To numerically approximate the second partial derivatives, and get the Hessian matrix, one may use "2-point" central finite difference formulas: $$H_{xy}(x,y) \approx \frac{f(x+h,y+h)-f(x+h,y-h)-f(x-h,y+h)+f(x-h,y-h)}{4h^2}\tag{1}$$ for the off-diagonal elements, and $$H_{xx}(x,y) \approx \frac{f(x+h,y)-2f(x,y)+f(x-h,y)}{h^2}\tag{2}$$ formulas for the diagonal elements. (I am simplifying here, this generalizes for any number of dimensions, and uses the same displacement for all variables)

It may be sometimes desirable to further improve the accuracy of these central difference formulas. For univariate functions this is conveniently done via the 4-point formula (aka. 5-point stencil in one dimension): $$f''(x) \approx \frac{-f(x+2 h)+16 f(x+h)-30 f(x) + 16 f(x-h) - f(x-2h)}{12 h^2}$$

However, I have been unable the find the "4-point" equivalent of $(1)$ and $(2)$ anywhere online.


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