A Hypothetical Algorithm…

One of the ideas which I’ve written about often is, that when certain Computer Algebra Software needs to compute the root of an equation, such as of a polynomial, an exact Algebraic solution, which is also referred to as the analytical solution, or symbolic Math, may not be at hand, and that therefore, the software uses numerical approximation, in a way that never churned out the Algebraic solution in the first place. And while it might sound disappointing, often, the numerical solution is what Engineers really need.

But one subject which I haven’t analyzed in-depth before, was, how this art might work. This is a subject which some people may study in University, and I never studied that. I can see that in certain cases, an obvious pathway suggests itself. For example, if somebody knows an interval for (x), and if the polynomial function of (x), that being (y), happens to be positive at one end of the interval, and negative at the other end, then it becomes feasible to keep bisecting the interval, so that if (y) is positive at the point of bisection, its value of (x) replaces the ‘positive’ value of (x) for the interval, while if at that new point, (y) is negative, its value for (x) replaces the ‘negative’ value of (x) for the interval. This can be repeated until the interval has become smaller than some amount, by which the root is allowed to be inaccurate.

But there exist certain cases in which the path forward is not as obvious, such as what one should do, if one was given a polynomial of an even degree, that only has complex roots, yet, if these complex roots nevertheless needed to be found. Granted, in practical terms such a problem may never present itself in the lifetime of the reader. But if it does, I just had lots of idle time, and have contemplated an answer.

(Updated 1/30/2019, 13h00 … )

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An Observation, about How My Blog Gets Spidered

If a blog was just to consists of a bunch of Web-pages, then one side-effect of that would be, that the blogger would add pages on a daily basis, but that his latest pages would not be spidered by the major search engines, potentially for a long time.

If we use ‘WordPress.org’ as our blogging engine, then one feature is, the very first time we add a new posting, doing so is announced or broadcast, to a service that lets search engines know, that a posting has been added to the blog. But one way in which this service does not work, is to reannounce, every time an existing posting has been edited by the blogger. This is fair, because bloggers like me may edit postings as many as 10 or even 20 times eventually. Reannouncing these edits would put an unfair burden on a free service.

But as it happens, the way this service works can have side-effects for me. For example, one of my recent postings links to the following URL:

http://dirkmittler.homeip.net/Plot_Complex_Points.pdf

The problem with this URL is, that in the original posting, it ended with ‘.html’, and not with ‘.pdf’ . I edited that posting after I had first created it, to use a ‘…pdf’ URL, instead of an ‘…html’ URL. What tends to happen is that Google will spider my new postings within seconds of their being created, while certain other search-engines will take maybe a few hours to spider the same posting. And this can entirely be a performance issue, with each search engine. But unfortunately for me, this suggests that Google caught a version of this earlier posting, that contains a broken link, just because the ‘…html’ URL no longer exists on my server. And if the posting contains any apparent, broken links, obviously, the search engine penalizes their ranking.

So that may be one reason, why the posting in question, has not received as many clicks, as it should according to how often other postings of mine receive clicks.

Oops.

Dirk

 

Complex Roots

One question which has fascinated me in recent years, was the question of what exactly happens, if we start with a complex number as a base, and if we then raise that base to either a rational exponent, or an irrational exponent.

In the following worksheet, I explored that subject for all to see:

http://dirkmittler.homeip.net/Plot_Complex_Points.pdf

(Note : )

The worksheet above was made using a graphical front-end to the open-source Computer Algebra System named ‘Maxima’. That graphical front-end is named ‘wxMaxima’, and adds as features, to be able to convert textual output from Maxima, into typeset equations, as well as to make certain presentations and plots nicer.

Enjoy,

Dirk

 

The Myth of Wave / Particle Duality

This posting describes some of the History, which many people may be bypassing, in their appreciation of Quantum Mechanics.

About until the 1920s, ‘light’ was largely thought to consist of waves. But a problem with that was, how to explain, why light can travel through apparently empty space. After all, the light that reaches us from distant stars is not fundamentally different, from light that originates on Planet Earth. And until the 1920s, it was believed that there exists a mysterious “Aether“, which transmitted light through space.

A basic premise of wave-propagation, such as in the case of sound-waves, is that there must first be some sort of medium, to conduct the waves, which in the case of sound may be air. But the need for the existence of a medium, also explains why there is no sound in space.

But during the 1920s, the existence of an aether was disproved. Decisively. And so another explanation was needed, of what constitutes light. And the thought seemed more logical, that particles can easily travel through empty space – hence, photons. Even though this was not actually the first form in which photons were theorized.

But then obviously, this raises questions, about how these particles are supposed to relate to waves, where waves were at first easier to observe.

I think that the way many people today are presented, what Quantum-Mechanics consists of, is just, “Wave / Particle Duality”. But then what many students believe – and what I once believed myself – is, that Quantum Mechanics holds some sort of secret key, as to how Matter and Energy might simultaneously consist of particles and waves. And in reality, QM holds no such decisive, secret answers. The only real secret which QM may hold, is a detail that could be embarrassing to the present way in which QM works.

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