I'm trying to create an application in java which does several matrix modifications like calculating the invereses and determinants.
Now I would also like to include the option for the application to calculate the eigenvalues and the eigenvectors for matrices.
Since the only 'solid' way to calculate eigenvalues, by my knowledge, is by using the characteristic formula given by:
det(A-λI) = 0
Where A is an nxn matrix and λ a real number.
To my knowledge, there is no simple, maybe none at all, way to use algebra in Java. Also I would like to program this myself, so I would like not to use external packages like Jama or others.
Can someone explain me how I can program this equation in Java or maybe tell me another way of doing it?
One way you could do it is have a look at Jama and see how it is calculated in there and interpret that. And don't just Copy and Paste :P we all know who tempting that can be.
Finding eigenvalues and eigenvectors is a bit tricky, and there are many algorithms with varying positives and negatives. I'll suggest a few that are quite good and that are not that difficult to implement.
First, compute the characteristic polynomial and then find the roots using. Then you have the eigenvalues. Then you can solve a set of equations to find the eigenvectors given the eigenvalues.
Related
I'm trying to find a way to compute roots of a polynomial with complex coefficients in Java (i.e. an equivalent of what is ridiculously easily done with roots() in MATLAB).
I'm ready to recode a root finding algorithm that builds the companion matrix and then uses generalized eigenvalue decomposition to find the roots, but for this I would need a library that handles complex-valued matrix operations.
I browsed for a while and nothing convincing seems to be available out there, which I think is rather weird. Then, I'd like to ask you:
Do you know a (stable) Java library that performs root finding on polynomials defined by COMPLEX coefficients?
Do you know a (stable) Java library that performs evd, svd, inverse, etc. on COMPLEX-valued matrices?
Note: I already looked at JAMA (doesn't handle complex), Michael Thomas Flanagan's Java Scientific Library (not available anymore), colt (doesn't seem to handle complex), Efficient Java Matrix Library (no complex either), DDogleg Numerics (does not handle polynomial with complex coefficients), JScience (not clear if evd is available) and common-math from Apache (not clear if they allow for complex matrices, and if yes, if evd is available).
The Durand-Kerner method also works for complex coefficients and does not rely on matrix computations.
It's quite simple to implement, you could google up an implementation (Stackoverflow forbids me to link the one I found) or make one of your own. You could use the jscience library for the complex data types, not for the algorithm itself.
EDIT: Didn't see that you need evd too, never mind my mention of jscience as an option to do the complex matrix math.
If one wants to keep it real, use the Bairstow method. If the polynomial has odd degree, use first Newton's method to find a real root and reduce the polynomial to even degree. This avoids an odd singularity of the Bairstow method where it converges towards a quadratic polynomial that has infinity as one root. Information of good quality can be found at the usual places. Some of it written or edited by yours truly.
Determine an inner root radius r and use z^2-2r*cos(phi)*z+r^2 with random angle phi as initial factor for Bairstow's method. It produces in each step a quadratic factor, always in and with real coefficients, containing either a pair of real roots or a conjugate pair of complex roots.
Check in each step for speed of convergence and restart with a different initial point if necessary. Find new roots after deflation, and polish the roots or quadratic factors by executing the method with the original polynomial and the factors as starting point.
I have some function (for example, double function(double value)), and some range (for example, from A to B). I need to calculate max value of function in this range. Are there existed libraries for it? Please, give me advice.
If the function needs to handle floating-point values, you're going to have to use something like Golden section search. Note that for this specific method, there are significant limitations regarding the functions that can be handled (specifically it must be unimodal). There are some adjustments you can make to the algorithm which extend it to more functions, specifically these modifications will allow it to work for continuous functions.
Is this a continuous function, or a set of discrete values? If discrete values, then you can either iterate over all values, and set max/min flags as 808sound suggests, or you can load all values into an array.
If it's a continuous function, then you can either populate an array with the function's value at discrete inputs, and find the max as above, or if it's differentiable, then you can use basic calculus to find the points at which df(x)/dx are 0. The latter case is a little more abstract, and probably more complicated than you want, though?
A quick google search led me to this:
http://code.google.com/p/javacalculus/
But I've never used it myself, so I don't know if that implements the required functionality. It does differential equations, though, so I assume they'd have "baby stuff" like basic differentiation.
I do not know if there are any librairies in Java for your problem.
But I know you can easily do that with MatLab (or Octave for the OpenSource equivalent).
If you do not have any indication of what the functions inner workings are (i.e. the function is a black box that accepts an input and produces an output), there is no "easy" way to find the global maximum.
There are an infinite number of points to choose for your input (technically) so "iterating over all possible inputs" is not feasible mathematically.
There are various algorithms that will give you estimated maximum values ina function like this:
The hill climbing algorithm, and the firefly algorithm are two, but there are many more. This is a fairly well documented/studied computer science problem and there is a lot of material online for you to look at. I suggest starting with the hill climbing algorithm, and maybe expanding out to other global optimization algorithms.
Note: These algorithms do not guarantee that the result is the maximum, but provide an estimate of its value.*
I have a bunch of sets of data (between 50 to 500 points, each of which can take a positive integral value) and need to determine which distribution best describes them. I have done this manually for several of them, but need to automate this going forward.
