Recently I was looking for information about multivariable interpolation. I was curious about interpolation with two and three independent variables. My first stop was Wikipedia. The algorithm presented there dealt with interpolation based on the coordinates of the independent variables forming a square. That is, the bilinear interpolation used points (x1,y1), (x1,y2), (x2,y1), and (x2,y2).

Further search on the Internet located a paper that dealt with bilinear and trilinear interpolation. The paper showed more elegant equations for square-shaped coordinates for the bilinear interpolation. The paper also included equations for the cube-shaped trilinear interpolation. Implementing these algorithms in BASIC, RPN, or RPL should be easy.

After implementing the bilinear and trilinear interpolations, I asked myself the question, “What if the anchor coordinates for the bilinear interpolation do not form a square? Likewise, what if the anchor coordinates for the trilinear interpolation do not form a cube?”

I proceeded to calculate the coefficients for the multi-variable polynomials used for the bilinear and trilinear interpolations. After a little bit of math wrangling, I realized that my best bet was to use Vandermonde types of matrices. Such matrices would allow me to calculate the coefficients of the interpolating polynomials. In addition, you need fewer anchor points than the square or cube approaches. One advantage of calculating the interpolating polynomial coefficients is that you can reuse the coefficients for additional interpolations that use the same anchor points. This is an advantage that is not available when using interpolation equations that simply yield the final answer.

Using the Vandermonde types of matrices allowed me to easily implement quadratic interpolations for two and three independent variables. If the implementation uses a language that works with matrices, then the calculations are greatly simplified! Such is the case with Matlab, RPL and the HP-71B that has the MATH ROM.

Namir