A driven oscillator's steady-state amplitude is measured at twenty-five frequencies…
PHYSICS · Multi-part · Fermi University Physics · Scientific Practice & Experimental Methods · Mathematical Methods for Physics · Data analysis and model fitting in the laboratory
Computational Physics, Ch Finding the best parameters: minimisation and fitting
Problem
A driven oscillator's steady-state amplitude A is measured at twenty-five frequencies f near its resonance and fitted to the Lorentzian modelA(f)=(f−f0)2+γ2A0γ2where A0 is a known calibration constant, f0 is the resonant frequency and γ is the half-width at half maximum. Every measured amplitude carries the same uncertainty σA. Fitting is done by gradient descent on χ2(f0,γ) starting from (f0,γ)=(950Hz,2Hz).Someone reports the following working."At η=1×10−6, twenty thousand iterations of gradient descent moved (f0,γ) by less than one part in 103. At η=3×10−6 the same run diverged, with χ2 reaching 1015 within twenty steps. Since the step length that works is tiny and the one three times larger blows up, the finite-difference gradient itself must be unreliable, so it was recomputed with forward differences and a larger step h=0.1Hz instead of central differences. The crawl did not improve, which shows the amplitude data are simply too noisy to fit well. The run at η=1×10−6 was stopped there and the resonance width is reported as γ=6.1Hz."
((a)) Identify the two errors of reasoning in this working: what the crawl-then-divergence pattern actually indicates, and why changing the finite-difference scheme was the wrong response to it.
((b)) State, without recomputing the fit, what a correctly reported value of γ would need to include that this one does not.