Difference between revisions of "Numerical Differentiation"
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<center><math>f^{\prime}(x) = \sum_{k=0}^n y_k L_{k}^{\prime}(x),</math></center> | <center><math>f^{\prime}(x) = \sum_{k=0}^n y_k L_{k}^{\prime}(x),</math></center> | ||
where <math>L_{k}^{\prime}(x)</math> is given as | where <math>L_{k}^{\prime}(x)</math> is given as | ||
− | <center><math>L_{k}^{\prime}(x) = \left[ \sum_{j=0}^{n} \left( \prod_{ {i=0}\atop{i \ne j,k} }^n (x-x_i) \right) \right] \left[ \prod_{{i=0}\atop{i\ne k}}^{n} (x_i | + | <center><math>L_{k}^{\prime}(x) = \left[ \sum_{j=0}^{n} \left( \prod_{ {i=0}\atop{i \ne j,k} }^n (x-x_i) \right) \right] \left[ \prod_{{i=0}\atop{i\ne k}}^{n} (x_k-x_i) \right]^{-1}. </math></center> |
Here <math>n</math> is the order of the polynomial and we require <math>n_p=n+1</math> points to form the Lagrange polynomial. | Here <math>n</math> is the order of the polynomial and we require <math>n_p=n+1</math> points to form the Lagrange polynomial. | ||
Revision as of 12:11, 15 July 2008
Contents
Introduction
Taylor Series
Lagrange Polynomials
Lagrange polynomials, which are commonly used for interpolation, can also be used for differentiation. The formula is
where is given as
Here is the order of the polynomial and we require points to form the Lagrange polynomial.
Tables for Derivatives on Uniform Grids
Derivative at point | Discrete Representation (uniform mesh) | Order |
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Forward Difference | ||
Backward Difference | ||
Central Difference |
Derivative at point | Discrete Representation (uniform mesh) | Order |
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