maxima-interface

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#+TITLE: Simple interface between Common Lisp and Maxima #+AUTHOR: Juan M. Bello-Rivas #+EMAIL: [email protected] #+STARTUP: showall

• Introduction

Maxima is a featureful and versatile symbolic computation package. While it runs on most (possibly all) Common Lisp (CL) implementations, using it as an external CL library is not entirely straight-forward despite the fact that the functionality exists. The purpose of this package is to provide a simple interface between Common Lisp and Maxima and facilitate its use.

• Installation

The simplest way to install this package is to compile and install Maxima from source and add the directory =/path/to/maxima/src= to the ASDF registry. Next, add =maxima-interface='s directory to the ASDF registry (see [[https://asdf.common-lisp.dev/asdf/Configuring-ASDF-to-find-your-systems.html][ASDF documentation]] for details).

• High level interface

The aim of this interface is to translate and evaluate expressions back and forth from Common Lisp to Maxima's internal representation as seamlessly as possible.

** =diff=

Takes the (partial) derivative of an s-expression with regard to a variable, possibly more than once.

*** Examples

#+begin_src lisp MAXIMA-INTERFACE-USER> (diff '(sin x) 'x) (COS X) #+end_src

#+begin_src lisp MAXIMA-INTERFACE-USER> (diff '(* (expt x 2) (expt y 3)) 'x 2) (* 2 (EXPT Y 3)) #+end_src

** =integrate=

Evaluates the definite or indefinite integral of an s-expression with respect to a symbol.

*** Examples

#+begin_src lisp MAXIMA-INTERFACE-USER> (integrate '(* (exp x) (sin (exp x))) 'x) (* -1 (COS (EXPT (EXP 1) X))) #+end_src

#+begin_src lisp MAXIMA-INTERFACE-USER> (integrate '(/ x) 'x 1 '(exp 1)) 1 #+end_src

** =expand=

Expands products.

*** Examples

#+begin_src lisp MAXIMA-INTERFACE-USER> (expand '(expt (+ x y) 2)) (+ (EXPT X 2) (* 2 X Y) (EXPT Y 2)) #+end_src

** =simplify=

Applies trigonometric and rational simplifications.

*** Examples

#+begin_src lisp MAXIMA-INTERFACE-USER> (simplify '(/ (1- (1+ x)) 1)) X #+end_src

#+begin_src lisp MAXIMA-INTERFACE-USER> (simplify '(+ (expt (cos x) 2) (expt (sin x) 2))) 1 #+end_src

** =limit=

Takes the limit of an s-expression with regard to a symbol.

*** Examples

#+begin_src lisp MAXIMA-INTERFACE-USER> (limit '(/ (+ (expt x 3) x) (* 2 (expt x 5))) 'x 'inf) 0 #+end_src

#+begin_src lisp MAXIMA-INTERFACE-USER> (limit '(/ (sin x) x) 'x 0 'plus) 1 #+end_src

• Low level interface

** =maxima-init=

This function is called when the module is loaded. It instructs Maxima to set the path names for the packages that will be loaded on-demand when carrying out different types of operations (e.g., simplification, eigendecompositions, etc.).

** =maxima-run=

Evaluates a Maxima expression passed as a string and prints the result.

*** Examples

#+begin_src lisp MAXIMA-INTERFACE-USER> (maxima-run "assume(sigma > 0)$" :display2d t) [sigma > 0] #+end_src

#+begin_src lisp MAXIMA-INTERFACE-USER> (maxima-run "integrate(exp(-x^2/(2*sigma^2)), x, -inf, inf);" :display2d t) sqrt(2) sqrt(%pi) sigma #+end_src

#+begin_src lisp MAXIMA-INTERFACE-USER> (maxima-run "exp(-x^2/(2sigma^2));" :display2d nil :return-expression t) %e^-(x^2/(2sigma^2)) ((MAXIMA::MEXPT MAXIMA::SIMP) MAXIMA::$%E ((MAXIMA::MTIMES MAXIMA::SIMP) ((MAXIMA::RAT MAXIMA::SIMP) -1 2) ((MAXIMA::MEXPT MAXIMA::SIMP) MAXIMA::$SIGMA -2) ((MAXIMA::MEXPT MAXIMA::SIMP) MAXIMA::$X 2))) #+end_src

