lsqcurvefit not working properly

Hi all, I have multiple sets of data that are to be fitted by my fitting function. Since these data sets are kind of related, what I did was to use an inital guess to find the parameter values for the first dataset, and then use the parameter values as the guess for the second dataset so on and forth. However, the fitting was really bad. May I ask if there is another way around this kind of problem involving fitting of multiple datasets?

tic
%% Preparation 
clear; clc

data = importdata("Experimental data\Transient Absorption\FCPIB-293K-2.5mW-400nm-Jan072021 -ibg -bg -chirp.csv"); % insert file path within parenthesis 


%% Preamble 

% Fundamental constants
h = 4.0135667696*10^-15; % units: eV/ Hz
c = 3*10^8; % SI units
kB = 8.617333268*10^-5; % units: eV/ K

% Data
wavelength = data(1:end, 1); % contains all of the wavelengths
delay_t = data(1, 1:end); % conatains all of the delay times
E = (h*c)./(wavelength*10^-9); % contains all of the probe energies

% Range of data needed
Range_E = E>=1.5 & E<=2.2; 
Range_W = wavelength >= (h*c)/(2.2*10^-9) & wavelength <= (h*c)/(1.5*10^-9);
Range_T = delay_t>=0.5 & delay_t<=1000;

E_p = E(Range_E); % probe energies for scatterplot
w_p = wavelength(Range_W); 
delay_T = delay_t(Range_T);

Range_Efit = E_p>=1.62 & E_p<=max(E_p); 
Range_Wfit = w_p >= (h*c)/(2.2*10^-9) & w_p <= (h*c)/(1.62*10^-9);
E_fit = E_p(Range_Efit); % probe energies for fitting 
W_fit = w_p(Range_Wfit);
t_min = min(delay_T);
t_max = max(delay_T);
w_min = min(w_p);
w_max = max(w_p);
[row1, col1] = find(data == w_min);
[row2, col2] = find(data == w_max);
[row3, col3] = find(data == t_min);
[row4, col4] = find(data == t_max);


% New cleaned up data
data_new = data(row1:row2, col3:col4);% new data containing required delta A

% for n = 1:length(delay_T)
%     delta_Abs(:,n) = -1*data_new(:,n);
%     delta_Abs_norm(:,n) = delta_Abs(:,n)/max(delta_Abs(:,n));
%     delta_Abs_fit(:,n) = data_new(Range_Wfit,n);
%     delta_Abs_norm_fit(:,n) = delta_Abs_fit(:,n)/max(delta_Abs_fit(:, n));
%     % plot command
%     plot(E_p, delta_Abs_norm)
%     xlabel('Probe Energy (eV)')
%     ylabel('Normalised \Delta A (a.u.)')
%     legend('Experimental Data')
% end

% Fitting function: Maxwell-Boltzmann distribution 
function F = MB(y, e_fit)
data = importdata("Experimental data\Transient Absorption\FCPIB-293K-2.5mW-400nm-Jan072021 -ibg -bg -chirp.csv"); 
h = 4.0135667696*10^-15; 
c = 3*10^8; 
kB = 8.617333268*10^-5; 
delay_t = data(1, :);
Range_T = delay_t>=0.5 & delay_t<=1000;
delay_T = delay_t(Range_T);
wavelength = data(1:end, 1); 
E = (h*c)./(wavelength*10^-9);
Range_E = E>=1.5 & E<=2.2; 
E_p = E(Range_E);
Range_Efit = E_p>=1.62 & E_p<=max(E_p); 
E_fit = E_p(Range_Efit);
    for i = 1:length(E_fit)
        E_fit = e_fit(i);
        F(i) = y(1).*exp(-(E_fit./(kB.*y(2)))) + y(3);
    end
    F = F(:);
end 


% For loop to create required datasets for plotting

for n = 2:length(delay_T)
    delta_Abs(:,1) = -1*data_new(:,1);
    delta_Abs_norm(:,1) = delta_Abs(:,1)./max(abs(delta_Abs(:,1)));
    delta_Abs_fit(:,1) = data_new(Range_Wfit,1);
    delta_Abs_norm_fit(:,1) = delta_Abs_norm(Range_Wfit);

    delta_Abs(:,n) = -1*data_new(:,n);
    delta_Abs_norm(:,n) = delta_Abs(:,n)./max(abs(delta_Abs(:,n)));
    delta_Abs_fit(:,n) = data_new(Range_Wfit,n);
    delta_Abs_norm_fit(:,n) = delta_Abs_norm(Range_Wfit);

