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This all pertains to a piece of software I am writing but figured I'd get a better answer here than in Stackoverflow. I have no problem migrating the question if needed.

Disclaimer: I am a software developer - my math terminology might not be spot on. Correct me please!

I have implemented a basic exact polynomial curve fit, typically second or third order (user driven). The coordinate space and precision/scale of the indep/dep variables can be orders of magnitude apart and I found the best results by first normalizing the coordinate space and then solving the curve. The problem I have is that the coefficients of the polynomial curve are only useful in the normalized space and the users want to manually use the coefficients outside of the application (say in Excel, Matlab, by hand, or in a report). The coefficients are only useful if you transform the input indep value into normalized space, evaluate the equation, and then transform the result from normalized space to dependent space.

The normalized space is because I'm dealing with Y/dependent space measured in 10^6 and X/indep space measured in 10^-9. Normalization is only required because I can't easily represent infinite precision.

Example: Quadratic case, where A,B,C previously found from solving a linear system

$$ [\begin{matrix} X & 1 \end{matrix}] [\begin{matrix} S_x\\ T_x \end{matrix} ] = X' $$ $$ [\begin{matrix} A & B & C \end{matrix}] [\begin{matrix} X'^2\\ X'\\ 1 \end{matrix}] = Y' $$ $$ [\begin{matrix} X' & 1 \end{matrix}] [\begin{matrix} S_y^-1\\ T_y^-1 \end{matrix} ] = Y $$

Is there any way to transform the equation coefficients so that the input transform and output transform are not needed? I want new coefficients, [A', B', C'] such that the input X and output Y transformations are not needed.

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I'm not quite sure what you meant by "normalization"; could you maybe include a worked example of a polynomial fitting problem where your "normalization" is required? –  J. M. Dec 9 '11 at 13:37
P.S. What algorithm(s) are you using for this? –  J. M. Dec 9 '11 at 13:37
Somehow, I think your scaling problems are better solved by a change in units than by mucking around with normalizations. It's a bit like using ångströms instead of parsecs to measure the distances between galaxies... –  J. M. Dec 9 '11 at 13:49

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