Mixima

Mathematica compatibility functions and translator for the Maxima computer algebra system.

This software is independent from Wolfram and is neither supported by, nor endorsed by Wolfram Research.

Mathematica is a registered trademark of Wolfram Research, Inc.

Help

If you have any difficulty, please file an issue.

Introduction

This software has four parts:

1. Compatibility functions for Maxima that try to duplicate Mathematica (Mma) functions, eg Table. Used alone, they provide a quasi-Mma environment. Mixima is designed to extend the maxima environment more or less like any other package-- not to replace it, as does Mockmma. The compatibility functions are Maxima functions called with Maxima syntax from within Maxima.

2. A translator that translates a subset of Mma code to Maxima code. The compatitiblity functions can be used by themselves or with this translator. But the translator produces code that depends on the compatibility functions.

3. A mock Mma mode that allows using Mathematica syntax at the command line. This is sort of a Mma clone. The parser is a modified version of the Richard Fateman's Mockmma project. The allowed syntax and semantics is limited to that available when the Mockmma parser was written.

4. A translation of Robert Griffiths' Quantum Information Programs in Mathematica to Maxima is included here. This was the original motivation for Mixima. It is installed together with the mixima compatibility functions.

There are more READMEs at the top level and some information can be found in doc/mixima_user_doc. See also the section on online help at the bottom of this document.

Motivation and context

1. I wrote a quantum-information package, qinf, for Maxima to support reproducible research. It was able to verify computations done by hand in a particular already-published paper.

2. I found that progress could be made by translating existing quantum-information packages for Mathematica (Wolfram) to Maxima. This translation, in turn, is much easier if one writes compatibility functions for the basic functions called by the Mathematica package. A package by Robert Griffiths was a great candidate because it a) Supports the calculations done in the papers I am interested in, and b) Uses a restricted subset of the Mathematica language. In particular, it does not use advanced pattern matching. This package, Mixima, is the result of this process. It successfully translates and runs Griffiths' quantum-information code, as well as an existing package for studying many-body physics. These are both found in the applications folder.

3. The translated application code strong_deco runs correctly but is algorithmically inefficient, which becomes important when the length of the objects studied is large enough. This is due to a fundamental difference in the way expressions are represented in the two languages. So, I implemented an alternate, efficient, representation for expressions in Maxima and supporting code. This code and other third-party packages are organized in the mext repository. The compatibiliy functions and tooling in mext are far more developed than the code here. But, in mext, there is more deviation from the Mathematica API, and there is no translator.

4. In the meantime a language, Julia, has arrived that is suitable for symbolic computation. Its ecosystem is far more developed than those of Maxima and Common Lisp, and is growing. This allows much greater productivity. So I wrote Symata in Julia. Symata is similar in many ways to Mathematica and Maple.

Install

MS Windows and other OSs

If you have maxima installed, this software should work in principle. It is tested a bit on Win32; the testsuite passes. For Windows, copy the entire folder .\build\mixima to your maxima share folder. For instance, copy the folder to c:\Program Files\Maxima-5.25.1-gcl\share\maxima\5.25.1\share Then you should have a folder c:\Program Files\Maxima-5.25.1-gcl\share\maxima\5.25.1\share\mixima with all the mixima files in it.

Linux/unix system

Edit the file (config_install.pl) to choose where the files are installed. By default, the source is installed in ~/.maxima/mixima/ and the executables in ~/bin/. Then type the shell command ./mkdistcompat.pl. Then type the shell command ./install.pl.

If this does not work, try copying ./build/mixima to a location where your maxima installation can see it.

Testing

If mixima is installed, give the maxima commands

load("mixima/mixima");
mixtest();

Or in this toplevel directory, start maxima and give the following commands: load("./tests/mixima_testsuite.mac"); or load("./tests/mixima_testsuite_slow.mac");.

Known problems

• The mockmma shell does not currently work well with xmaxima nor wxmaxima. The compatibility functions seem to work ok.

• wxmaxima has a bug that will print an error message involving STRIPDOLLAR. Mixima can still be used, but the state of maxima is changed and and some things are broken. To avoid this, remove the line that loads loadlast.mac in the file mixima.mac. The offending function is Clear().

