in profiling performance bottlenecks
Coordinate transformations in geospatial applications
In scientific computing, project requirements are often stated in algebraic form, especially for computationally intensive code
And that's just off the top of my head.
And of course, anything involving "pure" computer science (algorithms, computational complexity, lambda calculus) tends to look more and more like math the deeper you go.
In answering this image-comparison-algorithm question, I drew on lots of knowledge of math, some of it from other answers and web searches (where I had to apply my own knowledge to filter the information), and some from my own engineering training and lengthy programming background.
General Mindforming
Solving Problems - One fundamental method of math, independent of the area, is transofrming an unknown problem into a known one. Even if you don't have the same problems, you need the same skill. In math, as in programming, virtually everything has different representations. Understanding the equivalence between algorithms, problems or solutions that are completely different on the surface helps you avoid the hard parts.
(A similar thing happens in physics: to solve a kinematic problem, choice of the coordinate system is often the difference between one and ten pages full of formulas, even though problem and solution are identical.)
Precision of Language / Logical reasoning - Math has a very terse yet precise language. Learning to deal with that will prepare you for computers doing what you say, not what you meant. Also, the same precision is required to analyse if a specification is sufficient, to check a piece of code if it covers all possible cases, etc.
Beauty and elegance - This may be the argument that's hardest to grasp. I found the notion of "beauty" in code is very close to the one found in math. A beautiful proof is one whose idea is immediately convincing, and the proof itself is merely executing a sequence of executing the next obvious step.
The same goes for an elegant implementation.
(Most mathematicians I've encountered have a faible for putting the "Aha!" - effect at the end rather than at the beginning. As have most elite geeks).
You can learn these skills without one lesson of math, of course. But math ahs perfected this for centuries.
Applied Skills
Examples:
- Not having to run calc.exe for a quick estimation of memory requirements
- Some basic statistics to tell a valid performance measurement from a shot in the dark
- deducing a formula for a sequence of values, rather than hardcoding them
- Getting a feeling for what c*O(N log N) means.
- Recursion is the same as proof by inductance
(that list would probably go on if I'd actively watch myself for items for a day. This part is admittedly harder than I thought. Further suggestions welcome ;))
Where I use it
The company I work for does a lot of data acquisition, and our claim to fame (comapred to our competition) is the brain muscle that goes into extracting something useful out of the data. While I'm mostly unconcerned with that, I get enough math thrown my way. Before that, I've implemented and validated random number generators for statistical applications, implemented a differential equation solver, wrote simulations for selected laws of physics. And probably more.
I wrote some hash functions for mapping airline codes and flight numbers with good efficiency into a fairly limited number of data slots.
I went through a fair number of primes before finding numbers that worked well with my data. Testing required some statistics and estimates of probabilities.
In machine learning: we use Bayesian (and other probabilistic) models all the time, and we use quadratic programming in the form of Support Vector Machines, not to mention all kinds of mathematical transformations for the various kernel functions. Calculus (derivatives) factors into perceptron learning. Not to mention a whole theory of determining the accuracy of a machine learning classifier.
In artifical intelligence: constraint satisfaction, and logic weigh very heavily.
I was using co-ordinate geometry to solve a problem of finding the visible part of a stack of windows, not exactly overlapping on one another.
There are many other situations, but this is the one that I got from the top of my head. Inherently all operations that we do is mathematics or at least depends on/related to mathematics.
Thats why its important to know mathematics to have a more clearer understanding of things :)
Infact in some cases a lot of math has gone into our common sense that we don't notice that we are using math to solve a particular problem, since we have been using it for so long!
Thanks
-Graphics (matrices, translations, shaders, integral approximations, curves, etc, etc,...infinite dots)
-Algorithm Complexity calculations (specially in line of business' applications)
-Pointer Arithmetics
-Cryptographic under field arithmetics etc.
