Pre-Engineering 220 Introduction to MatLab ® & Scientific Programming j kiefer
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- 1. Uniqueness
A. Polynomial InterpolationWith interpolation, the approximating function passes through the data points. Commonly, the unknown f(x) is approximated by a polynomial of degree n, pn(x), which is required to pass through all the data points, or a subset thereof. 1. UniquenessTheorem: Given {xi} and {fi}, i = 1, 2, 3, . . ., n + 1, there exists one and only one polynomial of degree n or less which reproduces f(x) exactly at the {xi}. Notes i) There are many polynomials of degree > n which also reproduce the {fi}. ii) There is no guarantee that the polynomial pn(x) will accurately reproduce f(x) for  . It will do so if f(x) is a polynomial of degree n or less.
Proof: We require that pn(x) = fi for all i = 1, 2, 3, . . ., n+1. This leads to a set of simultaneous linear equations 
which we’d solve for the {ai}. As long as no two of the {xi} are the same, the solution to such a set of simultaneous linear equations is unique. The significance of uniqueness is that no matter how an interpolating polynomial is derived, as long as it passes through all the data points, it is the interpolating polynomial. There are many methods of deriving an interpolating polynomial. Here, we’ll consider just one. Directory: physics physics -> Photometer and Optical Link Sensitivity issues, needs editing 1/2010 Purpose physics -> Photometer and Optical Link Purpose physics -> - physics -> Teacher Notes – Activity 14: Momentum in Collisions physics -> Teacher Notes – Activity 15: Impulse and Change in Momentum physics -> Linear Momentum Definition Download 8.69 Mb. Share with your friends:
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