clc;
clear All;
%To search for the minimum of an arbitrary function, MATLAB pro- MATLAB provides the function fminsearch. 
%To use it, you have to write an additional short function calculating the value of the function to be minimized. 
%If you would want,e. g., to ?t a Lorentzian y = a1/((x - a2)^2 + a3) to a data set Xi, Yi, 
%you should define in MATLAB a function resulting in the sum of the squared residuals
%function d = devsum(a)
%global X Y

%The start values for the parameters (a1, a2, a3) could of course be chosen in a better way when the data set is known. 
%An example for the ?t with a Lorentzian is shown in Fig. 2, the data set there was de?ned by

% a(1)=20, a(2)=20, a(3)=20
X = linspace(0,100,200);
%Y = 20./( 1 + ( (X-30)./20 ).^2+0.08*randn(size(X))); % wikipeida definition

b1 = 12;
b2 = 100000;
b3 = 50;
b4 = 10;
b5 = 1.5;
b=[b1,b2,b3,b4,b5];

%Y = 1/(sqrt(2*pi)*b1)*exp(-abs(X-b3).^b2/(2*b1.^b2) + 0.08*randn(size(X)));
Y = b(1)+b(2)*exp( -0.5*(abs(X-b(3))/(b(4))).^b(5) + 0.08*randn(size(X))); % super gaussian 
plot(X',Y','.');
hold on;
%dis = 1/(sqrt(2*pi)*b1)*exp(-abs(X-b3).^b2/(2*b1.^b2));
dis = b(1)+b(2)*exp( -0.5*(abs(X-b(3))/(b(4))).^b(5));
plot(X',dis','k');
hold on;

% a(1) = base
% a(2) = A; amplitude
% a(3) = x0; center
% a(4) = D; sigma_0
% a(5) = N; 

a1 = b1*0.8;
a2 = b2*1.4;
a3 = b3*1.2;
a4 = b4*1.2;
a5 = b5*1.2;

%SupGau = a(1)+a(2)*exp(-abs(X-a(3))/(a(4))).^a(5); % super gaussian 

a0 = [a1,a2,a3,a4,a5];

opts = optimset('TolX',1e-4,'MaxFunEvals',10000,'MaxIter',10000,'Display','on');
[fitpara,fval,flag,err] = fminsearch(@SupGau_devsum,a0,opts,X,Y);

display('sigma_0:');
fitpara(1)
display('N');
fitpara(2)
display('x0');
fitpara(3)

%y_fit = 1/(sqrt(2*pi)*fitpara(1))*exp(-abs(X-fitpara(3)).^fitpara(2)/(2*fitpara(1).^fitpara(2)));
y_fit = fitpara(1)+ fitpara(2)*exp( -0.5*(abs(X-fitpara(3))/(fitpara(4))).^fitpara(5));

plot(X',y_fit','r')

%The function fminsearch can be tuned by additional options, one can demand a certain accuracy, restrict the number of optimization steps etc. 
%For a detailed description consult Help