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Redefining a function with a rule for coefficients

• To: mathgroup at smc.vnet.net
• Subject: [mg59376] Redefining a function with a rule for coefficients
• From: Wonseok Shin <wssaca at gmail.com>
• Date: Sun, 7 Aug 2005 03:47:08 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

Suppose that there are two functions f and g:
f[x_, y_] := A x + B y;	
g[x_, y_] := A x - B y;
where the coefficients A and B are unknown.

We know some kind of boundary condition f[1, 1] == 3 and g[1, 1] == -1. 
  I want to solve for coefficients A and B using this boundary 
condition, and to put them back into f and g.

Of course, solving for A and B is very straightforward:
coeff = Solve[{f[1, 1] == 3, g[1, 1] == -1}, {A, B}]

This gives the solution in the form:
{{A -> 1, B -> 2}}
which is the rule assigned in the variable 'coeff.'

The next process is, of course, to put 'coeff' into f and g, and to 
redefine them to be x - 2y and x + 2y.  What is the most standard way 
of doing this?

Here is my solution:
f[x_, y_] = f[x, y] /. coeff[[1]];
g[x_, y_] = g[x, y] /. coeff[[1]];

Note that I used Set (=) instead of SetDelayed (:=).

It works but looks clumsy, and has a potential error.  Look at the 
following codes:
In[1]:=
f[x_, y_] := A x + B y;
g[x_, y_] := A x - B y;

In[3]:=
x = 1;
?f

Global`f
f[x_, y_] := A x + B y

(* The assignment x = 1 does not affect the definition of f[x_, y_]. *)

In[5]:=
coeff = Solve[{f[1, 1] == 3, g[1, 1] == -1}, {A, B}]

Out[5]=
{{A -> 1, B -> 2}}

In[6]:=
f[x_, y_] = f[x, y] /. coeff

Out[6]=
1 + 2 y

In Out[6] our expectation is x + 2 y, but since we've assigend 1 to x, 
this specific value is used for x in the Set procedure in In[4].  Using 
SetDelayed (:=) instead of Set (=) generates more serious problem, 
because it causes an infinite recursion when we evaluate f, for 
example, at (x, y) = (1, 1).

Since determining coefficients of functions using a boundary condition 
is very common situation, I believe there exists some standard and 
elegant way to do this.

Thanks,
-- 
Wonseok Shin
wssaca at gmail.com

• Follow-Ups:
  • Re: Redefining a function with a rule for coefficients
    • From: Pratik Desai <pdesai1@umbc.edu>
  • Re: Redefining a function with a rule for coefficients
    • From: Zhengji Li <zhengji.li@gmail.com>