hmwk_3_2010

# These definitions always ensure that exactly what we

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that specific basis function is associated with. These definitions always ensure that , exactly what we want. Note that we do not solve any system of equations and that each basis func- tion is developed as a sequence of products, simply implemented in do loops. Note that the interpolator that we derive using the Basis Function approach is identical to that derived using the power series approach. Method 3: Newton Forward Interpolation reformulates the interpolation function in terms of a sequence of polynomial terms mutiplied by increasingly higher order forward differences. This method has the advantage that each higher order polynomial term has the previous term embeded in it, making it quite inexpensive. In addition, to increase the order of the interpolant, simply requires adding the next term in the sequence, making the methodology hierarchal. Finally, the error is simply estimated by considering the subsequent term in the sequence. Again this method gives the identical interpolation function as the power series and Lagrange Basis function methods. Problem 1 Consider the function: on the interval [1, 4]. a) Develop a power series based interpolator for 4 equispaced points for the given function. i) Evaluate , , and using the specified function . ii) Define the appropriate polynomial expansion, g(x) .

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