Suppose we have applied trilinear interpolation technique over grid points in the space $A\times B \times C \times D={(x,y,z)|x\in A, y\in B,z \in C}$} . The interpolant, which is piecewise function, is denoted by $I(x,y,z)$. My aim is to calulcate the error of interpolation, that is $I(x,y,z)-f(x,y,z)$. For a linear interpolation, as i know the error is calulated based on the formula $$I(x)-f(x)=\frac{(x_1-x_0)^2}{8}max|f''(x)|.$$ Even for this case I don't grasp how to estimate the error, since the function $f(x)$ is unknown, moreover it is needed that $f(x)$ has second order derivative. Please guide me to understand error esimation for linear interpolation and for tilinear interpolation, when functional forms $f(x,y,z)$ and $f(x)$ are unknown.


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