Exact Inverse Polynomials

Exact Inverse Polynomials

             Welcome to PolyGen.exe & Polyize.exe

A Free, Powerful, Even-Order Polynomial Coefficient Generator: exact by default to: 8th order & 2,000+ coefficients & real (non-exact) by custom configuration to: 20th order & 2,000+ coefficients

The enabling math & tech:
Computing adjoints and determinants involve no division; hence any full-rank matrix with integer coefficients has an inverse with exact rational coefficients.
Real16 (128-bit) variables have 31 significant digits; which supports tracking integer values up to 1 octillion. Adjoint integers grow faster than the determinant integers. My (this) cookbook reverse-engineers the key 2,4,6,& 8th order integer divisors which exactify the real16 inversion results when rounded to the nearest integer. [PolyGen.f95 – source code lines #337-#340.]

The computed coefficients are decoupled from one another, so you can compute just one, or a few, if you so choose. The central accomplishment of matrix inversion itself is completely decoupled quantification wherein outputs ‘reverse engineer’ inputs. Consider that: `Higher powers` existing in exact -and- predictive harmony resembles a breath of fresh air in the world of social thought.

Measurement noise(s) will degrade higher power coefficients. Fortunately real8 and real16 math models bypass the noise problem.

Coefficients will vary wildly if you sample on a domain that has discontinuous derivatives, but they’ll still replicate their source data. Hence arbitrary datasets can be encoded as multi-variable polynomials.

The programs run within Windows XP & Windows 10; The content is posted in http://ftp.setterholm.com/ExactInvPoly/ and is free… e.g. post copyright. Change the .exf files to .exe files. The two .exe files and two .ini files drive the environment; the .zip file has far more content. Spreadsheets tend to mangle the exact coefficients, so use a text editor to view the .csv file(s).

Elsewhere on my website, in source file Tweak-Engine.f95 dated 2018.10.09, lines #895 – #903, you’ll see two examples which were my first intentional use of exact inverse polynomials.

2021.09.11: MS Windows 10 does not allow *.bin (binary) files to be written. The version 1.1 change writes a *.bim file instead (with the same binary information content), which runs. http://ftp.setterholm.com/ExactInvPoly/Use-Windows10-Vsn11 A .zip file of this subdirectory’s contents is also provided. Rename the two *.exf files to *.exe to run them.

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