Quick start

The compatibility functions are functions written in the Maxima language that emulate the behavior of Mathematica (Mma) functions. For example:

• Compatibility Functions

> maxima
(%i1) load("mixima/mixima");
(%o1)          /usr/share/maxima/5.21.1/share/mixima/mixima.mac
(%i2) Table( i*j, [i,3],[j,3]);
(%o2)                  [[1, 2, 3], [2, 4, 6], [3, 6, 9]]
(%i3) mixFuncs();       /* lists compatibililty functions */
(%i4) mixDoc(Table);    /* prints examples */

• entering the mockmma shell to enter Mma code at the command line

(%i5) tomma();
In[6]:= Array[#1^#2 &,{3,4}]

(%i6) Array(lambda( [[lambda_args]], (lambda_args[1]^lambda_args[2]) ),[3,4])
(%o6) [[1, 1, 1, 1], [2, 4, 8, 16], [3, 9, 27, 81]]

Out[6]:= {{1, 1, 1, 1}, {2, 4, 8, 16}, {3, 9, 27, 81}}
In[7]:= Exit
(%o7)                                true
(%i8) %o6;
(%o8)            [[1, 1, 1, 1], [2, 4, 8, 16], [3, 9, 27, 81]]

tomma() does not work with xmaxima and wxmaxima.

• Evaluating strings of Mma code.

(%i1) smmatomax(" a = 1 ");
(%o1)                                  1
(%i2) a;
(%o2)                                  1

smmatomax() should work with xmaxima and wxmaxima.

• translating code at the unix shell: mmatomax mma_code.m

Details

Mixima can be used several ways:

Functions that extend Maxima with functions similar to those in Mma

 > maxima
(%i1) load("mixima/mixima");
(%o1)          /usr/share/maxima/5.21.1/share/mixima/mixima.mac
(%i2) Table( i*j, [i,3],[j,3]);
(%o2)                  [[1, 2, 3], [2, 4, 6], [3, 6, 9]]

Note that the syntax used here is Maxima syntax, not Mma. For instance

lists: {} --> []
function call: [] --> ()
assignment:   = --> :  ,etc.

A list of the compatibility functions is included in the file function_list. Example code can be found in the (./tests) folder and in the (./applications) folder. The code in the tests folder contains comments for some Mma functions concerning how much of those functions are implemented.

As a convenience, the executable mixima loads the mixima code automatically but does not currently allow other initialization scripts):

> mixima
...
(%i1) Tuples([0,1],2);
(%o1)                  [[0, 0], [0, 1], [1, 0], [1, 1]]

Mixima can use the Mockmma shell.

For example

(%i1) tomma();
 ...
In[1]:= Table[ i j,{i,3},{j,3}]

Out[1]:= {{1, 2, 3}, {2, 4, 6}, {3, 6, 9}}

In[2]:= NIntegrate[1/Sqrt[x],{x,0,1},PrecisionGoal->12]

Out[2]:= 1.999999999999998

In[3]:= Exit

As a convenience, the executable miximamma starts Maxima and the Mockmma shell with the Mixima translator.

Here are commands that can be used to control whether the results are shown in Maxima or Mma or both.

• MmaShowOut[True] (or False) to control Mma output
• MmaShowTrans[True] (or False) to control maxima display of translation.
• MmaVerbose[True] (or False) to print more stuff during evaluation.
• MmaPrintEx[arg] print strings suitable for inserting in rtest files as tests and examples. If no arg is given , print no function name in the example. If a string is given, it is used as the function name for subsequent input. If arg is False, then disable.

The original Mockmma is included

Mockmma is modified slightly from the upstream Mockmma distribution. Usually, when we refer to mockmma, we don't mean this original, but rather the mockmma shell modified to use the compatibility functions.

(%i1) :lisp (mockmma)
 ...
In[1]:=

As a convenience, the executable mockmma starts maxima and Mockmma (code by Fateman et. al. modified by H.Monroe)

Note that for many inputs, Mockmma and Mixima with the Mockmma shell give identical output. But in fact there are important difference that are not explained here.

Using translator from maxima, Using stand alone translation scripts

For these, see README.translator

Compatibility Functions

Due in part to fundamental differences int the two CAS's, these compatibility functions reproduce the behavior of the original functions imperfectly and to varying degrees. Examples of ways in which they may be useful are

1. They can be used in Maxima projects by people who are already familiar with the Mma functions.
2. They can be used to more easily translate user Mma code to Maxima.
3. They may provide functionality not currently available in Maxima.

Use of Compatibility functions

Load the functions from the Maxima prompt with (%i1) load("mixima/mixima");.

The other source files will then be loaded automatically. The files must have been placed in your Maxima search path for this to work.

Examples of how to use the functions and some notes on limits in functionality are in the test file rtest_mixima.mac.

