From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: 10th order polynomial from 11 data points
Date: 22 Nov 1997 20:25:17 GMT

In article <6eguoCAcQxd0EwXw@lynagh.demon.co.uk>,
Ian Lynagh  <ian@lynagh.demon.co.uk> wrote:

>In an experiment I collected 2 sets of 11 data points
>and I now need to find the area under the graph. However,
>first I need a function which I can integrate. Can anyone
>recommend a program I can download which will find a 10th
>order polynomial if I give it the points?

Getting the polynomial is no problem for just 11 points. You simply compute
	Sum_i  y_i * Prod_{j <> i} (X-x_j)/(x_i-x_j)
which clearly takes the value  y_i  when we substitute  X = x_i.

But you do need to ask if this is "the graph". Do you have any reason to
suspect that the function you're observing in this experiment is really
a degree-10 polynomial? Do you really know your points (x_i, y_i) are
accurate? If the answer to either question is "not really", then the
integral of this polynomial could well be far from the value you hope
you're computing. Much more reasonable, I would think, would be to do
one of the following:

1. Decide, on the basis of the theory of whatever field your experiment is
in, what kind of relationship  x  and  y  should be in  -- linear, perhaps,
or  y  quadratic in  x,  or  y  an exponential in  x. Then use some
statistical technique to choose the "best" curve of this type to fit the
data. _Then_ you have a function to integrate.

2. Or, assume the 11 measurements are accurate, but assume something else
about the function than that it be a degree-10 polynomial. Two reasonable
suggestions come to mind: assume either that  f  is locally constant
or piecewise linear. (Those are the assumptions involved in estimating an
integral via a Riemann sum or via the trapezoidal rule, after all).

dave