We have loosely tossed about the term “domain”. We shall now look at this in a bit (but not much) more detail. Domains are fundamental to the way in which MuPAD works, and we need to have a basic understanding of them in order to use MuPAD effectively.
A domain in MuPAD is either an algebraic structure (such as finite field or permutation group) or a data type (such as Matrix, Polynomial or Fraction), for which overloaded operators or functions defined on that domain always return results in the domain (or the result FAIL if no result exists).
To give some examples, suppose we investigate matrices over the integers modulo 29. Since 29 is prime, these integers form a Galois field, and so the matrices should respond to all standard arithmetic operations.
First the definition:
>> M29:=Dom::Matrix(Dom::IntegerMod(29)); Dom::Matrix(Dom::IntegerMod(29))
We have two domains being used here: Dom::Matrix, which creates a matrix domain, and Dom::IntegerMod(29), which creates the field of integers modulo 29.
>> A:=M29([[100,200,-30],[47,-97,130],[13,33,-1001]]); +- -+ | 13 mod 29, 26 mod 29, 28 mod 29 | | | | 18 mod 29, 19 mod 29, 14 mod 29 | | | | 13 mod 29, 4 mod 29, 14 mod 29 | +- -+Notice that the result returned by MuPAD is automatically normalized so that the matrix elements are in the required field. If we enter values which can't be normalized (say, decimal fractions), MuPAD will return an error message.
>> 1/A; +- -+ | 3 mod 29, 8 mod 29, 15 mod 29 | | | | 28 mod 29, 9 mod 29, 22 mod 29 | | | | 12 mod 29, 19 mod 29, 13 mod 29 | +- -+Here the inverse operator returns a suitable result. Let's check this.
>> %*A; +- -+ | 1 mod 29, 0 mod 29, 0 mod 29 | | | | 0 mod 29, 1 mod 29, 0 mod 29 | | | | 0 mod 29, 0 mod 29, 1 mod 29 | +- -+This is the identity for our particular matrix ring. Now we can try a few other matrix operations.
>> linalg::det(A); 12 mod 29 >> linalg::gaussElim(A); +- -+ | 13 mod 29, 26 mod 29, 28 mod 29 | | | | 0 mod 29, 12 mod 29, 2 mod 29 | | | | 0 mod 29, 0 mod 29, 9 mod 29 | +- -+ >> linalg::gaussJordan(A); +- -+ | 1 mod 29, 0 mod 29, 0 mod 29 | | | | 0 mod 29, 1 mod 29, 0 mod 29 | | | | 0 mod 29, 0 mod 29, 1 mod 29 | +- -+ >> A^10; +- -+ | 22 mod 29, 10 mod 29, 3 mod 29 | | | | 3 mod 29, 21 mod 29, 16 mod 29 | | | | 4 mod 29, 18 mod 29, 5 mod 29 | +- -+ >> exp(A); FAILThe matrix exponential exp(X) is defined as 1 + X + (X^2)/2 + (X^3)/6 + (X^4)/24 + . . . + (X^n)/n! + . . . As you might expect, this is not defined for matrices over our field. For another example, consider polynomials over the integers modulo 2. The definition is similar to the matrix definition above.
>> PK:=Dom::Polynomial(Dom::IntegerMod(2)); Dom::Polynomial(Dom::IntegerMod(2))Now we'll create a polynomial in this domain.
>> p1:=PK(x^17+1); 17 x + 1For good measure, we'll create a second polynomial which looks the same, but is not in our domain.
>> p2:=x^17+1; 17 x + 1Even though they look the same on the screen, MuPAD knows all about them; the type command will tell us.
>> type(p1); Dom::Polynomial(Dom::IntegerMod(2)) >> type(p2); "_plus"(The result of this last command is that p2 is an object formed by adding things together.)
>> Factor(p1); 3 4 5 8 2 4 6 7 8 1 (x + 1) (x + x + x + x + 1) (x + x + x + x + x + x + 1) >> Factor(p2); 2 3 4 5 6 7 8 9 10 11 12 13 (x + 1) (x - x - x + x - x + x - x + x - x + x - x + x - x 14 15 16 + x - x + x + 1)The domains package is part of MuPAD which is very much in a state of constant revision and enhancement. For example, at present, it is not possible to perform polynomial division in a polynomial domain.
Practical Task Scheduling Deployment
July 20, 2016 12:00 pm CDT
One of the best things about the UNIX environment (aside from being stable and efficient) is the vast array of software tools available to help you do your job. Traditionally, a UNIX tool does only one thing, but does that one thing very well. For example, grep is very easy to use and can search vast amounts of data quickly. The find tool can find a particular file or files based on all kinds of criteria. It's pretty easy to string these tools together to build even more powerful tools, such as a tool that finds all of the .log files in the /home directory and searches each one for a particular entry. This erector-set mentality allows UNIX system administrators to seem to always have the right tool for the job.
Cron traditionally has been considered another such a tool for job scheduling, but is it enough? This webinar considers that very question. The first part builds on a previous Geek Guide, Beyond Cron, and briefly describes how to know when it might be time to consider upgrading your job scheduling infrastructure. The second part presents an actual planning and implementation framework.
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With all the industry talk about the benefits of Linux on Power and all the performance advantages offered by its open architecture, you may be considering a move in that direction. If you are thinking about analytics, big data and cloud computing, you would be right to evaluate Power. The idea of using commodity x86 hardware and replacing it every three years is an outdated cost model. It doesn’t consider the total cost of ownership, and it doesn’t consider the advantage of real processing power, high-availability and multithreading like a demon.
This ebook takes a look at some of the practical applications of the Linux on Power platform and ways you might bring all the performance power of this open architecture to bear for your organization. There are no smoke and mirrors here—just hard, cold, empirical evidence provided by independent sources. I also consider some innovative ways Linux on Power will be used in the future.Get the Guide