    % lsqcurvefit 
    optim_lsq = optimoptions('lsqcurvefit', 'Algorithm', 'levenberg-marquardt', 'MaxFunctionEvaluations',10^20, 'MaxIterations', 10^20, 'FunctionTolerance',10^-20, 'StepTolerance', 10^-20);
    lb = [0, 293, -1]; ub = [Inf, 6000, 1];
    lb1 = [0, 293, -1]; ub1 = [Inf, 2000, 1];
    lb2 = [0, 293, -1]; ub2 = [Inf, 1200, 1];
    lb3 = [0, 293, -1]; ub3 = [Inf, 1000, 1];
    lb4 = [0, 293, -1]; ub4 = [Inf, 750, 1];
    y1 = [2*10^7, 1000, 0.5];
    y2 = [2*10^7, 800, 0.5];
    y3 = [2*10^10, 800, 0.5];
    y4 = [2*10^11, 600, 0.5];
    y5 = [2*10^14, 600, 0.5];
    y(1,:) = lsqcurvefit(@MB, y1, E_fit, delta_Abs_norm_fit(:, 1), lb1, ub1, optim_lsq);
    y(n,:) = lsqcurvefit(@MB, y(n-1, :), E_fit, delta_Abs_norm_fit(:, n), lb, ub, optim_lsq);
    
    % plot command
    plot(E_p, delta_Abs_norm)
    hold on
    plot(E_fit, MB(y(1,:), E_fit), 'red')
    plot(E_fit, MB(y(n,:), E_fit), 'red')
    xlabel('Probe Energy (eV)')
    ylabel('Normalised \Delta A (a.u.)')
    legend('Fitted curve')
end
carrier_T = y(:,2);
disp(carrier_T)
toc
%% Miscellaneous

% For a rectangular pic: 
% drag horizontally until it just covers the letter 'c' in the word 'col'

% For a square pic: 
% drag horizontall until it just covers the letter 'o' in the word 'Contribution' in the next line)
% legend('Experimental Data', 'Fitted Curve', 'Carrier Contribution','Excitonic Contribution')

Answer 1

[Star Strider]

It is difficult for me to understand what you are doing in this version of your code.

You have a curve that is in part an exponential decay, so one option to estimate tthe initial parameters would be to linearise it, do a linear regression on it, re-transform the estimated parameters to use with your nonlinear regression function.

Lv = any(delta_Abs_norm_fit > 0,2);
B = [log(E_fit(Lv)) ones(size(E_fit(Lv)))] \ log(delta_Abs_norm_fit(Lv,:));

figure
plot(E_fit, delta_Abs_norm_fit)
hold on
plot(E_fit(Lv), exp([log(E_fit(Lv)) ones(size(E_fit(Lv)))]*B(:,1)))
hold off
title('Linear Regression Parameter Estimate Results')

The transformation would then be:

y1 = [exp(B(2,1)) B(1,1) rand]

The purpose of the ‘Lv’ calculation is to avoid taking the logarithms of negative numbers.

When I experimented with that approach with your other version of your code, it worked. The upside is that it’s reasonably efficient.

Experiment with it to see if it works in this version of your code.

.

[Jack]

Hi Star, thanks for the Ans! But do I need the for loop to do the fittings? I'm asking this becasue my original data file (named data in my code) is very large and each of the column corresponds to one set of data that needs to be used for fitting. I tried using the for loop and the computational time is really long. Is there another way around it?

[Torsten]

Within the loop

y(1,:) = lsqcurvefit(@MB, y1, E_fit, delta_Abs_norm_fit(:, 1), lb1, ub1, optim_lsq);

You fit your first data set again and again. Why ?

[Jack]

Yeah I think I have made a mistake. I should fit the first data set outside the loop.

[Star Strider]

My pleasure!

I believe your code coiuld be more efficient, however I¹m not certain how it works, so I’m hesitant to alter it. I would do all data transformations that need to be done first, just after reading the file. Fittinbg the columns would be relatively straightforward after that.