#+begin_src lisp MAXIMA-INTERFACE-USER> (maxima-run "trigsimp(cos(x)^2 + sin(x)^2);") 1 #+end_src

** =maxima-read=

Takes a Maxima expression (represented as a string) as input and returns its internal representation in (Maxima) Lisp.

*** Example

#+begin_src lisp MAXIMA-INTERFACE-USER> (maxima-read "x^2$") ((MAXIMA::MEXPT) MAXIMA::$X 2) #+end_src

** =maxima-eval=

Evaluates the internal Lisp representation of a Maxima expression and returns the internal Lisp representation of its result.

*** Example

#+begin_src lisp MAXIMA-INTERFACE-USER> (maxima-eval '((maxima::$expand) ((maxima::mexpt) ((maxima::mplus) maxima::$x maxima::$y) 2))) ((MAXIMA::MPLUS MAXIMA::SIMP) ((MAXIMA::MEXPT MAXIMA::SIMP) MAXIMA::$X 2) ((MAXIMA::MTIMES MAXIMA::SIMP) 2 MAXIMA::$X MAXIMA::$Y) ((MAXIMA::MEXPT MAXIMA::SIMP) MAXIMA::$Y 2)) #+end_src

** =maxima-print=

Prints the internal Lisp representation of a Maxima expression in human-readable form to =output-stream= (which is =standard-output= by default).

The keyword argument =display2d= is a boolean indicating whether the representation should be done in 2D or not. The keyword argument =return-expression= is another boolean that determines whether the original expression should be returned by =maxima-print=.

*** Example

#+begin_src lisp MAXIMA-INTERFACE-USER> (maxima-print '((maxima::%integrate maxima::simp) ((maxima::mexpt) maxima::$%e ((maxima::mtimes) -1 ((maxima::mexpt) maxima::$x 2))) maxima::$x 0 maxima::$inf) :display2d t) inf / 2 [ (- 1) x I %e dx ] / 0 #+end_src

** Tips

The document [[https://maxima.sourceforge.io/misc/Fateman-Salz_Simplifier_Paper.pdf][Macsyma’s General Simplifier: Philosophy and Operation by R. Fateman]] is a useful guide to better understand the inner workings of Maxima.

It is sometimes useful to execute =:lisp (trace meval)= inside a regular Maxima session (i.e., the REPL you get when invoking =maxima= from the command line) to see how commands are processed.

• LaTeX rendering

The expressions returned by Maxima can be rendered in LaTeX using either Emacs, Jupyter notebooks, or printing the LaTeX strings to a stream. The precise output is governed by the special variable =latex-output= which can be =:emacs=, =:jupyter=, or =:console=.

The function to accomplish this is named =latex=. The way a symbol is represented in LaTeX can be controlled by the =maxima::texword= property of the symbol's property list (see examples below).

** Jupyter

Using [[https://github.com/yitzchak/common-lisp-jupyter][=common-lisp-jupyter=]] it is easy to interface with a Jupyter notebook. Just run the Common Lisp kernel and start using =maxima-interface=.

#+CAPTION: LaTeX rendering in a Jupyter notebook [[file:screenshot-jupyter.png]]

** Emacs

LaTeX can be rendered in a resizable fashion within Emacs using =tex2svg= in order to convert LaTeX strings to SVG files. This requires the custom patch to SLIME included in the file [[file:slime.patch]] as well as the [[https://github.com/mathjax/MathJax-demos-node/blob/master/component/tex2svg][=tex2svg= utility]] from MathJax.

#+CAPTION: Emacs rendering [[file:screenshot-emacs.png]]

** Console

This is the default setting. It outputs the LaTeX string corresponding to the expression.