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MathGroup Archive 1998

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Re: mg[14566]: Follow-up on Piecewise Continuous Functions

  • To: mathgroup at smc.vnet.net
  • Subject: [mg14587] Re: mg[14566]: Follow-up on Piecewise Continuous Functions
  • From: Jack Goldberg <jackgold at math.lsa.umich.edu>
  • Date: Mon, 2 Nov 1998 01:50:59 -0500
  • Sender: owner-wri-mathgroup at wolfram.com

Hi Group:

I remarked in my earlier post that manipulating PC functions  was not
difficult and I promised to show why and how in  subsequent posts. 
Here I explain the basic ideas. 

(1).  Let us agree on a standard form for PC functions. I prefer using
Which[]. Examples:

tent[x_] := 
	Which[ -Infinity<x<0, 0, 0<=x<1, x, 1<=x<2, 2-x, 2<=x<Infinity, 0 ]   

square[x_] := 
	Which[ -Infinity<x<0, 0, 0<=x<1, 1, 1<=x<Infinity, 0 ]

noname[x_] := 
	Which[ -Infinity<x<0, x, 0<=x<Infinity, 1 ]


(a)  All functions are defined for all x.  If necessary we use 
"-Infinity < x < a, 0"   and   "b<= x < Infinity < 0"  to insure  that
the functions are defined everywhere.  

(b)  The functions are continuous from the right, that is, 
 	f(a) = Lim f(x), x->a, x>a

(c)  In  "Which[..., a<=x<b, f(x), ...]"  f(x) is assumed to  be
analytic in  a<x<b.  (So  f[x]  is not itself a piecewise continuous
function.)

Now the question.  The function 

(A)	noname[x] + tent[x]*square[x]

for example, is a PC function.  What is its canonical Which form? To
construct the which form for this function, I rely on a new function
called "step".  (Not to be confused with UnitStep[]  which appears in 
many packages but is not built-in at least in Version 3.0 and earlier.)


(2).  Defining "step"

step[x_, a_, f_]  is not available to the user.  It is a function  used
only in the private part of the package as an intermediate 
computation.  It is a numerical function of  x, a  and  f(x) but is
never evaluated.  Its meaning is this:

step( x, a, f(x) )  returns 0 if x<a, returns  f(x)  if  x>=a. 

Various simplification rules are defined for "step"  (as of now  I have
9) Here are some samples.

step  /:  step[ x_,a_,f_ ]*step[ x_, b_, g_ ] := step[ x,Max[a,b], f*g ]
 
step /:   step[ x_,a_,f_ ]^n_Integer?Positive := step[ x,a,f^n ]

step /:   step[ x_,a_,f_ ]+step[ x_,a_,g_ ] := step[ x,a,f+g ]


(3).  The central idea.

The private function  "tostep[]"  takes as an argument any combination 
of canonical Which[]'s  such as the one given in (A) above, and
produces  a sum of step functions.  The private function  "fromstep[]"
converts output of "tostep' to the canonical  "Which[]".  The rules
that define "step" when used in "tostep" always lead to a sum of this
kind,

step[ x,a1,f1[x] ] + step[ x,a2,f2[x] ] + ... + step[ x,an,fn[x] ]

where  a1 < a2 < ... < an.  From such sums,  "fromstep[]"  easily 
converts to a single canonical  "Which[]".  Handling ordinary functions
combined with PC functions is easy, since a separate program writes 
f(x)  as  Which[ -Infinity<x<Infinity,f]  or, if necessary,
 
 	Which[ -Infinity<x<a, 0,  a<=x<b, f(x), b<=x<Infinity, 0 ] 

This latter case occurs for functions such as  Sqrt[1-x^2]  which is 
not defined for  |x|>1.  

The actual implementation of each of these functions (step, tostep,
fromstep, etc)  is not difficult but is a bit tiresome.  The code
itself is less than 100 lines and has printing and tabulation options
that  make for easier visualization.  For example, the user function 

ToPiecewise[f_,x_,opts] 

calls on the private functions to return the canonical PC representation
for any (?)  f(x).  An optional to  ToPiecewise  sketches the graph; a 
second option writes the PC function in a convenient  ColumnForm.

The code will be posted (I hope!) next week.

Jack



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