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manipulating equations
Posted:
Jan 8, 1998 11:54 PM


==_Exmh_1940392000
this little utility was originally developer for DMUG, the German Mathematica User Group (http://www.mathematica.ch/). 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, please find attached the little package EqualThread.m that 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
Roman Mdder
 MathConsult Dr. R. Mdder Samstagernstrasse 58a ComputerAided Mathematics 8832 Wollerau, Switzerland
T: +411687 4051 mailto:maeder@mathconsult.ch F: +411687 4054 http://www.mathconsult.ch/  ==_Exmh_1940392000 ContentDescription: EqualThread.m
(* :Title: make equations behave like lists *)
(* :Author: Roman E. Maeder *)
(* :Summary: make listable functions thread over equations as they do over lists. Allows for easy manipulation of equations. *)
(* :Context: EqualThread` *)
(* :Package Version: 1.1 *)
(* :Copyright: ) Copyright 1997, Roman E. Maeder.
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. *)
(* :History: Version 1.1 for mathgroup and MathSource, January 1998. Version 1.0 for DMUG (German Mathematica User Group), January 1998. *)
(* :Keywords: Equal, equation solving, threading *)
(* :Warning: Adds definitions to the builtin symbol System`Equal. *)
(* :Mathematica Version:3.0 *)
BeginPackage["EqualThread`"]
(* no exports *)
Begin["`Private`"]
listableQ[f_] := MemberQ[Attributes[f], Listable]
protected = Unprotect[Equal]
Equal/: lhs:f_Symbol?listableQ[___, _Equal, ___] := Thread[ Unevaluated[lhs], Equal ]
Protect[Evaluate[protected]]
End[]
EndPackage[]
==_Exmh_1940392000



