Then, we also compute the cross-derivative along $xy$ for each grid node. 17-1, p.18-20, 1974.įor example, to interpolate on a 2-D grid $\left(x,y\right)$, we first apply the 'makima' derivative formula separately along $x$ and $y$ to obtain two directional derivatives for each grid node. Akima, "A Method of Bivariate Interpolation and Smooth Surface Fitting Based on Local Procedures", Communications of the ACM, v. Akima noticed this property in his 1974 paper. Generalization to n-D gridsĪkima's formula and our modified 'makima' formula have another desirable property: they generalize to higher dimensional n-D gridded data. Therefore, 'makima' still preserves Akima's desirable properties of being a nice middle ground between 'spline' and 'pchip' in terms of the resulting undulations. In fact, the results are so similar that it is hard to tell them apart on the plot. Notice that 'makima' closely follows the result obtained with Akima's formula. Indeed, 'makima' does not produce an overshoot if the data is constant for more than two nodes ($v_5=v_6=v_7=1$ above).īut what does this mean for the undulations we saw in our first example? compareCubicPlots(x1,v1,xq1,true, 'NE') Let's try the 'makima' formula on the above overshoot example: compareCubicPlots(x2,v2,xq2,true, 'SE') For this special case of constant data, we set $d_i =0$. Modified Akima interpolation - 'makima'Īkima piecewise cubic Hermite interpolationįor each interval $[x_i~x_$.Akima piecewise cubic Hermite interpolation.
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