How to get fine fitting of B-M equation with high nonliearity?

I have tried to fit Birch's equation like below:

but I could not get fine fit..

This is my code.

clc
clear all

data=[208.75 -171.04204447
    227.72 -195.21101066
    247.80 -212.78884133
    269.03 -224.97316547
    291.44 -232.75405432
    315.06 -236.95077221
    339.92 -238.24362657
    366.05 -237.19727153
    393.50 -234.28678793
    422.28 -229.90809458
    452.43 -224.39439701
    483.98 -218.02503783
    516.97 -211.03510699
    ];



V=data(:,1);
E=data(:,2);
ft=fittype( 'E0+9/16*(V0*B0)*Bp*((V0/V)^(2/3)-1)^3+(((V0/V)^(2/3)-1)^2)*(6-4*((V0/V)^(2/3)))', 'independent', {'V'}, 'dependent', {'E'}','coefficients',{'E0','V0','B0','Bp'})
[fitresult, gof] = fit( V, E, ft,'StartPoint',[-1 1 4 1])%, opts );
figure( 'Name', ' B-M' );
h = plot( fitresult,  V, E );
legend( h, 'Energy vs Volume', ' B-M Equation of State', 'Location', 'NorthEast' );
xlabel('Volume (A^3)');
ylabel('Energy (eV)');
grid on

Please check my code for good fitting and wether there is error for a fit.

please someone give me any comments.

Best regards

[Alex Sha]

Hi, L world, the function code you used is:

ft(E0,V0,B0,Bp,V) = E0+9/16*(V0*B0)*Bp*((V0/V)^(2/3)-1)^3+(((V0/V)^(2/3)
                    -1)^2)*(6-4*((V0/V)^(2/3)))

However, according the function picture your poeted, the right one should be:

ft(E0,V0,B0,Bp,V) = E0+9/16*(V0*B0)*(Bp*((V0/V)^(2/3)-1)^3+(((V0/V)^(2/3) 
                    -1)^2)*(6-4*((V0/V)^(2/3))))

The result will then be perfect like below:

Sum Squared Error (SSE): 0.0510334009736824
Root of Mean Square Error (RMSE): 0.0626549776852949
Correlation Coef. (R): 0.999994415208434
R-Square: 0.999988830448059

Parameter	Best Estimate    
---------	-------------    
e0       	-238.182608653109
v0       	340.622561557616 
b0       	1.19504392025662 
bp       	3.92181712158994                     

or:

Sum Squared Error (SSE): 0.0510334009320099
Root of Mean Square Error (RMSE): 0.0626549776597137
Correlation Coef. (R): 0.999994415208439
R-Square: 0.999988830448068

Parameter	Best Estimate    
---------	-------------    
e0       	44157.2760056618 
v0       	4.44032780297811 
b0       	-29880.4231743995
bp       	5.41151658492575

The latter is slightly better than the former。

[L world]

Dear Alex Sha

Thanks for your comment a lot.

I fixed the equation! thanks to you. :)

Best regard

[Alex Sha]

Hi, L world, the more possible reason is that the initial start-values your given are not very suitable, of course, it is always hard jobs for guessing reasonable and correct initial values。

[Star Strider]

The originally posted code:

clc
clear all

data=[208.75 -171.04204447
    227.72 -195.21101066
    247.80 -212.78884133
    269.03 -224.97316547
    291.44 -232.75405432
    315.06 -236.95077221
    339.92 -238.24362657
    366.05 -237.19727153
    393.50 -234.28678793
    422.28 -229.90809458
    452.43 -224.39439701
    483.98 -218.02503783
    516.97 -211.03510699
    ];



V=data(:,1);
E=data(:,2);
ft=fittype( 'E0+9/16*(V0*B0)*Bp*((V0/V)^(2/3)-1)^3+(((V0/V)^(2/3)-1)^2)*(6-4*((V0/V)^(2/3)))', 'independent', {'V'}, 'dependent', {'E'}','coefficients',{'E0','V0','B0','Bp'})
[fitresult, gof] = fit( V, E, ft,'StartPoint',[-1 1 4 1])%, opts );
figure( 'Name', ' B-M' );
h = plot( fitresult,  V, E );
legend( h, 'Energy vs Volume', ' B-M Equation of State', 'Location', 'NorthEast' );
xlabel('Volume (A^3)');
ylabel('Energy (eV)');
grid on

I had that open intending to give it ago with ga (that I still may do) so I have it available.

.

[Rena Berman]

(Answers Dev) Restored edit