5x5: Polynomial Convergence of 5 Constants - Combined Newton-Raphson Iteration Technique - Standard Approach

The Algorithm of post 9/5/2022 new-algorithm-idea-multi-variate CONVERGES when holding the proton mass and electron mass (thus the $m_p\over m_e$ mass ratio) constant and iterating on these 5 constants:

$$e$$

$$\epsilon0$$

$$h$$

$$c$$

$$R_H$$

The initial values and tolerances need to be corrected and re-run the algorithm. Here is a table of the constants NIST/CODATA and what I used in the Newton_Raphson Iteration NRI method.

Image 1. Constants: NIST/CODATA & NRI

NRI method showing electron charge error getting smaller and smaller
(i.e., more negative on the log Y-scale)

https://phxmarker.blogspot.com/2024/10/showing-99-iterations-on-plots-e-e0-h-c.html 

This link is to the post that shows the color points on the graph with the inputs shown above in Image 1.

The algorithm is described in this post, only modification was to hold the proton and electron mass constant: https://phxmarker.blogspot.com/2022/09/new-algorithm-idea-multi-variate.html

A re-run is needed with the corrected inputs - I didn't use all the digits from NIST/CODATA.

The interesting thing is it appears to converge to a stable solution.  I need to investigate if it is UNIQUE by using different starting values and verfiy if it always converges to these same values.

Another interesting thing is a MAJOR NOTE: One does not need the proton radius solution to perform this NRI method of determining the constants. Simply don't drop the term, the reduced mass approximation (effective masses) when determining the constants. 

There may be some reason the $m_e\over m_p$ term was dropped from the polynomial - as I understand it, that assumption allows one to proceed with an analytic solution to Schrodinger's wave equations for solid-state theory. And that assumption does not have to interfere with determining the coefficients like anyone would do for their boundary value problems.

When you zoom in on the solutions, the last series of iterations, it curves back, very similar to the Lambert W function that is used to solve the iterative Widlar current source circuit.

Zoom in on last iterations of electron charge

https://en.wikipedia.org/wiki/Lambert_W_function

The dream lives! The dream of solving for the constants. 

I'm looking into applying the constants to analyzing the "Island of Stability" - i.e., 

™Resonance & Harmony™ and MetaMaterials†™:

https://en.wikipedia.org/wiki/Island_of_stability 

https://en.wikipedia.org/wiki/Island_of_stability

The Surfer, OM-IV

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Showing the 99 Iterations on the Plots: e, e0, h, c, r_h

5 Constants Solved: Document Write Results On Page for first 99 Iterations

5 Constants e, e0, h, c, r_h - Hold Electron Mass and Proton Mass, Iterate 5 Other Constants; See Console for Error (~=1e-9) & Constants

1st Iteration: Electron Mass(m_e), Charge(e), Permittivity (e0), Planck's Constant (h), Speed of Light (c), Rydberg Constant (Rh) - Plotly Polts with CODAT/NIST and Minimum Error 1st Iteration Values in Console

Plotly: 6 Constants - Electron mass, charge; e0-permitivity; h-Plank Constant; c-speed of light;r_h -Rydberg Constant;

Electron Mass: Newton-Raphson

Electron Mass Polynomial Error Plot HTML/JS

If you click on the file link and download the file and then open it in a browser, it plots the error function vs. electron mass as the error goes from negative to positive as it passes through zero:

https://drive.google.com/file/d/12m6kXmHgOkZWCWuiJojzySNCyl9jR_Xi/view

This is using the nominal NIST/CODATA values for the other constants in the polynomial:

Y-axis is error function polynomial, X-axis is the electron mass: 

Axis labeling is crude, looking into plotly.js and d3.js or some other js library.

Plotly.js for Electron Mass Error (all other constants at default NIST/CODATA values):

Red dot is default NIST/CODATA electron mass value
(Plotly is much easier to read!)

Looking at LOG error (log(abs(error)) plotting with fine steps allows one to visually see the root for both electron mass and electron charge (assuming all other constants are constant and correct):

HTML/Javascript for electron mass and charge

Electron Mass

Electron Charge

e0 (permittivity);  HTML/JS added e0

HTML/JS for electron, mass, charge, e0, h, c, & r_h:

h, Plankck's constant

c, speed of light

Rh, r_h, Rydberg constant

r_p and alpha cannot be fine tuned until after the other constant are corrected enough to correct for the initial errors. Then, limited regions of stability around nominal values can be checked. Ongoing.

Added the Newton-Raphson method to the electron mass error vs. electron mass code and plotted both the NIST/CODATA value in RED/orange and my value in GREEN* from the polynomial to an error of less than 0.00000000001e-31 kg:

Newton-Raphson code for electron mass
(shows the new electron mass used for the GREEN point on plot)
(the new mass is calculated using the polynomial and Newton_Raphson method)

Some refinements are needed to tolerance as it converges in like 1-2 iterations due to linear nature of error of interval, thus it is simple to calculate the electron mass required to get a value of a specified tolerance (near zero).

The basic pieces of a Javascript approach for minimizing error and solving for the values of the constants that give minimum error have been developed... Time to implement some of the algorithms mentioned previosly in the blog, like the one to iterate to find the value of each constant to give minimum error and iterate through and check for convergence...

*GREEN MEANS GO!

The Surfer, OM-IV

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Universe* as a Unity Gain (?) Feedback System

Using 

${S\over O}=1$ ; justification: energy and mass are conserved, lossless system

${S\over O}={A\over{1+A\beta}}$

$1={R_\infty\over R_H}-{m_e\over m_p}$ & (using $m_e\over m_p$ instead of ${{\pi r_pR_H}\over \alpha^2}$)

$R_\infty={m_ee^4\over 8{\epsilon_0}^2h^3c}$

(will fill in math steps later, refer to The Universe is a Feedback System)

$A_v={S\over O}={A\over{1+A\beta}}$

$A_v=1$

$1+A\beta=A$

$1=A-A\beta$

$A={m_ee^4\over 8{\epsilon_0}^2h^3cR_H}$

$\beta={{8{\epsilon_0}^2h^3cR_H}\over{m_pe^4}}$

A, open loop forward gain:

https://www.google.com/search?q=%28m_e*%28elementary+charge%29%5E4%29%2F%288*e_0%5E2*h%5E3*c*%28Rydberg+constant%29%29%3D

This is approximately equal to 1 due to artificial tuning of constants

$\beta$, reverse gain:

https://www.google.com/search?q=%288*e_0%5E2*h%5E3*c*%28Rydberg+constant%29%29%2F%28m_p*%28elementary+charge%29%5E4%29%3D

* Universe of a Hydrogen atom

Moar later.

These values of $A$ and $\beta$ are reminders that this equation and this problem is rooted in the physical single hydrogen atom - a dynamic between a proton and electron.  This mathematical tool of transforming the equation to an analogous "system", one can pull out key features (with good math tools/models). The dynamic between the proton and electron captured in a simple diagram. The hydrogen atom, 1H, has a full analytical solution (full Wave Equation or combined Schrödinger wave equation for both proton and electron or however it's done).

The Surfer, OM-IV

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Idea of Visualizing Multi-Dimensional Data Using #Tempest

https://cds.cern.ch/record/343250/files/p189.pdf

Some ideas for multidimensional data viewing.

The Surfer, OM-IV

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BTC Value of 4th Order 8D Polynomial

BTC Value of 4th Order 7D Polynomial

Donate your spare bitcoin today if this polynomial has any value to you!

The Surfer, OM-IV

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