Difference TablesDate: 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/ |
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