Difference Tables

Date: 05/09/97 at 09:49:17
From: Terah DeJong
Subject: Math Project Questions

Dear Dr. Math,

I am learning how to make equations from a bunch of x and y values by 
using difference tables and matrices. I understand how to do it, but 
I just have a few questions relating to how it works.

Why is it that with almost any values for the difference table, a 
common number is always reached to determine the degree of the 
equation?

I thank you in advance for taking the time to look over this.

Sincerely,
Terah DeJong

Date: 05/09/97 at 13:29:47
From: Doctor Jerry
Subject: Re: Math Project Questions

Hi Terah,

If the x values are increased by a fixed amount and the y values are 
calculated using an nth degree polynomial, then, as you know, the nth 
differences will be constant.  For example, if x = 1,2,3,4,... and 
y = 1,4,9,16,..., then the first differences of the y values are 
3,5,7,9,... and the second differences are 2,2,2,2,....  

If the x values are increased by a fixed amount, say, a, a+h, a+2h and 
so on and the y values are calculated as sin(a), sin(a+h), sin(a+2h), 
a constant difference will never be reached, no matter how far you 
continue the differencing.  This is true since the sine function is, 
in a certain sense, a polynomial of infinite degree.  What I mean is, 
if t is in radians, sin(t) = t-t^3/3! + t^5/5! and so on.

However, in practice we can't do the differences exactly - we must 
round the numbers to some fixed number of decimal places. When we do 
this, depending on the number of places, we get differences that are 
approximately constant.

Charles Babbage (of Difference Engine or Analytical Engine fame), in 
the last century, used differences to calculate tables, which was a 
very important idea at the time. These days, we calculate tables in 
different ways, or, rather, we can use a calculator or computer. 
Usually, they don't use differences. The idea of differences is still 
important. Because they are simpler (in some ways) than derivatives, 
some applied mathematicians model various phenomena with difference 
equations.

As late as the 1950s, mathematicians used differencing techniques to 
check tables that had been calculated in other ways.

If just one error has been made in a long list of calculations, the 
size of the error can be inferred by looking at differences. The 
error propagates through the differences in a binomial way. You could 
try this yourself. Generate some numbers from 2x^2 + 5x + 3. Let 
x = 1,1.5,2,...,4. Change the 28 to a 29. Take third differences.  
Observe the 1,-3,3,-1, which are binomial coefficients.  From this you 
can infer the error. Do this more generally and work out the theorem.

-Doctor Jerry,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/