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Posted 09 January 2011 - 01:11 PM
Posted 10 January 2011 - 03:35 PM
Posted 10 January 2011 - 07:20 PM
Yes, I have an undergraduate algebra book covering all of this stuff. I'm definitely trying my best to learn from it, but learning from books can be very difficult.You might want to check out out a linear algebra book; I like Gilbert Strang's (Chapter 6 is eigenvectors and eigenvalues). I know it's harder to learn math on your own, but it sounds like linear algebra isn't actually out of your league; I think that with a proper introduction you'd probably "get" eigenvalues and eigenvectors, and be able to understand Davenport.
You can actually find the full book by digging through a couple of link on his homepage (http://people.bath.ac.uk/masjhd/), but for convenience: http://staff.bath.ac.uk/masjhd/masternew.pdfGoogle has no preview of Davenport's book, so I can't be sure exactly what he's doing, but I can make some guesses.
The eigenvalues and eigenvectors actually come in with Berlekamp's algorithm as described in the book. Theres a lot more besides Berlekamp's algorithm in the factoring algorithm, but I figured I have to narrow it down before I can move on.1.) My first guess (this post, before edits) was that he's looking at the Companion Matrix for a polynomial. The eigenvalues of the companion matrix are the roots of the polynomial. This only really helps for polynomials of one variable though.
2.) My second guess is that he's explaining multivariate quadratic polynomials in terms of the eigenvectors and eigenvalues of matrices. [EDIT: On second thought, probably not...]
Posted 11 January 2011 - 04:15 AM
cos^{4}(x) + 2cos^{2}(x)sin^{2}(x) + sin^{4}(x)
Posted 11 January 2011 - 08:41 AM
Yes, I would match that particular expression against the polynomial a^2+2ab+b^2, factor that, and then substitute back in for a and b. I was only using it as a practical example, since you can raise sin^2(x)+cos^2(x) to any integer power and it should still reduce to 1. However, yes, the book only describes an algorithm for working with integer coefficients (which, by extension, will work for any rational coefficients by factoring out the the smallest denominator). I know that the big-boy computer algebra systems like Mathematica can factor polynomials with irrational coefficients, so I suppose I should look for a different algorithm.Technically, an expression like
cos^{4}(x) + 2cos^{2}(x)sin^{2}(x) + sin^{4}(x)
is not a polynomial, but a transcendental function. In this particular case, you can simply consider sin^{2}(x) and cos^{2}(x) as indeterminate and then use a polynomial factorization algorithm to simplify it, but it is not always possible or useful to use such an algorithm. Sometimes, it is better to use some trigonometric identity for example. Moreover, the algorithm in that book is designed for polynomials with rational coefficients (actually integer) and you may have expressions with transcendental numbers like pi or with square roots. You have to consider them as "indeterminate" if you want to use that algorithm, but the results may not be particularly good. A factorization algorithm which works on reals or complex numbers may however introduce bad looking factorizations and also contains some approximations of the real roots.
I'm not an expert of this field, but I know there are some ACM journal on mathematical softwares. You may search for papers on expression simplification and polynomial factorization at the ACM portal (http://portal.acm.org/) to find additional material on the subject. To read the paper there you have to pay, but the preprints of several papers are freely available on internet. Note however that some papers may require advanced knowledge of algebra (in particular of field or ring theory).
Posted 11 January 2011 - 01:26 PM
The eigenvalues and eigenvectors actually come in with Berlekamp's algorithm as described in the book. Theres a lot more besides Berlekamp's algorithm in the factoring algorithm, but I figured I have to narrow it down before I can move on.
This sounds straightforward enough, though there are some subtleties.[2] Calculate the matrix Q.
[3] Find a basis of its eigenvectors for the eigenvalue 1. One eigenvector
is always the vector [1; 0; 0;:::; 0],
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