Some of the sets are completely modal (every datum has the value of 15), some are strongly modal or bimodal, some are bell-curves (often skewed and with differing degrees of kertosis/pointiness), some are roughly flat, and there are any number of other possible distributions (possion, power-law, etc.). I need a way to determine which distribution best describes the data and (ideally) also provides me with a fitness metric so that I know how confident I am in the analysis.
Existing open-source libraries would be ideal, followed by well documented algorithms that I can implement myself.
Looking for a distribution that fits is unlikely to give you good results in the absence of some a priori knowledge. You may find a distribution that coincidentally is a good fit but is unlikely to be the underlying distribution.
Do you have any metadata available that would hint at what the data means? E.g., "this is open-ended data sampled from a natural population, so it's some sort of normal distribution", vs. "this data is inherently bounded at 0 and discrete, so check for the best-fitting Poisson".
I don't know of any distribution solvers for Java off the top of my head, and I don't know of any that will guess which distribution to use. You could examine some statistical properties (skew/etc.) and make some guesses here--but you're more likely to end up with an accidentally good fit which does not adequately represent the underlying distribution. Real data is noisy and there are just too many degrees of freedom if you don't even know what distribution it is.
This may be above and beyond what you want to do, but it seems the most complete approach (and it allows access to the wealth of statistical knowledge available inside R):
use JRI to communicate with the R statistical language
use R, internally, as indicated in this thread
Look at Apache commons-math.
What you're looking for comes under the general heading of "goodness of fit." You could search on "goodness of fit test."
Donald Knuth describes a couple popular goodness of fit tests in Seminumerical Algorithms: the chi-squared test and the Kolmogorov-Smirnov test. But you've got to have some idea first what distribution you want to test. For example, if you have bell curve data, you might try normal or Cauchy distributions.
If all you really need the distribution for is to model the data you have sampled, you can make your own distribution based on the data you have:
1. Create a histogram of your sample: One method for selecting the bin size is here. There are other methods for selecting bin size, which you may prefer.
2. Derive the sample CDF: Think of the histogram as your PDF, and just compute the integral. It's probably best to scale the height of the bins so that the CDF has the right characteristics ... namely that the value of the CDF at +Infinity is 1.0.
To use the distribution for modeling purposes:
3. Draw X from your distribution: Make a draw Y from U(0,1). Use a reverse lookup on your CDF of the value Y to determine the X such that CDF(X) = Y. Since the CDF is invertible, X is unique.
I've heard of a package called Eureqa that might fill the bill nicely. I've only downloaded it; I haven't tried it myself yet.
You can proceed with a three steps approach, using the SSJ library:
Fit each distribution separately using maximum likelihood estimation (MLE). Using SSJ, this can be done with the static method getInstanceFromMLE(double[] x,
int n) available on each distribution.
For each distribution you have obtained, compute its goodness-of-fit with the real data, for example using Kolmogorov-Smirnov: static void kolmogorovSmirnov (double[] data, ContinuousDistribution dist, double[] sval,double[] pval), note that you don't need to sort the data before calling this function.
Pick the distribution having the highest p-value as your best fit distribution
I have data I would like to plot, and more importantly, do a least squares regression on using cosines (instead of using polynomials):
Any recommendations? Thanks.
Probably the following page solves the regression part of your aim:
http://www.teneighty.org/software/index.html?f=fft&c=e98b8
You might find this demo Least Squares & Data Fitting helpful since it solves a few of your problems.
Just a bit of cautionary advice. Using a Fourier series makes sense if you think your underlying function has a cosine series as a basis; however, if you are using it as a basis for an arbitrary function (with unknown shape), you may do better trying to guess at a more specific underlying function type (polynomial, exponential, etc).
I did some constrained optimization on such a series, and the function wiggled around so much it was hard to say if my fit was meaningfull; my fit function had great number of local maxima.
MathGL can plot, fit (by help of GSL) and show fitting result - see this sample
I asked a question before but duffymo said it is not clear so i am going to post it again here.
I am using Jama api for SVD calculation. I know very well about jama and SVD.
Jama does not work if your column are more than rows. I have this situation. What should I do?? any help?
I can't transpose the matrix too as it can produce wrong results.
Thanks.
P.S: I am calculating LSI with the help of jama. I am going like column(docs) and rows ( terms )
Why not use transpose? If X = USV', then X' = VS'U'. Right?
Transpose your matrix. Get U, S and V. Transpose everything back.
If I understand correctly you are trying to compute the SVD of a matrix which is not square, and you have the library JAMA which only works on square matrices? If I have understood you correctly then the answer to your question is obvious: Get a library which does compute SVD for non-square matrices. If I remember correctly Numerical Recipes contains such an algorithm, I expect you can find many other sources with Google.
Since you're doing LSI, you could use SVDLIBJ, which is the Java equivalent of SVDLIBC, which is one of the most scalable SVD implementations that is freely available. The S-Space package has a command-line tool for SVDLIBJ set up already. Also, you can use their Matrix libraries and avoid the command-line if that fits your needs better.
I now it is a really late reply . But i thought it is better late than never
But i am aware that jblas performs svd in a effective manner.