A better way to use “Maxima” to solve cubic equations…

One of the facts which I’ve pointed out in earlier postings was, that if I give the free Computer Algebra System (‘CAS’) named ‘Maxima’, a cubic equation to solve, as in, to find the exact analytical solutions to, it will fail to produce intelligible output, if the usage was naive.

More specifically, cubic equations exist that have 3 distinct, real, irrational roots, and which a ‘CAS’ should be able to solve, just because their general solution is publicly known. That solution boils down, to deriving a second cubic, which is called a ‘depressed cubic equation’, and then performing a trigonometric substitution. (:1)

A fact which I’ve also known for some time is that, especially if a person is using a free or open-source CAS, then in some cases its behaviour has not been made particularly user-friendly, in that work needs to be done by the user, to set up his or her problem for the ‘CAS’ to solve. This latter observation casts a shadow of doubt, over the question of whether a ‘CAS’ will ultimately lead experienced Mathematicians to new discoveries in Algebra, or whether this can only reduce the workload in certain situations.

In this posting I’m going to show, how ‘Maxima’ can be coerced into giving correct answers, by users who know how. What I tend to use is a Graphical Front-End to ‘Maxima’, that is itself named ‘wxMaxima’, but which has equal capabilities, except for the abilities to typeset its solutions, as well as to export its Worksheets to PDF as well as HTML format, using LaTeX.

The following embedded worksheet will only display properly in the reader’s browser, if

  • The reader has allowed JavaScript from my blog to run on his browser, and
  • The reader has also allowed JavaScript to run from a CDN named ‘mathjax.org’.

 

 


What’s observable here is the fact that the package ‘odes’ can be loaded, which is mainly used to solve Ordinary Differential Equations, and that afterwards, the function ‘solvet()’ can be used, even to solve certain polynomials – better than what Maxima can solve on its own, with the built-in ‘solve()’ function. (:3)

(Updated 6/14/2020, 0h30… )

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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 3/2/2019, 16h30 … )

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