Fitting a Gaussian absorption line profile

PHYSICS · Subjective · Fermi University Physics · Scientific Practice & Experimental Methods · Computational Physics · Scientific programming for physicists

Scientific Computing for Physicists, Ch Minimisation with guarantees, with constraints and with randomness

Problem

Fitting a Gaussian absorption line profile, the Gauss-Newton normal-equation matrix at some iterate is JTWJ=(100202050),JTWr=(−30−10),J^{\mathsf T}WJ = \begin{pmatrix}100 & 20\\ 20 & 50\end{pmatrix}, \qquad J^{\mathsf T}W\mathbf r = \begin{pmatrix}-30\\ -10\end{pmatrix}, where the parameters have been made dimensionless by scaling. Levenberg-Marquardt damping with μ=1.5\mu = 1.5 replaces the Gauss-Newton normal equations by (JTWJ+μ diag(JTWJ))δ=−JTWr.\left(J^{\mathsf T}WJ + \mu\,\mathrm{diag}(J^{\mathsf T}WJ)\right)\boldsymbol\delta = -J^{\mathsf T}W\mathbf r. Find the parameter update δ=(δ1,δ2)\boldsymbol\delta = (\delta_1,\delta_2).

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