Then, I would loop through the rows, and do the regression on each column. I do nott understand your data. If I did, I would attemp a column-wise parameter estimation in a loop.

data = readmatrix('FCPIB-293K-2.5...bg -chirp.csv');
wavelength = data(:,1);

goodcols = any(~ismissing(data(2:end,2:end)),1);
numel(goodcols)

ans = 300

goodcolsn = find(goodcols);

figure
% plot(data(2:end,1),data(2:end,2:end))
plot(data(2:end,1), data(2:end,goodcolsn(3:50:end)))
xlabel('Wavelength')
ylim([min(ylim) 0.025])
grid

function F = MB(y, E_fit)
    F = y(1).*exp(-(E_fit./(8.617333268*10^-5.*y(2)))) + y(3);
end 

.

[Jack]

Hi Star, appreciate your ans! I hope this will make clear of how to read my data:

1. The first row contains quantities called delay time and I am only interest in the delay times within the range 0.5 - 1000
2. The first column contians quantities called wavelength (with their units in nano-meters)
3. For each entry in the first row, the remaining entries in the column is this quantity called "Absorbance" and hence each column corresponds to one dataset
4. What I am doing now is that I will transform the first column (wavelength) into this quantity call energy (given by the formula (h*c)/(wavelength*10^-9)) the values for h and c may be found in my code in the header fundamental constants
5. And again, I will only need energies ranging between 1.5 to 2.2 for my scatter plot
6. However, for the fitting of my data, I will only need energies ranging between 1.65 to 2.2

may I please ask for you kind advice in the approach to column wise parameter estimation within a loop?

[Star Strider]

%% Preparation 
clear;clc 

data = importdata("FCPIB-293K-2.5...bg -chirp.csv"); % insert file path within parenthesis 



%% Preamble 

% Fundamental constants
h = 4.0135667696*10^-15; % units: eV/ Hz
c = 3*10^8; % SI units
kB = 8.617333268*10^-5; % units: eV/ K

% Clean up of data to select range of values 
wavelength = data(1:end, 1);
delay_t = data(1, 1:end); % conatains all of the delay times
E = (h*c)./(wavelength*10^-9); % contains all of the probe energies
Range_E = E>=1.5 & E<=2.2;
Range_T = delay_t>=0.5 & delay_t<=1000;

% for one delay time
T = find(Range_T);
T_min = min(T)

T_min = 56

T_max = max(T)

T_max = 241

% Fitting function
function F = MB(y, E_fit)
    F = y(1).*exp(-(E_fit./(8.617333268*10^-5.*y(2)))) + y(3);
end 

for t = T_min:T_max
% t = 57; % choose an integer b/w T_min and T_max
delaytime = delay_t(1, t);
disp(delaytime)

% Initial parameter guess and bounds 
lb = [0, 293, -1]; ub = [Inf, 1200, 1];
y0 = [2*10^9, 1000, 0.5];


% Data for fitting 
E_p = E(Range_E); % selected probe energies
delta_Abs = -1*data(Range_E,t);
delta_Abs_norm = delta_Abs./max(abs(delta_Abs)); % normalised delta_Abs
Range_Efit = E_p>=1.62 & E_p<=max(E_p);
E_fit = E_p(Range_Efit);
delta_Abs_norm_fit = delta_Abs_norm(Range_Efit);

Lv = any(delta_Abs_norm_fit > 0,2);
B = [log(E_fit(Lv)) ones(size(E_fit(Lv)))] \ log(delta_Abs_norm_fit(Lv,:));


%% Curve fitting options 

% % Initial parameter guess and bounds 
% lb = [0, 293, -1]; ub = [Inf, 800, 1];
% y0 = [1.2*10^9, 700, 0.5];


% lsqcurvefit and choose between different algorithm that lsqcurvefit employs (3C1, comment those lines that are not choosen and uncomment the line that is choosen, if not, matlab will take the last line of "optim_lsq" by default)
optim_lsq = optimoptions('lsqcurvefit', 'Algorithm', 'levenberg-marquardt', 'MaxFunctionEvaluations',10^10, 'MaxIterations', 10^10, 'FunctionTolerance',10^-10, 'StepTolerance', 10^-10);
% optim_lsq = optimoptions('lsqcurvefit', 'Algorithm', 'trust-region-reflective', 'MaxFunctionEvaluations',10^10, 'MaxIterations',10^10, 'FunctionTolerance',10^-20, 'StepTolerance', 10^-20);
% optim_lsq = optimoptions('lsqcurvefit', 'Algorithm', 'interior-point', 'MaxFunctionEvaluations',1000, 'MaxIterations', 1000, 'FunctionTolerance',10^-20, 'StepTolerance', 10^-20);

