Could anybody, please, point me on the algorithm of how matrix multiplication can be done (Identifying the order of multiplying). I tried really different way which I could design, but nothing works as expected, unfortunately :( For example I have one of the following matrix chains to multiply:
((((0(((((((((((((((1 2)3)4)5)6)7)8)9)10)11)12)13)14)15)16))17)18)19
((((((((((((0 1)2)3)4)5)6)7)8)9)10)(11(12(13(14(15(16(1718))))))))19
(((((0 1)(2(((3 4)(5 6))(((7 8)(9(10 11)))((((12 13)14)15)16)))))17)18)19
(((((((0 1)2)3)4)5)(6(7(8(9(10(11(12 13))))))))(((((14 15)16)17)18)19)
Can somebody share a universal algorithm for all the cases?
NOTE:
1.I need multiplication only.
2.Matrices have different dimension, so they are not square. (Naturally, Matrices' dimensions agree)
3.Maybe this topic can be helpful, but I really have no idea how it can be applied here.
Thanks in advance to everybody who is trying to help!
What you are looking for is called "Matrix chain multiplication". http://en.wikipedia.org/wiki/Matrix_chain_multiplication
Related
I have 3 points [x0 y0], [x1 y1], [x2 y2] with strict conditional x0<x1<x2, y0<y1<y2. All this points lay on some exponentional functions y=ae^(bx)+c. I need to find a,b,c... It's not possible to solve system of 3 equations precisely, therefore I need to approximate it. Is there some math library in java that will help me solve this problem? I find something similar on mathcad
https://help.ptc.com/mathcad/en/index.html#page/PTC_Mathcad_Help/exponential_regression.html but not find in java.
Other way - how to solve system of 3 equations and 3 values approximately.
ae^(bx_0)+c=y_0
ae^(bx_1)+c=y_1
ae^(bx_2)+c=y_2
You have to solve a system of non-linear equations, for which only an approximate solution is possible but can be done using the Newton Raphson's Multivariate method.
The algorithm is, quite frankly, a notational pain but you can go through it here -
http://fourier.eng.hmc.edu/e176/lectures/NM/node21.html.
What is happening essentially is you have a function whose derivative lead you to an 'equilibrium' from an initial random point (which you guess as a possible root)
If you are not willing to write the code yourself this repo can give you a starter of sorts - https://github.com/prasser/newtonraphson.
But AFAIK, no ready library exists for this purpose. You can use Wolfram's Mathematica or MATLAB/OCTAVE for ready libraries though.
That said, here are a few other (more complicated) things you can look into
https://en.wikipedia.org/wiki/Levenberg%E2%80%93Marquardt_algorithm
https://www1.fpl.fs.fed.us/optimization.html
http://icl.cs.utk.edu/f2j/
http://optalgtoolkit.sourceforge.net/
http://scribblethink.org/Computer/Javanumeric/index.html
https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.fmin_l_bfgs_b.html
Hope this helps!
I'm looking for a Java library to solve this problem:
We know X is sparse(most of it's entries are zero), so X can be recovered by solving this:
variable X;
minimize(norm(X,1)+norm(A*X - Y,2));
It's a MATLAB code, matrix A and vector Y are known and I want the best X.
I saw JOptimizer, but I couldn't use it. (Doesn't have good documentation or examples).
What you need is a reasonably good LP Solver.
Possible Java LP Solver Options
Apache Commons (Math) Simplex Solver.
See this blog post.
If you have access to CPLEX (not-free), its Java API would work great.
Also, you can look into SuanShu, a Java numerical and statistical library
lpSolve has a Java wrapper which can do the job.
Finally, JOptimizer is indeed a good option. Not sure if you looked at this example.
Hope at least one of those help.
As far as I can tell, you're trying to solve a binary integer program for feasibility
Ax = b, x in {0,1}.
I'm not completely sure, but it seems that you might be interested in the optimization problem
min 1'*x
s.t. Ax = b, x in {0,1}
where 1 is a vector of 1's of the same dimension as x.
