# 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 lsqfit currently.

using EquationsOfState.Collections
using EquationsOfState.NonlinearFitting

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> lsqfit(BirchMurnaghan3rd(40, 0.5, 4, 0)(Energy()), volumes, energies)
BirchMurnaghan3rd{Float64}(40.989265727925826, 0.5369258245608038, 4.1786442319302015, -10.842803908298968)

julia> lsqfit(Murnaghan(41, 0.5, 4, 0)(Energy()), volumes, energies)
Murnaghan{Float64}(41.13757924894751, 0.5144967655882123, 3.912386317519504, -10.836794511015869)

julia> lsqfit(PoirierTarantola3rd(41, 0.5, 4, 0)(Energy()), volumes, energies)
PoirierTarantola3rd{Float64}(40.86770643567383, 0.5667729960008705, 4.331688934942696, -10.851486685029547)

julia> lsqfit(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.NonlinearFitting.lsqfitMethod
lsqfit(eos(prop), xdata, ydata; debug = false, kwargs...)

Fit an equation of state using least-squares fitting method (with the Levenberg-Marquardt algorithm).

Arguments

• eos::EquationOfState: a trial equation of state. If it has units, xdata and ydata must also have.
• prop::PhysicalProperty: a PhysicalProperty instance. If Energy, fit $E(V)$; if Pressure, fit $P(V)$; if BulkModulus, fit $B(V)$.
• xdata::AbstractVector: a vector of volumes ($V$), with(out) units.
• ydata::AbstractVector: a vector of energies ($E$), pressures ($P$), or bulk moduli ($B$), with(out) units. It must be consistent with prop.
• debug::Bool=false: if true, then an LsqFit.LsqFitResult is returned, containing estimated Jacobian, residuals, etc.; if false, a fitted EquationOfState is returned. The default value is false.
• kwargs: the rest keyword arguments are the same as that of LsqFit.curve_fit. See its documentation and tutorial.
source

## References

1. A. Otero-De-La-Roza, V. Luaña, Computer Physics Communications. 182, 1708–1720 (2011), doi:10.1016/j.cpc.2011.04.016.