Lv = any(delta_Abs_norm_fit > 0,2);
B = [log(E_fit(Lv)) ones(size(E_fit(Lv)))] \ log(delta_Abs_norm_fit(Lv));

y0 = [exp(B); rand];

% figure
% plot(E_fit, delta_Abs_norm_fit)
% hold on
% plot(E_fit(Lv), exp([log(E_fit(Lv)) ones(size(E_fit(Lv)))]*B))
% hold off
% title("Linear Regression Parameter Estimate Results "+t)

% Solver for lsqcurvefit
[y, residualnorm, residual, exitflag, output, lambda, jacobian] = lsqcurvefit(@MB, y0, E_fit, delta_Abs_norm_fit, lb, ub, [], [], [], [], [], optim_lsq);

y_mtx(:,t-55) = y;
r_nrm(T-55,:) = residualnorm;
t_vct(t-55,:) = t;

%% Plot command 

if rem(t-56,10) == 0

    figure
    plot(E_p, delta_Abs_norm,'Black')
    hold on 
    plot(E_fit, MB(y, E_fit), 'LineWidth', 1.0, 'Color', 'red')
    xlabel('Probe Photon Energy (eV)')
    ylabel('Normalised \Delta A (a.u.)')
    legend('Experimental Data', 'Fitted Curve')
    title("t = "+t)
    textx = min(xlim)+mean(xlim)/15;
    texty = min(ylim)+mean(ylim)/2;
    text(textx,texty, sprintf('F = %9.3E \\cdot e^{(-(E\\_fit/(8.617\\cdot10^{-5}\\cdot%.1f)))} + %.3f',y))
    % disp(y)
    % disp(y(1,1))
    % disp(y(1,2))
    % disp(y(1,3))
end

end

    0.5342

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.

lsqcurvefit stopped because the final change in the sum of squares relative to 
its initial value is less than the value of the function tolerance.

    0.5976

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum found.

Optimization completed because the size of the gradient is less than
1e-4 times the value of the function tolerance.

    0.6635

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.

lsqcurvefit stopped because the final change in the sum of squares relative to 
its initial value is less than the value of the function tolerance.

    0.7311

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.

lsqcurvefit stopped because the final change in the sum of squares relative to 
its initial value is less than the value of the function tolerance.

    0.8003

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.

lsqcurvefit stopped because the final change in the sum of squares relative to 
its initial value is less than the value of the function tolerance.

    0.8727

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.

lsqcurvefit stopped because the final change in the sum of squares relative to 
its initial value is less than the value of the function tolerance.

    0.9484

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.

lsqcurvefit stopped because the final change in the sum of squares relative to 
its initial value is less than the value of the function tolerance.

