How to fit to an infinite series function?

qt is a dependent variable; t is an independent variable; qe B are undetermined parameters.

syms n t;
x=[0 5 10 15 20 30 45 60 75 90 105 120];
y=[0 3.87 4.62 4.98 5.21 5.40 5.45 5.50 5.51 5.52 5.54 5.53];
plot(x,y,'bo');
hold on
beta0=[39,0.002];
fun=@(beta,xdata) beta(1)*(1-6/(pi^2)*symsum((1/n^2)*exp(-beta(2)*(n^2)*t),n,1,inf))
betafit = nlinfit(x,y,fun,beta0);
plot(x,y,fun,beta0)

However, it does not work well. How to do it in MATLAB? Help me. Many thanks.

Answer 1

[Vladimir Sovkov]

An iterative solution instead of the symbolic one can be more productive this case, like this one

x=[0 5 10 15 20 30 45 60 75 90 105 120];
y=[0 3.87 4.62 4.98 5.21 5.40 5.45 5.50 5.51 5.52 5.54 5.53];
plot(x,y,'bo');
hold on
pause(0.1);
beta0=[39,0.002];

% syms n t
% fun=@(beta,t) beta(1)*(1-6/(pi^2)*symsum((1./n.^2).*exp(-beta(2)*(n.^2).*t),n,1,Inf));
% betafit = nlinfit(x,y,fun,beta0);

beta1=beta0;
delta = 1e-8; % desired objective accuracy
R0=Inf; % initial objective function

for K=1:10000
    fun=@(beta,t) beta(1)*(1-6/(pi^2)*sum((1./(1:K)'.^2).*exp(-beta(2)*((1:K)'.^2).*t),1));
    [betafit,R] = nlinfit(x,y,fun,beta1);
    R = sum(R.^2);
    if abs(R0-R)<delta
        break;
    end
    beta1=betafit;
    R0 = R;
end

plot(x,fun(betafit,x),'.-r');

xlabel('x');
ylabel('y');
legend('experiment','model');
title(strcat('\beta=[',num2str(betafit),'];----stopped at--','K=',num2str(K)));

[Qili Hu]

Dear Sovkov,

Thanks for your solution. After running, it does not work. So can you help me again?

Best wishes.

Yours sincerely,

Qili Hu

[Vladimir Sovkov]

What messages do you get? I have rerun this exact code with my Matlab 2019b at Windows-10 system, and it has worked fairly well

[Qili Hu]

I want to get this figure as you give. But I does not obtain it after running. I does not know why. I run the following code with my Matlab 2016a at Windows-10 system, and it does not work.

x=[0 5 10 15 20 30 45 60 75 90 105 120];
y=[0 3.87 4.62 4.98 5.21 5.40 5.45 5.50 5.51 5.52 5.54 5.53];
plot(x,y,'bo');
hold on
pause(0.1);
beta0=[39,0.002];
beta1=beta0;
delta = 1e-8;
R0=Inf;
for K=1:10000
       fun=@(beta,t) beta(1)*(1-6/(pi^2)*sum((1./(1:K)'.^2).*exp(-beta(2)*((1:K)'.^2).*t),1));
       [betafit,R] = nlinfit(x,y,fun,beta1);
       R = sum(R.^2);
       if abs(R0-R)<delta
             break;
       end
       beta1=betafit;
       R0 = R;
end
plot(x,fun(betafit,x),'.-r');
xlabel('x');
ylabel('y');
legend('experiment','model');
title(strcat('\beta=[',num2str(betafit),'];----stopped at--','K=',num2str(K)));

[Vladimir Sovkov]

I have tried Matlab 2017a. It works the same way as Matlab 2019b, no problem. Maybe, you should update your Matlab installation to a newer version.

When I am in St. Petersburg University, I use Matlab 2019b; when I am in Shanxi University, I use Matlab 2017a. Both work fairly well.

The other way could be replacing the "sum" construction with an equivalent loop but this would slow down the computation significantly.

[Qili Hu]

Dear Sovkov,

Thank you very much. I have solved this problem. Thanks for your help again.