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 … )

Continue reading A Hypothetical Algorithm…

How the general solution of polynomials, of any degree greater than 2, is extremely difficult to compute.

There are certain misconceptions that exist about Math, and one of them could be, that if a random system of equations is written down, the Algebraic solution of those equations is at hand. In fact, equations can arise easily, which are known to have numerical answers, but for which the exact, Algebraic (= analytical) answer, is nowhere in sight. And one example where this happens, is with polynomials ‘of a high degree’ . We are taught what the general solution to the quadratic is in High-School. But what I learned was, that the general solution to a cubic is extremely difficult to comprehend, while that of a 4th-degree polynomial – a quartic –  is too large to be printed out. These last two have in fact been computed, but not on consumer-grade hardware or software.

In fact, I was taught that for degrees of polynomials greater than 4, there is no general solution.

This seems to fly in the face of two known facts:

  1. Some of those equations have the full number of real roots,
  2. Computers have made finding numerical solutions relatively straightforward.

But the second fact is really only a testament, to how computers can perform numerical approximations, as their main contribution to Math and Engineering. In the case of polynomials, the approach used is to find at least one root – closely enough, and then, to use long division or synthetic division, to divide by (1 minus that root), to arrive at a new polynomial, which has been reduced in degree by 1. This is because (1 minus a root) must be a factor of the original polynomial.

Once the polynomial has arrived at a quadratic, computers will eagerly apply its general solution to what remains, thereby perhaps also generating 2 complex roots.

In the case of a cubic, a trick which people use, is first to normalize the cubic, so that the coefficient of its first term is 1. Then, the 4th, constant term is examined. Any one of its factors, or the factors of the terms it is a product of, positively or negatively, could be a root of the cubic. And if one of them works, the equation has been cracked.

In other words, if this constant term is the product of square roots of integers, the corresponding products of the square roots, of the factors of these integers, could lead to roots of the cubic.

(Updated 1/12/2019, 21h05 … )

Continue reading How the general solution of polynomials, of any degree greater than 2, is extremely difficult to compute.

Exploring the newer GUI front-end, for use with SageMath.

One of the subjects which I had written about only yesterday, is that the Computer Algebra / Numerical Tool System called ‘SageMath‘ was available in the repositories, for Debian / Stretch – which is in itself news – and that additionally, the default way to use it under Debian is through a Web-interface called ‘SageNB’. Well what I’ve now learned is that the SageMath developers no longer support SageNB, and are continuing their work with the graphical front-end called ‘Jupyter‘.

But, installing Jupyter under Debian is a bit of a chore, because unlike how it is with custom-compiles, Debian package maintainers tend to break major software down into little bits and pieces. At one point, I had Jupyter running, but with no awareness of the existence of SageMath. What finally did the trick for me today, was to install the following packages:

  • python-notebook
  • jupyter-nbextension-jupyter-js-widgets
  • sage-math-jupyter

Needless to say, that last package out of the three is the most important, and may even pull in enough of the other packages, to be selected by itself. It’s just that I did not know immediately, to install that last package.

So this is what SageMath 7.4 looks like, through Jupyter:


(Corrected 09/18/2018, 3h50 … )

(Updated 09/18/2018, 5h40 … )

(As of 09/16/2018, 20h10 : )

Frankly, I was a bit disappointed at first. My main disappointment seemed to be with the fact, that this GUI did not offer to typeset the Math. It does allow us to ‘download’ our Notebooks as PDF-Files, but when we do, we simply get the same, highlighted text, and graphics, only as a PDF – in code – or with whatever appearance the browser-view is already showing us. Also, the support for 3D plots is lackluster, as the plot above is non-interactive. At least with SageNB, I was able to select the ‘canvas3d’ viewer, which allowed the plot to be rotated. Also, if we use SageMath from the command-line, it defaults to using ‘JMol’ as its viewer, which is full-featured.

But as it turns out, I have discovered ‘the trick’, to getting Jupyter to typeset the users’ Math…

Continue reading Exploring the newer GUI front-end, for use with SageMath.

I just installed Sage (Math) under Debian / Stretch.

One of the mundane limitations which I’ve faced in past years, when installing Computer Algebra Systems etc., under Linux, that were supposed to be open-source, was that the only game in town – almost – was either ‘Maxima’ or ‘wxMaxima’, the latter of which is a fancy GUI, as well as a document exporter, for the former.

Well one fact which the rest of the computing world has known about for some time, but which I am newly finding for myself, is that software exists called ‘SageMath‘. Under Debian / Stretch, this is ‘straightforward’ to install, just by installing the meta-package from the standard repositories, named ‘sagemath’. If the reader also wants to install this, then I recommend also installing ‘sagemath-doc-en’ as well as ‘sagetex’ and ‘sagetex-doc’. Doing this will literally pull in hundreds of actual packages, so it should only be done on a strong machine, with a fast Internet connection! But once this has been done, the result will be enjoyable:


I have just clicked around a little bit, in the SageMath Notebook viewer, which is browser-based, and which I’m sure only provides a skeletal front-end to the actual software. But there is a feature which I already like: When the user wishes to Print his or her Worksheet, doing so from the browser just opens a secondary browser-window, from which we may ‘Save Page As…’ , and when we do, we discover that the HTML which gets saved, has its own, internal ‘MathJax‘ server. What this seems to suggest at first glance, is that the equations will display typeset correctly, without depending on an external CDN. Yay!

I look forward to getting more use out of this in the near future.

(Update 09/15/2018, 21h30 : )

Continue reading I just installed Sage (Math) under Debian / Stretch.