    1.0257

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    1.1071

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    1.1901

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    1.2763

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    1.3634

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    1.4578

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    1.5522

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    1.6523

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    1.7524

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    1.8590

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    1.9673

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    2.0804

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    2.1976

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    2.3180

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    2.4417

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    2.5727

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    2.7062

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    2.8445

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    2.9870

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    3.1351

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    3.2897

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    3.4476

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    3.6111

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    3.7804

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    3.9554

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    4.1377

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    4.3257

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    4.5193

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    4.7203

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    4.9287

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    5.1443

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    5.3673

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    5.5976

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    5.8353

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    6.0818

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    6.3366

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    6.6002

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    6.8737

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    7.1552

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    7.4474

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    7.7485

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    8.0618

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    8.3849

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    8.7194

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    9.0644

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    9.4233

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

    9.7936

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   10.1769

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   10.5724

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   10.9817

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   11.4057

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   11.8444

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   12.2993

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   12.7689

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   13.2522

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   13.7568

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   14.2768

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   14.8139

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   15.3698

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   15.9459

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   16.5416

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   17.1577

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   17.7949

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   18.4541

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   19.1352

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   19.8416

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   20.5716

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   21.3260

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   22.1072

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   22.9153

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   23.7511

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   24.6154

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   25.5089

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   26.4367

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   27.3937

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   28.3833

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   29.4087

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   30.4682

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   31.5644

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   32.6989

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   33.8724

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   35.0874

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   36.3431

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   37.6403

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   38.9871

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   40.3779

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   41.8167

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   43.3060

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   44.8457

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   46.4367

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   48.0846

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   49.7911

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   51.5538

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   53.3792

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   55.2648

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   57.2195

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   59.2394

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   61.3284

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   63.4883

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   65.7270

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   68.0407

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   70.4341

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   72.9088

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   75.4715

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   78.1220

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   80.8629

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   83.6990

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   86.6336

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   89.6699

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   92.8095

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   96.0582

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

   99.4176

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  102.8950

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  106.4910

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  110.2130

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  114.0620

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  118.0430

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  122.1630

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  126.4240

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  130.8340

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  135.3940

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  140.1130

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  144.9940

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  150.0440

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  155.2670

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  160.6720

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  166.2620

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  172.0470

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  178.0310

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  184.2210

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.

lsqcurvefit stopped because the final change in the sum of squares relative to 
its initial value is less than the value of the function tolerance.

  190.6230

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.

lsqcurvefit stopped because the final change in the sum of squares relative to 
its initial value is less than the value of the function tolerance.

  197.2500

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.

lsqcurvefit stopped because the final change in the sum of squares relative to 
its initial value is less than the value of the function tolerance.

  204.1020

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.

lsqcurvefit stopped because the final change in the sum of squares relative to 
its initial value is less than the value of the function tolerance.

  211.1930

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.

lsqcurvefit stopped because the final change in the sum of squares relative to 
its initial value is less than the value of the function tolerance.

  218.5270

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.

lsqcurvefit stopped because the final change in the sum of squares relative to 
its initial value is less than the value of the function tolerance.

  226.1150

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.

lsqcurvefit stopped because the final change in the sum of squares relative to 
its initial value is less than the value of the function tolerance.

  233.9640

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.

lsqcurvefit stopped because the final change in the sum of squares relative to 
its initial value is less than the value of the function tolerance.

  242.0860

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.

lsqcurvefit stopped because the final change in the sum of squares relative to 
its initial value is less than the value of the function tolerance.

  250.4860

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.

lsqcurvefit stopped because the final change in the sum of squares relative to 
its initial value is less than the value of the function tolerance.

  259.1780

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.

lsqcurvefit stopped because the final change in the sum of squares relative to 
its initial value is less than the value of the function tolerance.

  268.1680

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.

lsqcurvefit stopped because the final change in the sum of squares relative to 
its initial value is less than the value of the function tolerance.

  277.4710

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.

lsqcurvefit stopped because the final change in the sum of squares relative to 
its initial value is less than the value of the function tolerance.

  287.0930

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.

lsqcurvefit stopped because the final change in the sum of squares relative to 
its initial value is less than the value of the function tolerance.

  297.0470

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.

lsqcurvefit stopped because the final change in the sum of squares relative to 
its initial value is less than the value of the function tolerance.

  307.3460

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  318.0010

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.

lsqcurvefit stopped because the final change in the sum of squares relative to 
its initial value is less than the value of the function tolerance.

  329.0190

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  340.4220

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum found.

Optimization completed because the size of the gradient is less than
1e-4 times the value of the function tolerance.

  352.2180

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  364.4210

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  377.0430

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  390.1020

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  403.6140

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  417.5870

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  432.0490

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  447.0070

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  462.4820

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  478.4880

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  495.0520

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  512.1850

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  529.9080

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  548.2430

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  567.2120

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  586.8350

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  607.1370

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  628.1390

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  649.8640

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum found.

Optimization completed because the size of the gradient is less than
1e-4 times the value of the function tolerance.

  672.3390

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  695.5940

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  719.6490

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum found.

Optimization completed because the size of the gradient is less than
1e-4 times the value of the function tolerance.

  744.5340

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  770.2780

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  796.9110

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  824.4630

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  852.9630

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  882.4530

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  912.9580

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum found.

Optimization completed because the size of the gradient is less than
1e-4 times the value of the function tolerance.

  944.5160

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

  977.1620

Warning: Initial point is not feasible. Initializing x0 at nearest feasible point.

Local minimum found.

Optimization completed because the size of the gradient is less than
1e-4 times the value of the function tolerance.

