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| - | '''Lagrange Interpolating Polynomials''' are derived from the simple idea that at each point in a set, the value of the interpolating polynomial must line up with that point. In other words,
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| - | <math> f_n(x) = f(x_i) </math> if <math> x = x_i </math>
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| - | and
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| - | <math> f_n(x) = 0 </math> otherwise.
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| - | Making this true for all of the data points in a set, we come up with the following:
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| - | <math> f_n(x) = \sum_{i=0}^n{\prod_{j=0,j\neq{i}}^n{\frac{x-x_j}{x_i-x_j} f(x_i)}} </math>
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| - | This is often simplified by defining the Lagrangian polynomials, <math> L_i </math> as the interior product, so that the final form of the Lagrangian interpolating polynomial is:
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| - | <math> f_n(x) = \sum_{i=0}^n{L_i f(x_i)} </math>
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| - | where
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| - | <math> L_i = \prod_{j=0,j\neq{i}}^n{\frac{x-x_j}{x_i-x_j}} </math>
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| - | This form has the advantage of not having to calculate the divided differences necessary for computing the Newton interpolating polynomials, but is much farther from the traditional form of polynomial interpolation than other methods. For an example of Lagrange interpolating polynomials, see LagrangeExample.doc .
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| - | References
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| - | 1. Chapra, Steven C. and Canale, Raymond P. Numerical Methods for Engineers. Second ed. 1988 McGraw Hill.
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