How to Improve the Accuracy of Transcendental Equation Solutions: What Have I Missed?

Increasing Accuracy of Solution of Transcendental Equation

This issue involves estimating the parameters a0, y0, and z0 from a set of measurements, with the goal of improving the accuracy of the solution.

Question 1: How to further improve accuracy of the Solution?

• Weighting the deviations: Consider weighting the deviations according to the angular distance from 0 degrees. However, it's unclear whether this would significantly improve accuracy, as a(t) may not necessarily include 0 degrees.
• Increasing the Angular range: The angular range cannot be increased due to design limitations.
• Increasing the number of points: The accuracy improves with more points up to around 100, after which it becomes unstable for some radii.
• Using different recursion: Recursions above 6 have minimal impact and increasing accuracy beyond this is challenging.

Question 2: Is there something I have Missed?

• Measurement accuracy: Ensure the input measurements are as accurate as possible.
• Calibration tube accuracy: Use a calibration tube with precisely known radius to ensure accurate calibration.
• Machine vibrations and eccentricity: These factors can affect measurements and should be accounted for if possible.

Additional Considerations:

• Simulation results are generally more accurate with z0 measured. Increasing simulation accuracy beyond 6 recursions does not improve results, as the real input data will also have limited accuracy.
• It's recommended to analyze the inferred r0 and y values as functions of each other to gain insights into the limitations of the method.

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