least_sq_intrpol

least_sq_intrpol - MIT OpenCourseWare http:/ocw.mit.edu...

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MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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LS. Least Squares Interpolation 1. The least-squares line. Suppose you have a large number n of experimentally determined points, through which you want to pass a curve. There is a formula (the Lagrange interpolation formula) producing a polynomial curve of degree n - 1which goes through the points exactly. But normally one wants to find a simple curve, like a line, parabola, or exponential, which goes approximately through the points, rather than a high-degree polynomial which goes exactly through them. The reason is that the location of the points is to some extent determined by experimental error, so one wants a smooth-looking curve which averages out these errors, not a wiggly polynomial which takes them seriously. In this section, we consider the most common case - finding a line which goes approximately through a set of data points. I . Suppose the data points are and we want to find the line which "best" passes through them. Assuming our errors in measurement are distributed randomly according to the usual bell-shaped curve (the so-called "Gaussian distribution"), it can be shown that the right choice of a and b is the one for which the sum D of the squares of the deviations I i= l is a minimum. In the formula (2), the quantities in parentheses (shown by
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least_sq_intrpol - MIT OpenCourseWare http:/ocw.mit.edu...

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