-GIS (triangles, squares algorithms like delone, bounding boxes, and many many etc)
-Performance monitor counters and the functions they describe
-Functional Programming (simply that, not saying more :))
-......
I used Combinatorials to stuff 20 bits of data into 14 bits of space.
Machine Vision or Computer Vision requires a thorough knowledge of probability and statistics. Object detection/recognition and many supervised segmentation techniques are based on Bayesian inference. Heavy on linear algebra too.
As an engineer, I'm trying really hard to think of an instance when I did not need math. Same story when I was a grad student. Granted, I'm not a programmer, but I use computers a lot.
Games and simulations need lots of maths - fluid dynamics, in particular, for things like flames, fog and smoke.
As an e-commerce developer, I have to use math every day for programming. At the very least, basic algebra.
There are other apps I've had to write for vector based image generation that require a strong knowledge of Geometry, Calculus and Trigonometry.
Then there is bit-masking...
Converting hexadecimal to base ten in your head...
Estimating load potential of an application...
Yep, if someone is no good with math, they're probably not a very good programmer.
Modern communications would completely collapse without math. If you want to make your head explode sometime, look up Galois fields, error correcting codes, and data compression. Then symbol constellations, band-limited interpolation functions (I'm talking about sinc and raised-cosine functions, not the simple linear and bicubic stuff), Fourier transforms, clock recovery, minimally-ambiguous symbol training sequences, Rayleigh and/or Ricean fading, and Kalman filtering. All of those involve math that makes my head hurt bad, and I got a Masters in Electrical Engineering. And that's just off the top of my head, from my wireless communications class.
The amount of math required to make your cell phone work is huge. To make a 3G cell phone with Internet access is staggering. To prove with sufficient confidence that an algorithm will work in most all cases sometimes takes people's careers.
But... if you're only ever going to work with this stuff as black boxes imported from a library (at their mercy, really), well, you might get away with just knowing enough algebra to debug mismatched parentheses. And there are a lot more of those jobs than the hard ones... but at the same time, the hard jobs are harder to find a replacement for.
Examples that I've personally coded:
wrote a simple video game where one spaceship shoots a laser at another ship. To know if the ship was in the laser's path, I used basic algebra y=mx+b to calculate if the paths intersect. (I was a child when I did this and was quite amazed that something that was taught on a chalkboard (algebra) could be applied to computer programming.)
calculating mortgage balances and repayment schedules with logarithms
analyzing consumer buying choices by calculating combinatorics
trigonometry to simulate camera lens behavior
Fourier Transform to analyze digital music files (WAV files)
stock market analysis with statistics (linear regressions)
using logarithms to understand binary search traversals and also disk space savings when using packing information into bit fields. (I don't calculate logarithms in actual code, but I figure them out during "design" to see if it's feasible to even bother coding it.)
None of my projects (so far) have required topics such as calculus, differential equations, or matrices. I didn't study mathematics in school but if a project requires math, I just reference my math books and if I'm stuck, I search google.
Edited to add: I think it's more realistic for some people to have a programming challenge motivate the learning of particular math subjects. For others, they enjoy math for its own sake and can learn it ahead of time to apply to future programming problems. I'm of the first type. For example, I studied logarithms in high school but didn't understand their power until I started doing programming and all of sudden, they seem to pop up all over the place.
The recurring theme I see from these responses is that this is clearly context-dependent.
If you're writing a 3D graphics engine then you'd be well advised to brush up on your vectors and matrices. If you're writing a simple e-commerce website then you'll get away with basic algebra.
So depending on what you want to do, you may not need any more math than you did to post your question(!), or you might conceivably need a PhD (i.e. you would like to write a custom geometry kernel for turbine fan blade design).
One time I was writing something for my Commodore 64 (I forget what, I must have been 6 years old) and I wanted to center some text horizontally on the screen.
I worked out the formula using a combination of math and trial-and-error; years later I would tackle such problems using actual algebra.
