Automatic Threading of Equations

This little utility was originally developer for DMUG, the German Mathematica User Group. It answers a frequently asked question about the manipulation of equations. Mathematica does not "thread" arithmetic and other functions over equations, so the following naive approach to solving equations by rearranging terms does not work:


    In[1]:= 1 + x == 2

    In[2]:= %-1

    Out[2]= -1 + (1 + x == 2)

To make it work, the symbol Equal (the head of an equation) should behave like List with respect to listable functions. Just as {a, b} + 1 turns into {a+1, b+1}, one may want (a==b) + 1 to turn into a+1 == b+1. This can be achieved explicitly with Thread:


    In[3]:= Thread[(a==b) + 1, Equal]

    Out[3]= 1 + a == 1 + b

The automatic transformation of f[a, b, c, ...] into Thread[f[a, b, c, ...]] should happen whenever f has the attribute Listable and at least one of the arguments a, b, ... has head Equal. This definition is essentially what is needed:


Equal/: lhs:f_Symbol?listableQ[___, _Equal, ___] := Thread[ Unevaluated[lhs], Equal ]

listableQ[f_] := MemberQ[Attributes[f], Listable]

The use of Unevaluated prevents an infinite recursion. Together with the necessary framework, the little package EqualThread.m implements this functionality.

Now, you can solve equation as you did in school:

read the package:


    In[1]:= Needs["EqualThread`"]

the equation, to be solved for x:


    In[7]:= a == b Log[2 x]

divide by b:


    In[8]:= %/b
    
	    a
    Out[8]= - == Log[2 x]
	    b

exponentiate:


    In[9]:= Exp[%]

	     a/b
    Out[9]= E    == 2 x

divide by 2:


    In[10]:= %/2

	      a/b
	     E
    Out[10]= ---- == x
	      2

You can also add equations, etc.:


    In[11]:= (a==b) + (c==d)

    Out[11]= a + c == b + d

Notebook and Package Copyright

© 1997 by MathConsult Dr. R. Mäder
Permission is granted to use and distribute this file for any purpose except for inclusion in commercial software or program collections. This copyright notice must remain intact.

© 2012 MathConsult Ltd
http://www.mathconsult.ch/math/stuff/equalthread.html
Comments to the Webmaster; Last update: 19.04.2004