Simplifying the approach, to finding roots of polynomials.

In some cases, the aim of my postings is to say, ‘I am able to solve a certain problem – more or less – and therefore, the problem is solvable.’ It follows from this position that my solutions are not assumed to be better by any means, than mainstream solutions. So recently, I suggested an approach to finding the roots of polynomials numerically, again just to prove that it can be done. And then one observation which my readers might have made would be, that my approach is only accurate to within (10-12), while mainstream solutions are accurate to within (10-16). And one possible explanation for this would be, that the mainstream solutions polish their roots, which I did not get into. (:1)

(Edit 2/8/2019, 6h40 : )

A detail which some of my readers might have missed is, that when I refer to a ‘numerical solution’, I’m generally referring to an approximation.

(End of Edit, 2/8/2019, 6h40 . )

But another observation which I made, was that Mainstream Code Examples are much tighter, than what I suggested, which poses the obvious question: ‘Why can mainstream programmers do so much, with much less code complexity?’ And I think I know one reason.

The mainstream example I just linked to, bypasses a concept which I had suggested, which was to combine conjugate complex roots into quadratic terms, which could be factorized out of the original polynomial as such. What the mainstream example does is to assume that the coefficients of the derived polynomials could be complex, even though the original one only has real coefficients. And then, if a complex root has been found, factorizing it out results in such a polynomial with complex coefficients, after which to factorize out the conjugate, causes the coefficients of the quotient to become real again.

(Edited 1/30/2019, 8h50… )

(Updated 2/9/2019, 23h50… )

I’ve just written some source-code of my own, to test my premises…

Continue reading Simplifying the approach, to finding roots of polynomials.

How the EPUB2 and MOBI formats can be used for typeset Math.

According to This preceding posting, I was experiencing some frustration over trying to typeset Math, for publication in EPUB2 format. EPUB3 format with MathML support was a viable alternative, though potentially hard on any readers I might have.

Well a situation exists in which either EPUB2 or MOBI can be used to publish typeset Math: Each lossless image can claim the entire width of a column of text, and each image can represent an entire equation. That way, the content of the document can alternate vertically between Text and Typeset Math.

In fact, if an author was to choose to do this, he or she could also use the Linux-based solutions ‘LyX’ , ‘ImageMagick’ , and ‘tex4ebook’ .

(Edited 1/9/2019, 15h35 … )

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Trying to bridge the gap to mobile-friendly reading of typeset equations, using EPUB3?

One of the sad facts about this blog is, that it’s not very mobile-friendly. The actual WordPress Theme that I use is very mobile-friendly, but I have the habit of inserting links into postings, that open typeset Math, in the form of PDF Files. And the real problem with those PDF Files is, the fact that when people try to view them on, say, smart-phones, the Letter-Sized page format forces them to pinch-zoom the document, and then to drag it around on their phone, not getting a good view of the overall document.

And so eventually I’m going to have to look for a better solution. One solution that works, is just to output a garbled PDF-File. But something better is in order.

A solution that works in principle, is to export my LaTeX -typeset Math to EPUB3-format, with MathML. But, the other EPUB and/or MOBI formats just don’t work. But the main downside after all that work for me is, the fact that although there are many ebook-readers for Android, there are only very few that can do everything which EPUB3 is supposed to be able to do, including MathML. Instead, the format is better-suited for distributing prose.

One ebook-reader that does support EPUB3 fully, is called “Infinity Reader“. But if I did publish my Math using EPUB3 format, then I’d be doing the uncomfortable deed, of practically requiring that my readers install this ebook-reader on their smart-phones, for which they’d next need to pay a small in-app purchase, just to get rid of the ads. I’d be betraying all those people who, like me, prefer open-source software. For many years, some version of ‘FBReader’ has remained sufficient for most users.

Thus, if readers get to read This Typeset Math, just because they installed that one ebook-reader, then the experience could end up becoming very disappointing for them. And, I don’t get any kick-back from ImeonSoft, for having encouraged this.

I suppose that this cloud has a silver lining. There does exist a Desktop-based / Laptop-based ebook-reader, which is capable of displaying all these EPUB3 ebooks, and which is as free as one could wish for: The Calibre Ebook Manager. When users install this either under Linux or under Windows, they will also be able to view the sample document I created and linked to above.

(Updated 1/6/2019, 6h00 … )

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What I’ve learned about RSA Encryption and Large Prime Numbers – How To Generate

One of the ways in which I function, is to write down thoughts in this blog, that may seem clear to me at first, but which, once written down, require further thought and refinement.

I’ve written numerous times about Public Key Cryptography, in which the task needs to be solved, to generate 1024-bit prime numbers – or maybe even, larger prime numbers – And I had not paid much attention to the question, of how exactly to do that efficiently. Well only yesterday, I read a posting of another blogger, that inspired me. This blogger explained in common-sense language, that a probabilistic method exists to verify whether a large number is prime, that method being called “The Miller-Rabin Test”. And the blogger in question was named Antoine Prudhomme.

This blogger left out an important part in his exercise, in which he suggested some working Python code, but that would be needed if actual production grade-code was to generate large prime numbers for practical cryptography. He left out the eventual need, to perform more than just one type of test, because this blogger’s main goal was to explain the one method of testing, that was his posting subject.

I decided to modify his code, and to add a simple Fermat Test, simply because (in general,) to have two different probabilistic tests, reduces the chances of false success-stories, even further than Miller-Rabin would reduce those chances by itself. But Mr. Prudhomme already mentioned that the Fermat Test exists, which is much simpler than the Miller-Rabin Test. And, I added the step of just using a Seive, with the known prime numbers up to 65535, which is known not to be prime itself. The combined effect of added tests, which my code performs prior to applying Miller-Rabin, will also speed the execution of code, because I am applying the fastest tests first, to reduce the total number of times that the slower test needs to be applied, in case the candidate-number could in fact be prime, as not having been eliminated by the earlier, simpler tests. Further, I tested my code thoroughly last night, to make sure I’ve uploaded code that works.

Here is my initial, academic code:

http://dirkmittler.homeip.net/text/Generate_Prime_3.py

 

(Corrected 10/03/2018, 23h20 … )

(Updated 10/08/2018, 9h25 … )

Continue reading What I’ve learned about RSA Encryption and Large Prime Numbers – How To Generate