Drawing, moving, and guidance of missiles and guns and lasers and gravity bombs and whatnot in this little 2d video game I made: wordwarvi
Lots of uses sine/cosine, and their inverses, (via lookup tables... I'm old, ok?)
Any geo based site/app will need math. A simple example is "Show me all Bob's Pizzas within 10 miles of me" functionality on a website. You will need math to return lat/lons that occur within a 10 mile radius.
This is primarily a question whose answer will depend on the problem domain. Some problems require oodles of math and some require only addition and subtraction. Right now, I have a pet project which might require graph theory, not for the math so much as to get the basic vocabulary and concepts in my head.
If you're doing flight simulations and anything 3D, say hello to quaternions! If you're doing electrical engineering, you will be using trig and complex numbers. If you're doing a mortgage calculator, you will be doing discrete math. If you're doing an optimization problem, where you attempt to get the most profits from your widget factory, you will be doing what is called linear programming. If you are doing some operations involving, say, network addresses, welcome to the kind of bit-focused math that comes along with it. And that's just for the high-level languages.
If you are delving into highly-optimized data structures and implementing them yourself, you will probably do more math than if you were just grabbing a library.
Part of being a good programmer is being familiar with the domain in which you are programming. If you are working on software for Fidelity Mutual, you probably would need to know engineering economics. If you are developing software for Gallup, you probably need to know statistics. LucasArts... probably Linear Algebra. NASA... Differential Equations.
The thing about software engineering is you are almost always expected to wear many hats.
More or less anything having to do with finding the best layout, optimization, or object relationships is graph theory. You may not immediately think of it as such, but regardless - you're using math!
An explicit example: I wrote a node-based shader editor and optimizer, which took a set of linked nodes and converted them into shader code. Finding the correct order to output the code in such that all inputs for a certain node were available before that node needed them involved graph theory.
And like others have said, anything having to do with graphics implicitly requires knowledge of linear algebra, coordinate spaces transformations, and plenty of other subtopics of mathematics. Take a look at any recent graphics whitepaper, especially those involving lighting. Integrals? Infinite series?! Graph theory? Node traversal optimization? Yep, all of these are commonly used in graphics.
Also note that just because you don't realize that you're using some sort of mathematics when you're writing or designing software, doesn't mean that you aren't, and actually understanding the mathematics behind how and why algorithms and data structures work the way they do can often help you find elegant solutions to non-trivial problems.
In years of webapp development I didn't have much need with the Math API. As far as I can recall, I have ever only used the Math#min() and Math#max() of the Math API.
For example
if (i < 0) {
i = 0;
}
if (i > 10) {
i = 10;
}
can be done as
i = Math.max(0, Math.min(i, 10));
I was glancing through the contents of Concrete Maths online. I had at least heard most of the functions and tricks mentioned but there is a whole section on Special Numbers. These numbers include Stirling Numbers, Eulerian Numbers, Harmonic Numbers so on. Now I have never encountered any of these weird numbers. How do they aid in computational problems? Where are they generally used?
Harmonic Numbers appear almost everywhere! Musical Harmonies, analysis of Quicksort...
Stirling Numbers (first and second kind) arise in a variety of combinatorics and partitioning problems.
Eulerian Numbers also occur several places, most notably in permutations and coefficients of polylogarithm functions.
A lot of the numbers you mentioned are used in the analysis of algorithms. You may not have these numbers in your code, but you'll need them if you want to estimate how long it will take for your code to run. You might see them in your code too. Some of these numbers are related to combinatorics, counting how many ways something can happen.
Sometimes it's not enough to know how many possibilities there are because you need to enumerate over the possibilities. Volume 4 of Knuth's TAOCP, in progress, gives the algorithms you need.
Here's an example of using Fibonacci numbers as part of a numerical integration problem.
Harmonic numbers are a discrete analog of logarithms and so they come up in difference equations just like logs come up in differential equations. Here's an example of physical applications of harmonic means, related to harmonic numbers. See the book Gamma for many examples of harmonic numbers in action, especially the chapter "It's a harmonic world."
