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… )

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

Complex Roots

One question which has fascinated me in recent years, was the question of what exactly happens, if we start with a complex number as a base, and if we then raise that base to either a rational exponent, or an irrational exponent.

In the following worksheet, I explored that subject for all to see:

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

(Note : )

The worksheet above was made using a graphical front-end to the open-source Computer Algebra System named ‘Maxima’. That graphical front-end is named ‘wxMaxima’, and adds as features, to be able to convert textual output from Maxima, into typeset equations, as well as to make certain presentations and plots nicer.

Enjoy,

Dirk

 

The failings of low-end consumer software, to typeset Math as (HTML) MathML.

One of the features which HTML5 has, and which many Web-browsers support, is the ability to typeset Mathematical formulae, which is known as ‘MathML’. Actually, MathML is an extension of XML, which also happens to be supported when inserted into HTML.

The “WiKiPedia” uses some such solution, partially because they need their formulae to look as sharp as possible at any resolution, but also, because they’d only have so much capacity, to store many, many image-files. In fact, the WiKiPedia uses a number of lossless techniques, to store sharp images as well as formulae. ( :1 )

But from a personal perspective, I’d appreciate a GUI, which allows me to export MathML. It’s fine to learn the syntax and code the HTML by hand, but in my life, the number of syntax-variations I’d need to invest to learn, would be almost as great, as the total number of software-packages I have installed, since each software-package, potentially uses yet-another syntax.

What I find however, is that if our software is open-source, very little of it will actually export to MathML. It would be very nice if we could get our Linux-based LaTeX engines, to export to this format, in a way that specifically preserves Math well. But what I find is, even though I posses a powerful GUI to help me manage various LaTeX renderings, that GUI being named “Kile”, that GUI relies back on a simple command-line tool named ‘latex2html’. Whatever that command-line outputs, that’s what all of Kile will output, if we tell it to render LaTeX specifically to HTML. ‘latex2html’ in turn, depends on ‘netpbm’, which counts as very old, legacy software.

One reason ‘latex2html’ will fail us, is the fact that in general, its intent is to render LaTeX, but not Math in any specific way. And so, just to posses the .TEX Files, will not guarantee a Linux user, that his resulting HTML will be stellar. ‘latex2html’ will generally output PNG Images, and will embed those images in the HTML File, on the premise that aside from the rasterization, PNG Format is lossless. Further, if the LaTeX code was generated by “wxMaxima”, using its ‘pdfLaTeX’ export format, we end up with incorrectly-aligned syntax, just because that dialect of LaTeX has been optimized by wxMaxima, for use in generating .PDF Files next.

(Updated 05/27/2018 : )

Continue reading The failings of low-end consumer software, to typeset Math as (HTML) MathML.