How to fit to an infinite series function when the independent variable contains an undetermined parameter?

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

Procedure code is as follows.

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.^(-h)),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.^(-h)),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)));

Answer 1

[Alan Stevens]

fminsearch seems to do ok:

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];

b0 = [5, 1, -1]; % b = [qe, B, h]
b = fminsearch(@(b) fn(b,x,y), b0);
disp(b)

t = 0:120;
qt = qt_fn(t,b);

plot(x,y,'bo',t,qt), grid
xlabel('t'),ylabel('qt')

function F = fn(b,x,y)

        q = qt_fn(x, b);
        F = norm(y - q);
         
end

function qt = qt_fn(t, b)

         N = 100;  
         term = 0;
         qt = 1;
         err = 1;
         n = 1;
         while (err>10^-8) & (n<=N)
           oldterm = term;
           term = -(6/pi^2)*exp(-(b(2)*n^2.*t.^(-b(3))));
           qt = term + qt;
           n = n+1;
           err = abs(term-oldterm);
         end
         qt = b(1)*qt;

end