Parameters = table(t_vct, y_mtx(1,:).', y_mtx(2,:).', y_mtx(3,:).', r_nrm, 'VariableNames',{'t','y1','y2','y3','resid_norm'})

Parameters = 186x5 table
    t         y1          y2         y3        resid_norm
    __    __________    ______    _________    __________

    56    2.1481e+06    1198.8    -0.048583     0.15882  
    57    2.2743e+06      1200    -0.055101     0.15882  
    58     2.961e+06    1187.1    -0.064151     0.15882  
    59    6.2943e+06    1139.1     -0.07204     0.15882  
    60    3.0488e+07    1043.3    -0.076711     0.15882  
    61    3.3322e+08    923.88    -0.080706     0.15882  
    62    2.5387e+09     841.1    -0.086548     0.15882  
    63    4.5132e+10    746.28    -0.091154     0.15882  
    64    5.8557e+10     738.3    -0.099297     0.15882  
    65    7.4922e+10     730.4     -0.10548     0.15882  
    66    8.4157e+10     725.8     -0.11023     0.15882  
    67    9.2389e+10    721.81     -0.11345     0.15882  
    68    9.5319e+10    719.26     -0.11483     0.15882  
    69    9.7492e+10     716.9     -0.11578     0.15882  
    70     9.856e+10    714.81     -0.11617     0.15882  
    71    9.3988e+10    714.22     -0.11527     0.15882  

Some of these worked better than others. My approximation method for the initial parameter estimates is reasonably robust for the earlier data, however less so for the last several, probably because they do not follow the same sort of exponential curve that the earlier ones did, and may not be modeled by ‘MB’ as well as the earlier ones. I leave that consideration (and necessary tweaks) to you. I added the ‘Parameters’ table to summarise the results.

.

[Jack]

Thank you Star!

[Star Strider]

As always, my pleasure!

Answer 2

[William Rose]

@Jack,

It would be helpful to run your script in the window to show that the "really bad" fit looks like. Post only the minimum code and data necessary to illustrate the issue. Modify your script so it runs in the Matlab answers window.

The value of Planck constant in eV/Hz is incorrect by a few percent; fixed below.

I had an error when I tried to run. The error went away when I changed the filename to something more simple.

Then the code timed out ("Your code took longer than 235 seconds to run. Simplify code and then run again.").

Your data file has >300 columns and 1024 rows of data. Plus a header line (the delay times) and lines of metadata at the end. Much of the data is NaNs.

You select data at a range of wavelengths (i.e. a range of rows) for plotting, from 547.8 to 802.5 nm. You select a subset of this data for fitting. The dashed line in plot below shows the begigning of the range for fitting. I edited the data file to eliminate rows and columns outside this range of plotting interest. And I adjusted Range_T so that only 3 spectra (3 columns) would be fitted.

The fitting function has 3 parameters: p(1:3). The fitting function is a decaying exponential function of energy, with parameters p(1)=amplitude of exponential, p(2)=temperature [K], which determines the energy constant for decay of the exponential, and p(3)=vertical offset of the baseline.

The absorbance data, in the energy range of interest, looks like an upside-down decaying exponential. The absorbance data wavelength increases with row number, so the energy decreases with row number. So the highest energies are the lowest number rows, and the lowest energy is the last row.

For the three columns plotted below, it looks like a good initial guess for the parameters is p0(2)=0.3 eV/kB~=3500 K, p0(3)=+0.002, and p0(1)=(ydata(end)-p0(3))/exp(-E(end)/(kB*p0(2))). A better initial guess for p0(3) is the mean value of the absorbance at the highest energy range being fitted.

data=importdata('spectrum1a.csv');
fprintf('Size(data)=%d x %.d.\n',size(data))

Size(data)=430 x 247.

