A swept resonance curve y(x) = A ^2 (x-x_0)^2+ ^2 is fitted to forty measured points

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 swept resonance curve

y(x) = \frac{A\gamma^2}{(x-x_0)^2+\gamma^2}

is fitted to forty measured points, with

A

of order

200

and

\gamma

of order

0.05

in the units of the problem. Gradient descent, using a central-difference gradient, takes

6200

iterations to bring

\chi^2

within

10^{-6}

of its minimum from a reasonable starting guess.Someone argues: 'The finite-difference gradient must be the problem. I used the same step

h=10^{-3}

to differentiate

\chi^2

with respect to both

A

and

\gamma

, but

A

is of order

200

while

\gamma

is of order

0.05

, so the same

h

cannot resolve both derivatives to the same relative accuracy. If I shrink

h

10^{-6}

for every parameter, the gradient will be exact and the descent will converge in a handful of iterations, just as the Gauss-Newton step did for the counting data.' ((a)) Find the error in this reasoning. ((b)) State, in one sentence each, what actually causes the

6200

iterations, and what would cure it without touching

h

Browse the problem library