I have a number of small algorithms that I would like to write up in a paper. They are relatively short, and concise. However, instead of writing them in pseudo-code (à la Cormen or even Knuth), I would like to write an algebraic representation of them (more linear and better LaTeX rendering) . However, I cannot find resources as to the best notation for this, if there is anything: e.g. how do I represent a loop? If? The addition of a tuple to a list?
Has any of you encountered this problem, and somehow solved it?
Thanks.
EDIT: Thanks, people. I think I did a poor job at phrasing the question. Here goes again, hoping I make it clearer: what is the common notation for talking about loops and if-then clauses in a mathematical notation? For instance, I can use $acc \leftarrow acc \cup \langle i,i+1 \rangle$ to represent the "add" method of a list.
Don't do this. You are deviating from what people expect to see when they read a paper about algorithms. You should follow expected practices; your ideas are more likely to get the attention that they deserve. When in Rome, do as the Romans do.
Formatting code (or pseudocode as it may be) in a LaTeXed paper is very easy. See, for example, Formatting code in LaTeX.
I see if-expressions in mathematical notation fairly often. The usual thing for a loop is a recurrence relation, or equivalently, a function defined recursively.
Here's how the Ackermann function is defined on Wikipedia, for instance:
This picture is nice because it feels mathematical in flavor and yet you could clearly type it in almost exactly as written and have an implementation. It is not always possible to achieve that.
Other mathematical notations that correspond to loops include ∑-notation for summation and set-builder notation.
I hope this answers your question! But if your aim is to describe how something is done and have someone understand, I think it is probably a mistake to assume that mathematicians would prefer to see equations. I don't think they're interchangeable tools (despite Turing equivalence). If your algorithm involves mutable data structures, procedural code is probably going to be better than equations for explaining it.
I'd copy Knuth. Few know how to communicate better than him in a computer science setting.
A symbol for general loops does not exist; usually you will use the summation operator. "if" is represented using implications, and to "add a tuple to a list" you would use union.
However, in general, a bit of verbosity is not necessarily a bad thing - sometimes, especially for complex algorithms, it is best to spell it out in plain English, using examples and diagrams. This is doubly-true for non-coders.
Think about it: when you read a math text-book on Euclid's algorithm for GCD, or the sieve of Eratosthenes, how is it written? Usually, the algorithm itself is in prose, while the proof of the algorithm is where the mathematical symbols lie.
You might take a look at Haskell. Haskell formats well in latex, has a nice algebraic syntax, and you can even compile a latex file with Haskell in it, provided the code is wrapped in \begin{code} and \end{code}. See here: http://www.haskell.org/haskellwiki/Literate_programming. There are probably literate programming tools for other languages.
Lisp started out as a mathematical notation of a computing model so that the lecturer would have a better tool than turing machines. By accident, it turns out that it can be implemented in assembly - thus lisp, the programming language was born.
But I don't think this is really what you are looking for since the computing model that lisp describes doesn't have loops: recursion is used instead. The syntax derives from algebra where braces denote evaluate-this-and-substitute-the-result. Indeed, lisp's model of computing is basically substitution - what algebra essentially is.
Indeed, most functional languages like Lisp, Haskell and Erlang are derived from mathematics. Haskell is actually a result of proving that lambda calculus can be used to implement type systems. So Haskell, like Lisp was born out of pure mathematics. But again, the syntax is not what you would probably be used to.
You can certainly explain Lisp and Haskell syntax to mathematicians and they would treat it as a "game". Language constructs like loops, recursion and conditionals can be proven out of the rules of the game rather than blindly implemented like in other languages. This would lead you into the realms of combinatronics, another branch of mathematics. Indeed, in combinatronics, even the concept of numbers can be constructed out of the rules of the game rather than being a native part of the language (google Church Numerals).
So have a look at Lisp/Scheme, Erlang and Haskell if you want. Erlang especially has syntax close to what you want:
add(a,b) -> a + b
But my recommendation is to write in C-like pseudocode. It's sort of the lowest common denominator in programming languages. Has a syntax that is fairly easy to understand and clean. And the function syntax even derives from functions in mathematics. Remember f(x)?
As a plus, mathematicians are used to writing C, statisticians are used to writing C (though generally they prefer R), physicists are used to writing C, programmers are used to at least looking at C (I know a few who've never touched C).
Actually, scratch that. You mention that your target audience are statisticians. Write in R
Something like this website describes?
APL? The only problem is that few people can read it.
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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));
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I'm not sure if this is for SO or not. I am reading some of my old math textbooks and trying to understand math in general. Not how to figure something. I can do that but rather what is it that math is doing.
I'm sure this is painfully obvious but I never thought about it until I thought more about game programming. Is it right to think about math as the "language" that is used to explain, precisely explain, why things work?
I'm having a hard time asking it and again, I'm sure it's obvious to most, but after years of math I'm finally thinking when someone asks to "find the equation of a line" that people recognized certain characteristics of a line (y=mx+b) in space and found relationship. They needed something beside a huge paragraph (like this one) and something very precise. We call this math and at its base it's nothing more than a symbolic way to represent things.