%data = importdata("FCPIB-293K-2.5mW-400nm-Jan072021 -ibg -bg -chirp.csv"); 
%% Preamble 
% Fundamental constants
h = 4.135667696*10^-15; % units: eV/ Hz
%h = 4.0135667696*10^-15;
c = 3*10^8; % SI units
kB = 8.617333268*10^-5; % units: eV/ K
% Data
wavelength = data(1:end, 1); % wavelengths [nm]
delay_t = data(1, 1:end); % delay times
E = (h*c)./(wavelength*10^-9); % probe energies [eV]
% Range of data needed
Range_E = E>=1.5 & E<=2.2; % [eV]
%Range_T = delay_t>=0.5 & delay_t<=1000;
Range_T = delay_t>=0.5 & delay_t<=0.7;
E_p = E(Range_E); % probe energies for scatterplot
w_p = wavelength(Range_E); 
% w_p = wavelength(Range_W); 
delay_T = delay_t(Range_T);
%Range_Efit = E_p>=1.62 & E_p<=max(E_p); 
Emin=1.62;            % minimum energy for fitting [eV]
Range_Efit = E_p>=Emin; 
E_fit = E_p(Range_Efit); % probe energies for fitting 
t_min = min(delay_T);
t_max = max(delay_T);
w_min = min(w_p);
w_max = max(w_p);
[row1, ~] = min(find(data(:,1) == w_min));
[row2, ~] = max(find(data(:,1) == w_max));
[~, col1] = find(data == t_min);
[~, col2] = find(data == t_max);
% New cleaned up data
data_new = data(row1:row2, col1:col2);% new data containing required delta A
fprintf('Size(data_new)=%d x %.d.\n',size(data_new))

Size(data_new)=401 x 3.

fprintf('Size(E_Fit)=%d x %d.\n',size(E_fit))

Size(E_Fit)=340 x 1.

% Plot columns 1-3
figure
plot(E_p,data_new(:,1),'-r',E_p,data_new(:,2),'-g',E_p,data_new(:,3),'-b')
hold on; xline(Emin,'--k');
grid on; xlabel('Energy (eV)'); ylabel('\Delta Absorbance')
legend(['delay=',num2str(delay_T(1))],['delay=',num2str(delay_T(2))],['delay=',num2str(delay_T(3))],'Location','southeast')

% Fitting function: Maxwell-Boltzmann distribution 
function F = MB(p, e)
kB = 8.617333268*10^-5;     % [eV/K]
F = p(1).*exp(-(e./(kB.*p(2)))) + p(3);
end 

% For loop to fit the data
figure
p=zeros(length(delay_T),3);  % allocate array for fitted parameters
for n = 1:length(delay_T)
    ydata=data_new(Range_Efit,n);
    p0(3)=mean(ydata(1:10));                                % initial guess
    p0(2)=0.3/kB;                                           % initial guess [deg K]
    p0(1)=(ydata(end)-p0(3))/exp(-E_fit(end)/(kB*p0(2)));   % initial guess
    optim_lsq = optimoptions('lsqcurvefit', 'Algorithm', 'levenberg-marquardt', 'MaxFunctionEvaluations',10^20, 'MaxIterations', 10^20, 'FunctionTolerance',10^-20, 'StepTolerance', 10^-20);
    lb = [-Inf, 293, -1]; ub = [Inf, 6000, 1];
    p(n,:) = lsqcurvefit(@MB, p0, E_fit, ydata, lb, ub, optim_lsq);
    fprintf('n=%d: p0=[%.2e %.0f %.4f]; p=[%.2e %.0f %.4f]\n',n,p0,p(n,:))
end

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

n=1: p0=[-5.79e+00 3481 0.0014]; p=[-7.52e+09 710 0.0007]

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

n=2: p0=[-6.58e+00 3481 0.0019]; p=[-1.14e+10 703 0.0009]

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

n=3: p0=[-7.34e+00 3481 0.0022]; p=[-3.54e+09 739 0.0013]

% Plot data and fitted curves
figure
plot(E_fit,data_new(Range_Efit,1),'-r',E_fit,data_new(Range_Efit,2),'-g',E_fit,data_new(Range_Efit,3),'-b')
hold on
plot(E_fit,MB(p(1,:),E_fit),'--r',E_fit,MB(p(2,:),E_fit),'--g',E_fit,MB(p(3,:),E_fit),'--b')
grid on; xlabel('Energy (eV)'); ylabel('\Delta Absorbance')
legend(['delay=',num2str(delay_T(1))],['delay=',num2str(delay_T(2))],['delay=',num2str(delay_T(3))],...
    'fit 1','fit 2','fit 3','Location','southeast')

The three fits look reasonable. The fitted p(1) and p(2) are not close to the initial gueeses, but it seems to work, i.e. the fitting routine finds a reasonable fit.

[Jack]

Thank you William!

[Jack]

To add on, I think I should have been mroe specific when I say that my fit is not good. Basically, I tried to fit all the 186 delay times, the resulting fitting is far off from the data points. That's what I meant. Hope this is clear.