Polynomial interpolation

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Lagrange polynomial interpolation과 Newton polynomial table을 매트랩으로 계산

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Examine the Runge phenomenon for Lagrange polynomial interpolation of for . Let , where , and compare the graph of the tenth-degree Lagrange interpolant for f on with the graph of f. Investigate what happens when , where the are the Chebyshev abscissae, , . (Hint) To generate (n + 1) uniformly spaced nodes on an interval [a; b], use the following:
.

The Runge phenomenon is due to the fact that grows as the order of differentiation increases. As we can see from part (a), these large errors can be minimized by carefully choosing non-uniformly spaced nodes, such as the Chebyshev nodes. This is because the function , which appears below, depends strongly on the placement of the interpolation points. Plot the graph of the function for equally spaced interpolation points and Chebyshev points for the same interval. Comment on your results.

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