Really, I was thinking, "I know why they said 'find the equation of a line'."
So now I am thinking, not just googling for a formula that tells me how to turn a curve with a walking man or follow a path, but why and how do I represent this mathematically and then programatically.
Just hoping for comments on math in programming.
To my way of thinking, I create a "model" of some aspect of the world. Examples:
Profit = Income - Expenditure
I throw a ball it's path will be a parabola with equation ...
I then represent the model in a computer program. So some kind of abstaction underpins the program, sometimes the math is so "obvious" we hardly notice it, sometimes (eg. simulation games) it's both very clearly there and pretty darn tricky.
Key idea: math can be used to model reality, most business systems can be viewed as represented as a model of reality.
Having said that, in 30 years of programming the amount of true (algebra, calculus) maths I have done is negligable.
Steve Yegge wrote a very good article that you may find helpful: Math Every Day
I recommend that you look into materials related to the theory of computation. For example:
On Computable Numbers, with an Application to the Entscheidungsproblem - Alan Turing (1936)
The Mathematical Theory of Communication - Claude Shannon (1948)
The General and Logical Theory of Automata - John Von Neumann (1951)
These are not papers for the faint of heart, but they will give you insights into the beautiful relationship between mathematics and computer science.
You might want to start with a textbook on the subject of computation theory before you tackle the papers listed above, e.g.
Introduction to the Theory of Computation - Michael Sipser
Math for a programmer is like a hammer for a carpenter. The carpenter doesn't use the hammer for everything, but if he doesn't have one, there's a lot he can't do.
Not sure what your precise question is ...
Some thoughts:
Programming is nothing but math (Functional programming, Lambda calculus, programming == math)
Math is a kind of language - An abstract description/representation of an expression in thought
Math helps you to formalize expressions: Instead of For all integer numbers x from one to ten the square of x is less than 250 you can write ∀x ∈ {1..10} (x² < 250)
Programming (a programming language) does the same thing and helps to formalize algorithms.
The kind math that is commonly used in computer programms is numeric math, but with some efforts, you can also perform symbolic computations
I think math is really the concepts behind the symbols instead of the symbols themselves, but when most people speak of math, they're not making the distinction. They're just thinking of the symbols. Partly, this is because of they way math is taught in school, where the focus is on the mechanistic manipulation of the symbols to get correct results, rather than what the concepts are.
This is similar to the way non-programmers view programming. They look at a computer program and see gibberish, whereas a programmer in the given language (after more or less effort) understands the behavior the code represents.
Some people are better at retaining the meaning of such symbols than others. I think there are people who might appreciate math more than they think if they could get past that barrier to the concepts.
I agree with Taylor. Math inside computers is a very deep topic with numerical methods. The biggest issues is precision and the fact that 32 bits only get you so far. There are some really cool (and complicated) functions that describe how to find integrals and such with computers, but because we can't be exact with our answers, and because computers are limited with what they can do (add, multiply, etc) there are lots of methods of how to estimate math to a great deal of precision.
If you are interested in that topic, all the more power to you. That was one class I struggled through.
I'm looking at something similar (financial models) - similar in that we come up with mathematical models, and then implement these in code.
The main issue you face from a programming perspective is taking a model that is expressed in mathematical terms (which assume continuity, infinitely small time/space steps etc.) and then translate these into 'discrete' models, that assume finite time/space steps (e.g. the ball moves every 1mm, or every 1ms).
The translation of these models is not necessarily trivial, and you should have a look at appropriate references for these (Numerical Recipes is a classic). The implementation in code is often very different to how you might express the problem in mathematical terms.
I think Math in programming with time, silence and good food such that I have a lot of paper and a pen, friends-to-ask-help and a pile of books from Rudin to Bourbaki on the top of my Macbook on the floor.
I think why is a philosophical question.
As far as how I think of math/programming and the interplay between... I think of them as layers of modeling. At the lowest, 'truest' level there is some fundamental truth, whatever that may be. Then there is the mathematical modeling of this truth, upon which the 'language' of mathematics is developed (fortunately there is only one language?). Then there is another layer, that of modeling and approximations. In the case of y=mx+b, its only a line within one model, it could be anything. Being visual beings, the most beneficial is perhaps geometric (lines, surfaces, etc). Then upon this there is the computational modeling, the numerical methods/analysis if you will.
As to how do i think of things, I like to think in the modeling perspective. That is, I like to conceptually model some process, and then apply the math and then the numerical methods. Middle out development if you will (to draw an N-tier analogy).
As an afterthought, perhaps the modeling could be called engineering.
The best way to get the type of understanding that you're looking for is to work through "story problems" (i.e. problems stated in words rather than equations). From this and your other questions, you're mostly looking at trigonometry.
In short, I would recommend trying the trig book from the Schaum's Outline Series -- they are cheap (~$13) and have lots of problems with solutions.
There are other routes to finding problems in math to solve, such as just make up game design problems to solve. Here are two: 1) show an object moving around a circle at constant speed, and 2) show two object moving along to different lines that don't intersect, and draw a line between them. Or you could get a book that walks you through these types of things. But you've got to work out a number of problems to force you to think things through yourself.