The following linear programming problem is not of canonical form. I am really stuck when trying to put this in regular form and feed it into the normal lp() function.
Does someone has experience with such weird form?
B and A are the blocker and antiblocker, respectively, which are simply two sets of inequalities.
I don't know what the "normal lp() function" is. Let's assume this is the lp function from the LpSolve package.
This function does not expect a canonical form. (Canonical usually means each constraint has the same fixed sign, e.g. Ax<=b; lp() allows different signs for each constraint).
lp() just wants one big constraint matrix: each column is an individual variable and each row is an individual constraint. This is conceptual simple, but often tedious in practice. Best thing to do is to get a large piece of paper and draw the layout of the LP matrix: which variables and constraints go where.
For some classes of models there are easier-to-use tools to express an LP model, such as OMPR, CVXR.
Is there an efficient implementation of the set data structure in R?
In C++ I would use an std::set (which is implemented using red-black trees), in Python a set (which is implemented using hash tables), but I am not sure what I should use in R.
I have found this link, which describes some set operations, like union() and intersection(), that you can perform on vectors. So, I guess that since vectors are involved, the complexities would not be logarithmic, as you could have using the data structures mentioned above.
Fun fact, note how in this case the name of the language does not help, searching "r set" one finds many results concerning $\mathbb{R}$, and not the programming language :D
I'm currently working on a math library.
It's now supporting several matrix operations:
- Plus
- Product
- Dot
- Get & Set
- Transpose
- Multiply
- Determinant
I always want to generalize everything I can generalize
I was thinking about a recursive way to implement the transpose of a matrix, but I just couldn't figure it out.
Anybody help?
I would advise you against trying to write a recursive method to transpose a matrix.
The idea is easy:
transpose(A) = A(j,i)
Recursion isn't hard in this case. You can see the stopping condition: a 1x1 matrix with a single value is its own transpose. Build it up for 2x2, etc.
The problem is that this will be terribly inefficient, both in terms of stack depth and memory, for any matrix beyond a trivial size. People who apply linear algebra to real problems can require tens of thousands or billions of degrees of freedom.
You don't talk about meaningful, practical cases like sparse or banded matricies, etc.
You're better off doing it using a straightforward declarative approach.
Haskell use BLAS as its backing implementation. It's a more functional language than JavaScript. Perhaps you could crib some ideas by looking at the source code.
I'd recommend that you do the simple thing first, get it all working, and then branch out from there.
Here's a question to ask yourself: Why would anyone want to do serious numerical work using JavaScript? What will your library offer that's an improvement on what's available?
If you want to learn how to reinvent wheels, by all means proceed. Just understand that you aren't the first.
I would like to define a slightly more general version of a complex number in R. This should be a vector that has more than one component, accessible in a similar manner to using Re() and Im() for complex numbers. Is there a way to do this using S3/S4 classes?
I have read through the OO field guide among other resources, but most solutions seem focused around the use of lists as fundamental building objects. However, I need vectors for use in data.frames and matrices. I was hoping to use complex numbers as a template, but they seem to be implemented largely in C. At this point, I don't even know where to start.
If I have a function f(x) = y that I don't know the form of, and if I have a long list of x and y value pairs (potentially thousands of them), is there a program/package/library that will generate potential forms of f(x)?
Obviously there's a lot of ambiguity to the possible forms of any f(x), so something that produces many non-trivial unique answers (in reduced terms) would be ideal, but something that could produce at least one answer would also be good.
If x and y are derived from observational data (i.e. experimental results), are there programs that can create approximate forms of f(x)? On the other hand, if you know beforehand that there is a completely deterministic relationship between x and y (as in the input and output of a pseudo random number generator) are there programs than can create exact forms of f(x)?
Soooo, I found the answer to my own question. Cornell has released a piece of software for doing exactly this kind of blind fitting called Eureqa. It has to be one of the most polished pieces of software that I've ever seen come out of an academic lab. It's seriously pretty nifty. Check it out:
It's even got turnkey integration with Amazon's ec2 clusters, so you can offload some of the heavy computational lifting from your local computer onto the cloud at the push of a button for a very reasonable fee.
I think that I'm going to have to learn more about GUI programming so that I can steal its interface.
(This is more of a numerical methods question.) If there is some kind of observable pattern (you can kinda see the function), then yes, there are several ways you can approximate the original function, but they'll be just that, approximations.
What you want to do is called interpolation. Two very simple (and not very good) methods are Newton's method and Laplace's method of interpolation. They both work on the same principle but they are implemented differently (Laplace's is iterative, Newton's is recursive, for one).
If there's not much going on between any two of your data points (ie, the actual function doesn't have any "bumps" whose "peaks" are not represented by one of your data points), then the spline method of interpolation is one of the best choices you can make. It's a bit harder to implement, but it produces nice results.
Edit: Sometimes, depending on your specific problem, these methods above might be overkill. Sometimes, you'll find that linear interpolation (where you just connect points with straight lines) is a perfectly good solution to your problem.
It depends.
If you're using data acquired from the real-world, then statistical regression techniques can provide you with some tools to evaluate the best fit; if you have several hypothesis for the form of the function, you can use statistical regression to discover the "best" fit, though you may need to be careful about over-fitting a curve -- sometimes the best fit (highest correlation) for a specific dataset completely fails to work for future observations.
If, on the other hand, the data was generated something synthetically (say, you know they were generated by a polynomial), then you can use polynomial curve fitting methods that will give you the exact answer you need.
Yes, there are such things.
If you plot the values and see that there's some functional relationship that makes sense, you can use least squares fitting to calculate the parameter values that minimize the error.
If you don't know what the function should look like, you can use simple spline or interpolation schemes.
You can also use software to guess what the function should be. Maybe something like Maxima can help.
Wolfram Alpha can help you guess:
http://blog.wolframalpha.com/2011/05/17/plotting-functions-and-graphs-in-wolframalpha/
Polynomial Interpolation is the way to go if you have a totally random set
http://en.wikipedia.org/wiki/Polynomial_interpolation
If your set is nearly linear, then regression will give you a good approximation.
Creating exact form from the X's and Y's is mostly impossible.
Notice that what you are trying to achieve is at the heart of many Machine Learning algorithm and therefor you might find what you are looking for on some specialized libraries.
A list of x/y values N items long can always be generated by an degree-N polynomial (assuming no x values are the same). See this article for more details:
http://en.wikipedia.org/wiki/Polynomial_interpolation
Some lists may also match other function types, such as exponential, sinusoidal, and many others. It is impossible to find the 'simplest' matching function, but the best you can do is go through a list of common ones like exponential, sinusoidal, etc. and if none of them match, interpolate the polynomial.
I'm not aware of any software that can do this for you, though.