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 … )

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

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.

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.