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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Computing Pi

Nowadays, many people take the concept for granted, that they ‘know the meaning’ of certain Mathematical functions and constants, but that if they ever need a numerical equivalent, they can just tap on an actual calculator, or on a computer program that acts as a calculator, to obtain the correct result.

I am one such person, and an example of a Mathematical constant would be (π).

But, in the 1970s it was considered to be a major breakthrough, that Scientists were able to compute (π) to a million decimal places, using a computer.

And so the question sometimes bounces around my head, of what the simplest method might be, to compute it, even though I possess software which can do so, non-transparently to me. This is my concept, of how to do so:

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

http://dirkmittler.homeip.net/Pi_to_5000.pdf

(Update 08/28/2018 : )

The second link above points to a document, the textual contents of which were created simply, using the program ‘Yacas’. I instructed this program to print (π) to 5000 decimal places. Yet, if the reader was ever to count how many decimal places have been printed, he or she would find there are significantly more than 5000. The reason this happens is the fact that when ‘Yacas’ is so instructed, the first 5000 decimal places will be accurate, but will be followed by an uncertain number of decimal places, which are assumed to be inaccurate. This behavior can be thought related, to the fact that numerical precision, is not the same thing as numerical accuracy.

In fact, the answer pointed to in the second link above is accurate to 5000 decimal places, but printed with precision exceeding that number of decimal places.

Dirk