Five-point Stencil - Relationship To Lagrange Interpolating Polynomials | Relationship Lagrange Interpolating Polynomials

Relationship To Lagrange Interpolating Polynomials

As an alternative to deriving the finite difference weights from the Taylor series, they may be obtained by differentiating the Lagrange polynomials

where the interpolation points are

$\begin{align} x_0=x-2h,\quad x_1=x-h,\quad x_2=x,\quad x_3=x+h,\quad x_4=x+2h. \end{align}$

Then, the quartic polynomial interpolating ƒ(x) at these five points is

$\begin{align} p_4(x) = \sum\limits_{j=0}^4 f(x_j) \ell_j(x) \end{align}$

and its derivative is

$\begin{align} p_4'(x) = \sum\limits_{j=0}^4 f(x_j) \ell'_j(x). \end{align}$

So, the finite difference approximation of ƒ ′(x) at the middle point x = x₂ is

$\begin{align} f'(x_2) = \ell_0'(x_2) f(x_0) + \ell_1'(x_2) f(x_1) + \ell_2'(x_2) f(x_2) + \ell_3'(x_2) f(x_3) + \ell_4'(x_2) f(x_4) + O(h^4) \end{align}$

Evaluating the derivatives of the five Lagrange polynomials at x=x₂ gives the same weights as above. This method can be more flexible as the extension to a non-uniform grid is quite straightforward.

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