Fitting

• Fitting
• • Nonlinear fitting
  • Linear fitting
  • Usage
  • Public interfaces
  • References

Nonlinear fitting

The equations of state depend nonlinearly on 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 experimentally 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.^[1]

In EquationsOfStateOfSolids, 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.

Linear fitting

The linear fitting

Usage

julia> using EquationsOfStateOfSolids
julia> using EquationsOfStateOfSolids.Fitting
julia> 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,
       ];
julia> 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(volumes, energies, BirchMurnaghan3rd(40, 0.5, 4))BirchMurnaghan3rd{Float64}
 v0 = 40.9892657279331
 b0 = 0.5369258245616882
 b′0 = 4.178644231913714
 e0 = -10.842803908299864
julia> nonlinfit(volumes, energies, Murnaghan(41, 0.5, 4))ERROR: ArgumentError: unknown number of arguments 3.
julia> nonlinfit(volumes, energies, PoirierTarantola3rd(41, 0.5, 4))PoirierTarantola3rd{Float64}
 v0 = 40.86770643566527
 b0 = 0.5667729960008847
 b′0 = 4.3316889349555465
 e0 = -10.851486685029235
julia> nonlinfit(volumes, energies, Vinet(41, 0.5, 4))Vinet{Float64}
 v0 = 40.91687567400595
 b0 = 0.5493839427841869
 b′0 = 4.305192949386885
 e0 = -10.84616081098332

Then 4 different equations of state will be fitted.

They just work as well with units:

julia> using UnitfulERROR: ArgumentError: Package Unitful not found in current path.
- Run `import Pkg; Pkg.add("Unitful")` to install the Unitful package.
julia> volumes = volumes * u"angstrom^3"ERROR: LoadError: UndefVarError: @u_str not defined
in expression starting at REPL[2]:1
julia> energies = energies * u"eV"ERROR: LoadError: UndefVarError: @u_str not defined
in expression starting at REPL[3]:1
julia> nonlinfit(volumes, energies, BirchMurnaghan3rd(40u"angstrom^3", 1u"GPa", 4))ERROR: LoadError: UndefVarError: @u_str not defined
in expression starting at REPL[4]:1

Public interfaces

fit(volumes, energies, initial_params::FiniteStrainParameters, LinearFitting(); kwargs...)

fit(xs, ys, initial_params::Parameters, NonLinearFitting(); kwargs...)

References

1. A. Otero-De-La-Roza, V. Luaña, Comput. Phys. Commun. 182, 1708–1720 (2011).