The feasibility problem may be in practice much easier than the optimization problem - it all depends on a particular A and b.
If you can get a license of either CPLEX or Gurobi (if you're an academic), these are excellent integer programming solvers with good Java API's. If you don't have access to these, lpsolve may be a good option.
As far as I can tell, JOptimizer will not solve your problem since your variables are integers (although I have never used JOptimizer).
To solve convex optimization problems in java you can use the following library https://github.com/erikerlandson/gibbous
I am wanting to make a program that will when given a formula, it can manipulate the formula to make any value (or in the case of a simultaneous formula, a common value) the subject of the formula.
For example if given:
a + b = c
d + b = c
The program should therefore say:
b = c - a, d = c - b etc.
I'm not sure if java can do this automatically or not when I give the original formula as input. I am not really interested in solving the equation and getting the result of each variable, I am just interested in returning a manipulated formula.
Please let me know if I need to make an algorithm or not for this, and if so, how would I go about doing this. Also, if there are any helpful links that you might have, please post them.
Regards
Take a look at JavaCC. It's a little daunting at first but it's the right tool for something like this. Plus there are already examples of what you are trying to achieve.
Not sure what exactly you are after, but this problem in its general problem is hard. Very hard.
In fact, given a set of "formulas" (axioms), and deduction rules (mathematical equivalence operations), we cannot deduce if a given formula is correct or not. This problem is actually undecideable.
This issue was first addressed by Hilbert as Entscheidungsproblem
I read a book called Fluid Concepts and Creative Analogies by Douglas Hofstadter that talked about this sort of algebraic manipulations that would automatically rewrite equations in other ways attempting to join equations to other equations an infinite (yet restricted) number of ways given rules. It was an attempt to prove yet unproven theorems/proofs by brute force.
http://en.wikipedia.org/wiki/Fluid_Concepts_and_Creative_Analogies
Douglas Hofstadter's Numbo program attempts to do what you want. He doesn't give you the source, only describes how it works in detail.
It sounds like you want a program to do what highschool students do when they solve algebraic problems to move from a position where you know something, modifying it and combining it with other equations, to prove something previously unknown. It takes a strong Artificial intelligence to do this. The part of your brain that does this is the Neo Cortex, which does science, and it's operating principle is as of yet not understood.
If you want something that will do what college students do when they manipulate equations in calculus, you'll have to build a fairly strong artificial intelligence.
http://en.wikipedia.org/wiki/Neocortex
When we can do whole-brain emulation of a human neo cortex, I will post the answer here.
Yes, you need to write some algorithm to do this kind of computer algebra. At least
a parser to interpret the input
an algebra model to relate parsed operands ('a', 'b', ...) and operator ('+', '=')
implement any appropriate rule to support the manipulation you wish to do
Given a Matrix, I'm interested in the Eigenvalues and the corresponding Eigenvector.
Using Jama, I can get the Eigenvalues and the Eigenvectors, yet the correlation between the two is not defined: I want to map each Eigenvector to the corresponding Eigenvalue.
Can you please recommend me of a way to do so? I tried to implement it myself but it got nasty.
Thanks :)
I am trying to look for an authorized answer, yet for now, According to experiments and observation I performed, the eigenvectors and evigenValues seem to be corresponding.
Usually they are presented in corresponding order. But you can always multiply an eigenvector by the matrix and seeing what multiplier it applies to the vector. That's also your eigenvalue directly.
I asked the developer of the Weka by mail regrading the above issue and they confirm the assumption -
The eigenvectors are indeed provided in the same order as the eigenvalues.
Use a hashmap to store them? I'm not sure this answer is relevant given the question is a bit vague..
What would be the best approach to compare two hexadecimal file signatures against each other for similarities.
More specifically, what I would like to do is to take the hexadecimal representation of an .exe file and compare it against a series of virus signature. For this approach I plan to break the file (exe) hex representation into individual groups of N chars (ie. 10 hex chars) and do the same with the virus signature. I am aiming to perform some sort of heuristics and therefore statistically check whether this exe file has X% of similarity against the known virus signature.