These special numbers can help out in computational problems in many ways. For example:
You want to find out when your program to compute the GCD of 2 numbers is going to take the longest amount of time: Try 2 consecutive Fibonacci Numbers.
You want to have a rough estimate of the factorial of a large number, but your factorial program is taking too long: Use Stirling's Approximation.
You're testing for prime numbers, but for some numbers you always get the wrong answer: It could be you're using Fermat's Prime test, in which case the Carmicheal numbers are your culprits.
The most common general case I can think of is in looping. Most of the time you specify a loop using a (start;stop;step) type of syntax, in which case it may be possible to reduce the execution time by using properties of the numbers involved.
For example, summing up all the numbers from 1 to n when n is large in a loop is definitely slower than using the identity sum = n*(n + 1)/2.
There are a large number of examples like these. Many of them are in cryptography, where the security of information systems sometimes depends on tricks like these. They can also help you with performance issues, memory issues, because when you know the formula, you may find a faster/more efficient way to compute other things -- things that you actually care about.
For more information, check out wikipedia, or simply try out Project Euler. You'll start finding patterns pretty fast.
Most of these numbers count certain kinds of discrete structures (for instance, Stirling Numbers count Subsets and Cycles). Such structures, and hence these sequences, implicitly arise in the analysis of algorithms.
There is an extensive list at OEIS that lists almost all sequences that appear in Concrete Math. A short summary from that list:
Golomb's Sequence
Binomial Coefficients
Rencontres Numbers
Stirling Numbers
Eulerian Numbers
Hyperfactorials
Genocchi Numbers
You can browse the OEIS pages for the respective sequences to get detailed information about the "properties" of these sequences (though not exactly applications, if that's what you're most interested in).
Also, if you want to see real-life uses of these sequences in analysis of algorithms, flip through the index of Knuth's Art of Computer Programming, and you'll find many references to "applications" of these sequences. John D. Cook already mentioned applications of Fibonacci & Harmonic numbers; here are some more examples:
Stirling Cycle Numbers arise in the analysis of the standard algorithm that finds the maximum element of an array (TAOCP Sec. 1.2.10): How many times must the current maximum value be updated when finding the maximum value? It turns out that the probability that the maximum will need to be updated k times when finding a maximum in an array of n elements is p[n][k] = StirlingCycle[n, k+1]/n!. From this, we can derive that on the average, approximately Log(n) updates will be necessary.
Genocchi Numbers arise in connection with counting the number of BDDs that are "thin" (TAOCP 7.1.4 Exercise 174).
Not necessarily a magic number from the reference you mentioned, but nonetheless --
0x5f3759df
-- the notorious magic number used to calculate inverse square root of a number by giving a good first estimate to Newton's Approximation of Roots, often attributed to the work of John Carmack - more info here.
Not programming related, huh? :)
Is this directly programming related? Surely related, but I don't know how closely.
Special numbers, such as e, pi, etc., come up all over the place. I don't think that anyone would argue about these two. The Golden_ratio also appears with amazing frequency, in everything from art to other special numbers themselves (look at the ratio between successive Fibonacci numbers.)
Various sequences and families of numbers also appear in many places in mathematics and therefore, in programming too. A beautiful place to look is the Encyclopedia of integer sequences.
I'll suggest this is an experience thing. For example, when I took linear algebra, many, many years ago, I learned about the eigenvalues and eigenvectors of a matrix. I'll admit that I did not at all appreciate the significance of eigenvalues/eigenvectors until I saw them in use in a variety of places. In statistics, in terms of what they tell you about uncertainty of an estimate from a covariance matrix, the size and shape of a confidence ellipse, in terms of principal component analysis, or the long term state of a Markov process. In numerical methods, where they tell you about convergence of a method, be it in optimization or an ODE solver. In mechanical engineering, where you see them as principal stresses and strains.
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