Nonlinear fitting
From Ref. 1,
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.
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.
Usage
We provide API nonlinfit
currently.
using EquationsOfStateOfSolids
using EquationsOfStateOfSolids.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,
];
nonlinfit(EnergyEquation(BirchMurnaghan3rd(40, 0.5, 4, 0)), volumes, energies)
# BirchMurnaghan3rd{Float64}
# v0 = 40.98926572792904
# b0 = 0.5369258245610551
# b′0 = 4.178644231924164
# e0 = -10.84280390829923
nonlinfit(EnergyEquation(Murnaghan(41, 0.5, 4, 0)), volumes, energies)
# Murnaghan1st{Float64}
# v0 = 41.137579246216546
# b0 = 0.5144967654207855
# b′0 = 3.9123863218932553
# e0 = -10.836794510856276
nonlinfit(EnergyEquation(PoirierTarantola3rd(41, 0.5, 4, 0)), volumes, energies)
# PoirierTarantola3rd{Float64}
# v0 = 40.86770643566912
# b0 = 0.5667729960007934
# b′0 = 4.331688934950856
# e0 = -10.851486685029291
nonlinfit(EnergyEquation(Vinet(41, 0.5, 4, 0)), volumes, energies)
# Vinet{Float64}
# v0 = 40.91687567401044
# b0 = 0.5493839427843428
# b′0 = 4.305192949379345
# e0 = -10.846160810983534
Then 4 different equations of state will be fitted.
Public interfaces
EquationsOfStateOfSolids.Fitting.linfit
— Functionlinfit(eos::EnergyEquation{<:FiniteStrainParameters}, volumes, energies; kwargs...)
Fit an equation of state using linear algorithms.
Arguments
maxiter::Integer=1000
: .conv_thr::AbstractFloat=1e-12
: .root_thr::AbstractFloat=1e-20
: .verbose::Bool=false
: .
If you want to fit with BigFloat
data, you need to install GenericSVD.jl
and using GenericSVD
before fittting!
EquationsOfStateOfSolids.Fitting.nonlinfit
— Functionnonlinfit(eos::EquationOfStateOfSolids, xs, ys; kwargs...)
Fit an equation of state using nonlinear algorithms.
Arguments
xtol::AbstractFloat=1e-16
: .gtol::AbstractFloat=1e-16
: .maxiter::Integer=1000
: .min_step_quality::AbstractFloat=1e-16
: .good_step_quality::AbstractFloat=0.75
: .verbose::Bool=false
: .