The simplest and likely very wrong way I thought of doing this is, to compare exe[n, n-1] against virus [n, n-1] where each element in the array is a sub array, and therefore exe1[0,9] against virus1[0,9]. Each subset will be graded statistically.
As you can realize there would be a massive number of comparisons and hence very very slow. So I thought to ask whether you guys can think of a better approach to do such comparison, for example implementing different data structures together.
This is for a project am doing for my BSc where am trying to develop an algorithm to detect polymorphic malware, this is only one part of the whole system, where the other is based on genetic algorithms to evolve the static virus signature. Any advice, comments, or general information such as resources are very welcome.
Definition: Polymorphic malware (virus, worm, ...) maintains the same functionality and payload as their "original" version, while having apparently different structures (variants). They achieve that by code obfuscation and thus altering their hex signature. Some of the techniques used for polymorphism are; format alteration (insert remove blanks), variable renaming, statement rearrangement, junk code addition, statement replacement (x=1 changes to x=y/5 where y=5), swapping of control statements. So much like the flu virus mutates and therefore vaccination is not effective, polymorphic malware mutates to avoid detection.
Update: After the advise you guys gave me in regards what reading to do; I did that, but it somewhat confused me more. I found several distance algorithms that can apply to my problem, such as;
Longest common subsequence
Levenshtein algorithm
Needleman–Wunsch algorithm
Smith–Waterman algorithm
Boyer Moore algorithm
Aho Corasick algorithm
But now I don't know which to use, they all seem to do he same thing in different ways. I will continue to do research so that I can understand each one better; but in the mean time could you give me your opinion on which might be more suitable so that I can give it priority during my research and to study it deeper.
Update 2: I ended up using an amalgamation of the LCSubsequence, LCSubstring and Levenshtein Distance. Thank you all for the suggestions.
There is a copy of the finished paper on GitHub
For algorithms like these I suggest you look into the bioinformatics area. There is a similar problem setting there in that you have large files (genome sequences) in which you are looking for certain signatures (genes, special well-known short base sequences, etc.).
Also for considering polymorphic malware, this sector should offer you a lot, because in biology it seems similarly difficult to get exact matches. (Unfortunately, I am not aware of appropriate approximative searching/matching algorithms to point you to.)
One example from this direction would be to adapt something like the Aho Corasick algorithm in order to search for several malware signatures at the same time.
Similarly, algorithms like the Boyer Moore algorithm give you fantastic search runtimes especially for longer sequences (average case of O(N/M) for a text of size N in which you look for a pattern of size M, i.e. sublinear search times).
A number of papers have been published on finding near duplicate documents in a large corpus of documents in the context of websearch. I think you will find them useful. For example, see
this presentation.
There has been a serious amount of research recently into automating the detection of duplicate bug reports in bug repositories. This is essentially the same problem you are facing. The difference is that you are using binary data. They are similar problems because you will be looking for strings that have the same basic pattern, even though the patterns may have some slight differences. A straight-up distance algorithm probably won't serve you well here.
This paper gives a good summary of the problem as well as some approaches in its citations that have been tried.
ftp://ftp.computer.org/press/outgoing/proceedings/Patrick/apsec10/data/4266a366.pdf
As somebody has pointed out, similarity with known string and bioinformatics problem might help. Longest common substring is very brittle, meaning that one difference can halve the length of such a string. You need a form of string alignment, but more efficient than Smith-Waterman. I would try and look at programs such as BLAST, BLAT or MUMMER3 to see if they can fit your needs. Remember that the default parameters, for these programs, are based on a biology application (how much to penalize an insertion or a substitution for instance), so you should probably look at re-estimating parameters based on your application domain, possibly based on a training set. This is a known problem because even in biology different applications require different parameters (based, for instance, on the evolutionary distance of two genomes to compare). It is also possible, though, that even at default one of these algorithms might produce usable results. Best of all would be to have a generative model of how viruses change and that could guide you in an optimal choice for a distance and comparison algorithm.