A few days ago I wrote about my solution for factoring a polynomial into factors corresponding to complex roots, factorC.
One of the uses for that utility is to perform a partial fractions decomposition of rational functions. Here’s a complex partial fraction function that uses factorC and the built-in Maxima function partfrac.
partfracC(_f,_z):=block( [d,fd], d:denom(_f), fd:factorC(d,_z), partfrac(1/fd,_z) );
I’m getting ready for my fall Complex Variables class. I noticed that the built-in Maxima function residue doesn’t reliably do the right thing. My goal is to make some improvements to Maxima residue calculations in Maxima over the course of the next month.
As I started to look at some test cases, I realized I didn’t know how to factor a polynomial into complex factors. In the simplest case:
but I wanted to see
Maybe someone will find this and let me know that there’s a simple way to make that happen using existing Maxima commands. Until then, I’ve written a little utility to identify the roots (both real and complex) of a polynomial and return a factorization. Here’s an example, and the code. Notice that it only works as well as the root finder solve. I tried to upgrade to the more robust to_poly_solve, but I don’t yet know how to handle multiplicities in that case.
factorC(_f,_z):=block( [s,n,m,fp,j], fp:1, /* This commented code was meant to use the more robust solver to_poly_solve, but I couldn't understand how to handle multiplicities ss:args(to_poly_solve(_f,_z)), s:create_list(ss[k][1],k,1,length(ss)),*/ s:solve(_f,_z), m:multiplicities, n:length(s), for j:1 thru n do if lhs(s[j])#0 then fp:fp*(_z-(rhs(s[j])))^m[j], fp:fp*divide(_f,fp)[1], fp );
A few months ago, I posted my path integral functions, which are included in the MATH214 package. Recently, I came across something I’d been looking for: a Maxima utility for visualizing vector fields. Its in the Maxima/Share directory under drawutils.
Written in 2010 by Donald J Bindner, the commands plot_vector_field and plot_vector_field3d do almost everything I was looking for. The drawback is that I wanted to plot the vector fields along with the integration path. I modified those two commands slightly into versions called make_vector_field and make_vector_field3d to produce the lists of vectors for plugging into draw2d and draw3d, so that I could include the vector fields in bigger graphics calls. My modifications are available here. The package includes my home-baked help utility.
Here’s what the path integral command and the vector field generator look like on an example from chapter 13 of Stewart’s “Calculus Concepts and Contexts”
A few months ago, when I was very new to Maxima, I posted my solutions to several multivariable optimization problems, and included a standard constrained optimization example for which the built in command solve failed.
Here’s the problem:
In my multivariable calculus class, we’ve just encountered that problem again. What my students and I discovered was that the order of the variables as specified by the user in the argument list caused the command to either succeed or fail, as you can see from the first two lines below:
I wrote to the experts at maxima-discuss@lists.sourceforge.net. They replied en masse very quickly and were very generous with their time and advice. They verified the problem, pointing out that solving systems of equations is really complicated, and that the solve program seeks to solve for the variables one at a time and things can go wrong if an inauspicious order of variables is followed.
The experts also tried our problem with the package to_poly_solve, and happily it succeeded without the hassle of needing to specify the variables in the right order. For solve users who reference the solutions it produces as elements of a list, direct use of to_poly_solve presents a challenge due to output in a format different from that of solve — t0_poly_solve returns solutions as a set with %union.
I’ve written a little wrapper Ksolve.mac that is called like solve as illustrated above.
In Ksolve if an initial call to solve fails, the process automatically upgrades to to_poly_solve in several variations, and (hopefully) returns solutions in the same format as solve.
Together with the 3D graphing capabilities of Maxima, we can bring symbolic differentiation and the numerical solver to bear when we seek local extrema of a surface.
Here is a link to the html export of a wxMaxima session where I work on two examples from my multivariable calculus class. And here is the wxMaxima session.
The Maxima documentation says the numerical ODE solver in the command rk() is fourth order accurate. I ran a little test to confirm that.
Here’s the html export of the Maxima session, and a screenshot of the punchline:
I recently posted about : and := for defining functional expressions. I’m starting to enjoy these emoji-like constructions 😉
This is another colon-equals post. This time for defining functions involving the maxima differentiation command diff.
Notice below that if we define a function with :=, the naive use of :=diff doesn’t produce a derivative with the expected results upon evaluation.
In fact, it’s a good thing that :=diff works like that. The error with fp(3) above comes from the fact that we’ve actually defined an operator that differentiates the function with respect to the argument we pass…in the case above, differentiating with respect to the symbol u makes sense, while differentiating with respect to the constant 3 doesn’t.
So how to make the derivative function do what we want? Two ways, that are subtly different, in ways I’m not completely sure of. More about that when I learn more :-).
First is define,
Also you can use ”() quote-quote with parens around the whole right hand side:
I used define to write functions for vector valued 3D curves in an earlier post. In figuring this out, I also learned that the :=diff form is really useful. Below are three little functions in which I use :=diff to define the vector calculus operators grad, div and curl. Notice that we pass the function f as an argument, and the :=diff form allows Maxima to differentiate them behind the scenes and return the results of the grad, div, and curl operators as you’d expect. These versions of div, grad and curl behave differently, and for me more as expected, than the functions of those names included in the Maxima vect package. You can download the .mac file here.
/* Three Maxima functions for the multivariable calculus operators grad, div, and curl
TheMaximaList.org, 2016
*/grad(f,x,y,z):=[diff(f,x),diff(f,y),diff(f,z)]$
div(f,x,y,z):=diff(f[1],x)+diff(f[2],y)+diff(f[3],z)$
curl(f,x,y,z):=[ diff(f[3],y)-diff(f[2],z),
diff(f[1],z)-diff(f[3],x),
diff(f[2],x)-diff(f[1],y) ]$
Here is a screenshot showing how to call these functions:
I’m testing the latex math-mode utility in WordPress.com. The first thing to know is that based on the way the title of this post appears, the latex utility doesn’t work in post titles.
The rub here is that MathJax would be the natural solution, but WordPress.com isn’t compatible with JavaScript.
That leaves us with $, followed by the word latex, followed by my latex markup, followed by a closing $ , which I illustrate below with a screenshot because I don’t know how to quote a string verbatim in WordPress:
This works OK as inline style math 
and then back to inline for 
And there it is.
The official guide to the WordPress 
I’m really posting this for myself to act as a big paperweight. Whenever I forget about := and : for functions in Maxima, I can refer to this page.
The gist:
Check it out! Full-featured Maxima on my phone is pretty cool.
Typing is a little bit of a drag. I’ll look around for a math symbol friendly keyboard: I needed separate sub-keyboards for ^ and *. However, in a pinch, I *love* being able to do a quick integral on my Android phone 🙂
The MaximaList 