Secondary Polishing

When the project is undertaken to write programs, that would be sub-components to Computer Algebra Systems, but that produce floating-point numerical outputs, then an unwanted side effect of how those work is, they can be output in place of integers (whole numbers), but may differ from those integers by some very small fractional amount. Thus, instead of outputting (1) exactly, such a program might output:

0.9999999999999994

The problem is that such output can be visually misleading, and confusing because a Human user wants to know that the answer to a problem was (1). And so a possible step in the refinement of such programs is “Secondary Polishing”, which does not change the actual computations, but which makes the output ‘look nicer’.

I recently completed a project that approximates the roots of arbitrary polynomials, and also looked in to the need for secondary polishing. There was one specific situation in which this was not required: The root’s real or imaginary component could have an absolute of (1/10) or greater. In this case, the simple fact that I had set the precision of the printed output to (14), but that the roots found are more precise than to be within (10^-14), at least after the actual, primary polishing, that affects computed values, together with the way the standard output functions work in C++, will cause the example above to be output as a single-digit (1), even though what was stored internally might be different from that, by less than (10^-14). But a special case exists within the norms of C++, if the absolute of the numerical term to be output is less than (1/10).

(Updated 2/11/2019, 19h35 … )

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How an exact solution can sometimes be found, without using the general solution.

One of the facts which I’ve been writing about is, that the general solution to a polynomial of a degree higher than (4), that is expected to produce Algebraically exact results, cannot be used because none exists. At the same time, I make a distinction between an exact solution, and the general solution. This distinction can also be explained in greater detail…

We are sometimes given a polynomial, which has at least one “rational root”, meaning a root that can be stated either as a whole number, which is actually referred to as an “integer”, or as a fraction. The following is an example:

x^3 -3*x^2 -2*x + 6 = 0

In this case it can be observed, that the coefficient of (x^3), which is not stated, corresponds to a (1), and that the constant term, which is visible as (+6), is an integer. What can be done here, is that all the factors of (6) can be used positively and negatively – not only the prime factors – and plugged in to see whether they do in fact constitute one root. Again, they do if and only if the equation is satisfied as resulting in zero.

Thus, as potential candidates, ±1, ±2, ±3, ±6 can all be tried.

(Updated 2/9/2019, 19h40 … )

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My computer Plato is having a technical issue.

One of the main computers which I’ve been using, that is named ‘Plato’, that was running Debian / Stretch, has experienced a major technical problem. When I got home this afternoon, I found it was not running. And, when I pushed the power button, it did not turn on.

A basic, automatic idea which would pop into people’s heads is, ‘The power-supply burned out.’ If the only task which lies ahead really was, to replace the power supply, I’d have it easy. This is a tower-computer from the year 2011, with a Sabertooth X58 motherboard.

  • The correct power-supplies for this old MB may have become hard to find,
  • Even if I had a replacement power-supply, it would be very cumbersome to replace because the harnesses of the present one loops behind too many recessed compartments, within the case.

The only thing I’ve done so far, is to perform a diagnostic test. I disconnected all the jacks between the power-supply and the MB, and retried the power button. My purpose behind that was, the idea that modern power supplies will refuse to turn on, if they sense a short-circuit between their load, and ground. Thus, if the power supply had been able to resume, with the MB disconnected, I’d know it was the MB, and I’d also know there’s no point in replacing the power-supply. But thankfully, the power-supply also did not power up like that. So I reconnected the power-supply to the MB.

So as it stands, I don’t know the best way to proceed, but am without the use of that trusty computer for now.

(Update 2/7/2019, 14h15 : )

One reason this apparent loss is unfortunate is the fact that, being my only Debian / Stretch computer, that computer was also the only one, which had “SageMath” installed and working on it. So my available Computer Algebra Systems are reduced to “Maxima” and “Yacas” for now.

(Update 2/9/2019, 18h50 : )

Actually, I’ve learned that my so-called diagnostic test was pointless. The power button these days, does not have a direct connection to the power-supply, to signal that the power-supply should turn on. The power button has its connection to the M.B., which tells the power-supply to turn on. Therefore, with the M.B. disconnected from the power-supply, there was no way for the power-supply even to get the signal, to turn on.

A personal friend of mine has lent me a power-supply tester, so that I’ll next be able to test that more properly. And, hoping that it is just the power-supply which is faulty, I’ll look into replacing it.

(As of 2/7/2019, 14h15 … )

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DSL Problems, Downtime

I take the unusual approach of hosting my site and my blog, on a private PC at home, instead of on a professional hosting service. This means that the availability of the site is only as good as my DSL connection at home.

Since about 11h00 yesterday, January 30, I’ve been experiencing problems with my DSL. This means that my site has been sporadically unavailable in the meantime. Further, when the technician from my ISP comes to repair the problem, today in the afternoon (January 31), his work will require that he disconnect my DSL, which means more downtime.

I apologize to my readers, and hope that this problem will soon be fixed.

(Update 1/31/2019, 20h00 : )

The technician was able to resolve this problem, but the service interruption which was required, for him to complete his job, lasted from 17h40 until 20h00 this evening.

I thank the tech-support of Bell Canada, as well as the hard work of the technician.

My site and blog should now be reliable again.

Dirk

 

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