# 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.

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Real*16 (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 real*16 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 real*8 and real*16 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 *.bi**m** 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 *.ex**f** files to *.ex**e** to run them.