Difference between revisions of "Numerical Differentiation"
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== Lagrange Polynomials == | == Lagrange Polynomials == | ||
− | Lagrange polynomials, which are commonly used for [[interpolation]], can also be used for differentiation. | + | Lagrange polynomials, which are commonly used for [[Interpolation#Lagrange_Polynomial_Interpolation|interpolation]], can also be used for differentiation. The formula is |
− | + | <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 | |
− | + | <center><math>L_{k}^{\prime}(x) = \left[ \sum_{{i=0} \atop {i\ne k}}^{n} (x-x_i) \right] \left[ \prod_{{i=0}\atop{i\ne k}}^{n} (x_i-x_k) \right]^{-1}. </math></center> | |
== Tables for Derivatives on Uniform Grids == | == Tables for Derivatives on Uniform Grids == |
Revision as of 09:41, 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
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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