Many years ago, I encountered a problem involving four numerical data in a square array or a rectangular array. The standard method for interpolating that design is the bilinear equation.(adsbygoogle = window.adsbygoogle || []).push({});

For example, let the array be ACIG as below left. If A=1, C=3, G=7, I=9, then the bilinear equation yields z = (1+2x+6y) in the (x, y) = (0 .. 1, 0 ..1) coordinate system. However, if the data are the squares of

the cited data, the bilinear equation yields z = 35 at the center of the design. That is not a

G I good estimate of the true value z = 25. I sought help at universities but was rebuffed by

A C the remark that no other equation for the four-point rectangle is possible. A few years

later I happened on the relation (e^x)F(x) = F(x+h) where is 'e' is now an operator and F(x) is any function of (x).

This ultimately led to an interpolating equation that is exact on bilinear data and on the squares of bilinear data. That led to a nasty verbal interchange. It also resulted in two papers: "No Free Lunch: Comments on Silver" where the new equation was declared impossible. Read it and see: Quality Engineering 5(3) 369-373 (1993) by Norman R Draper and Dennis K J Lin. It also led to my rebuttal: Free Lunch, Bigger Menu, Better Food" by G. L. Silver in Quality Engineering 6(2) 307-310 (1994). Read it and see.

Draper and Lin were never heard from again. After I arrived at Los Alamos National Laboratory (New Mexico) I added to the altercation: "Operational equations for data in rectangular array" by G. L. Silver, Computational Statistics and Data Analysis 28 (1998) 211-215. That citation gives the complete equation in the -1 .. 1 coordinate system. It is exact on (positive) bilinear numbers and on the squares of such numbers.

For example, if A=1, C=3, G=7, I=9 then the equation is z = (5+x+3y) but if A=1, C=9, G=49, I=81 then z = (5+x+3y)^2. There are lots of operational equations. Most are "substitute and see" equations. They are probably pertinent to physics. You can join in the fun, too!

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# Do you know about operational equations?

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