Matrices and Sets

• Maxima has a special data type for matrices, while Mma implements matrices as ordinary lists. This is handled in the compatibility functions by allowing either a Maxima list or a Maxima matrix as an argument that is to be interpreted as a matrix. If the function returns a matrix, then, if possible, the output type is the same as the input type. For instace:

(%i29) Inverse( matrix( [1,2],[3,4]));
(%o29) matrix([-2,1],[3/2,-1/2])
(%i30) Inverse( [ [1,2],[3,4] ]);
(%o30) [[-2,1],[3/2,-1/2]]

• Likewise Mma implements sets as ordinary lists. These are handled by the compatibility functions in the same way that matrices are. For instance:

 (%i34) Union( {1,2}, {3,4});
 (%o34) {1,2,3,4}
 (%i35) Union( [1,2], [3,4]);
 (%o35) [1,2,3,4]

• If the arguments are a both sets and lists, then the return type is the same as the first argument.

• Use the compatibility function Dot and not the Maxima . if you want the the Mma function Dot, which can also be written in Mma in the . form. In other words do not use the Maxima . function for the Mma ., but rather use the compatibility Dot. The function Dot does not accept Maxima matrices (but they can be converted) because the two systems are not compatible. Read the documentation on matrix multiplication for both systems to see why.

• Useful functions are

  • ensuremat(m) Converts m to a maxima matrix if it is a list. Returns m unchanged if it is already a matrix.
  • ensurelist(m) makes matrix m to a list if not already a list.

Online Help

• mixFuncs(); lists compatibility functions and operators.

• mixFuncs(string); lists compatibility functions and operators whose names contain string.

• mixDoc(function); lists examples of a function, where function is a symbol or a string. ie. mixDoc("Integrate"); or mixima_doc(Integrate);.

• mixDoc(Integrate,bad); list unimplemented or failed examples.

Not all functions have examples.

Syntax

• @@ can be used for Apply

• -> can be used for Rule.

Mma shell

• Example of entering and using the shell.

Note below that the mma shell and maxima shell use the same line numbers. Use In[3], Out[3] to acess %i3, etc. The Mma parser can't read %i3 properly. Note that in the mixima-mockmma shell, the maxima input and output is printed.

(%i1) a : 1;
(%o1)                                  1
(%i2) tomma();

In[3]:= a
 (%i3) a
 (%o3) 1

Out[3]:= 1

In[4]:= b = 2
 (%i4) (b:2)
 (%o4) 2

Out[4]:= 2

In[5]:= In[1]
 (%i5) In(1)
 (%o5) a : 1

Out[5]:= MSETQ[a, 1]

In[6]:= Out[1]
 (%i6) Out(1)
 (%o6) 1

Out[6]:= 1

In[7]:= Exit
 (%i7) Exit
 (%o7) Exit
(%o7)                                true
(%i8) %o4;
(%o8)                                  2
(%i9)

• What happens when you use the shell ? You enter a line of Mma code. It is parsed as Mma code. This is translated to maxima code and evaluated by maxima. The resulting maxima expression is then translated back to Mma and printed.

You can turn off the printing of the intermediate results.

In[10]:= MmaShowTrans[False]
(%i10) MmaShowTrans(False)

Unsetting maxima translation

Out[10]:= False

In[11]:= N[Pi,12]

Out[11]:= 3.1415926535897931

In[12]:= InString[11]

Out[12]:= N(Pi,12)

In this case, using InString[11] gives the desired result. In[11] ( or %i11 in Maxima) give errors because they stored results passed throught the simplifier, which are returned in a new context.

You can read Mma expressions from a file with miximaEvalFile("filename");.

Or from withing the mockmma shell miximaEvalFile["./translator/t1.m"].

• Commands related to the mockmma shell

These "commands" are functions that can be usefully called from withing the maxima shell as well; eg In(3) rather than In[3].

• InString[n] The nth (eg 3rd) mma input string from user's input, returned by the parser, with whitespace and commas stripped. This is stored as %is3. Recall that the line numbers and input and output are the same in the two shells. Thus, if the nth line was entered in the maxima shell, there will be no corresponding mma string and InString[n] returns an atom, eg %is3.

• mInString[n] The nth (eg 3rd) maxima input string built by translating the mma input string. Also available as %ims3, but this is not valid syntax in the mockmma shell.

• In[n] The nth maxima input expression. Currently only printed in "2d" form. This expression is created by the parser from the user's mma input string. This is the correct expression to use in the mockmma shell. This is the same as %i3, etc. in the maxima shell. The expressions entered in the mockmma shell are copied to %i{n} so that they are available within the maxima shell; also via In(n).

• Out[n]

The nth maxima output expression. This is the correct way to access the output for further calculation from within the mockmma shell.

Acknowledgments

A lot of invaluable advice and code snippets were provided by Maxima developers. In particular, Robert Dodier, Richard J Fateman, Stavros Macrakis, and Barton Willis.