Nonlinear fitting
From Ref. 1,
The equations of state depend nonlinearly of a collection of parameters, $E_0$, $V_0$, $B_0$, $B_0'$, ..., that represent physical properties of the solid at equilibrium and can, in principle, be obtained expermentally by independent methods. The use of a given analytical EOS may have significant influence on the results obtained, particularly because the parameters are far from being independent. The number of parameters has to be considered in comparing the goodness of fit of functional forms with different analytical flexibility. The possibility of using too many parameters, beyond what is physically justified by the information contained in the experimental data, is a serious aspect that deserves consideration.
In EquationsOfState, the nonlinear fitting is currently implemented by LsqFit, a small library that provides basic least-squares fitting in pure Julia. It only utilizes the Levenberg-Marquardt algorithm for non-linear fitting. See its documentation for more information.
Usage
We provide API nonlinfit currently.
using EquationsOfState.Collections
using EquationsOfState.Fitting
volumes = [
25.987454833,
26.9045702104,
27.8430241908,
28.8029649591,
29.7848370694,
30.7887887064,
31.814968055,
32.8638196693,
33.9353435494,
35.0299842495,
36.1477417695,
37.2892088485,
38.4543854865,
39.6437162376,
40.857201102,
42.095136449,
43.3579668329,
44.6456922537,
45.9587572656,
47.2973100535,
48.6614988019,
50.0517680652,
51.4682660281,
52.9112890601,
54.3808371612,
55.8775030703,
57.4014349722,
58.9526328669,
];
energies = [
-7.63622156576,
-8.16831294894,
-8.63871612686,
-9.05181213218,
-9.41170988374,
-9.72238224345,
-9.98744832526,
-10.210309552,
-10.3943401353,
-10.5427238068,
-10.6584266073,
-10.7442240979,
-10.8027285713,
-10.8363890521,
-10.8474912964,
-10.838157792,
-10.8103477586,
-10.7659387815,
-10.7066179666,
-10.6339907853,
-10.5495538639,
-10.4546677714,
-10.3506386542,
-10.2386366017,
-10.1197772808,
-9.99504030111,
-9.86535084973,
-9.73155247952,
];
julia> nonlinfit(BirchMurnaghan3rd(40, 0.5, 4, 0)(Energy()), volumes, energies)
BirchMurnaghan3rd{Float64}(40.989265727925826, 0.5369258245608038, 4.1786442319302015, -10.842803908298968)
julia> nonlinfit(Murnaghan(41, 0.5, 4, 0)(Energy()), volumes, energies)
Murnaghan{Float64}(41.13757924894751, 0.5144967655882123, 3.912386317519504, -10.836794511015869)
julia> nonlinfit(PoirierTarantola3rd(41, 0.5, 4, 0)(Energy()), volumes, energies)
PoirierTarantola3rd{Float64}(40.86770643567383, 0.5667729960008705, 4.331688934942696, -10.851486685029547)
julia> nonlinfit(Vinet(41, 0.5, 4, 0)(Energy()), volumes, energies)
Vinet{Float64}(40.91687567368755, 0.5493839427734198, 4.30519294991197, -10.846160810968053)Then 4 different equations of state will be fitted.
Public interfaces
EquationsOfState.Fitting.nonlinfit — Methodnonlinfit(f, volumes, ydata; kwargs...)Fit an equation of state using least-squares fitting method (with the Levenberg-Marquardt algorithm).
If eos, volumes and ydata are all unitless, volumes must have the same unit as eos.v0, and ydata, if are energies, must have the same unit as eos.e0. The eos parameter b0 must have the unit of e0 / v0, and b′′0 must have the unit of v0 / e0, etc. Use with care. Better to use the Unitful version.
Arguments
eos::EOSParameters: a trial equation of state. If it has units,volumesandydatamust also have.prop::EquationOfState: anEquationOfStateinstance. IfEnergy, fit $E(V)$; ifPressure, fit $P(V)$; ifBulkModulus, fit $B(V)$.volumes: an array of volumes ($V$), with(out) units.ydata: an array of energies ($E$), pressures ($P$), or bulk moduli ($B$), with(out) units. It must be consistent withprop.kwargs: the rest keyword arguments are the same as that ofLsqFit.curve_fit. See its documentation and tutorial.