Encountering error with using lsqcurvefit

I have the following experimental data.

xdata = [1 1.01 1.12 1.24 1.39 1.61 1.89 2.17 2.42 3.01 3.58 4.03 4.76 5.36 5.76 6.16 6.40 6.62 6.87 7.05 7.16 7.27 7.43 7.50 7.61];
ydata = [0 0.03 0.14 0.23 0.32 0.41 0.50 0.58 0.67 0.85 1.04 1.21 1.58 1.94 2.29 2.67 3.02 3.39 3.75 4.12 4.47 4.85 5.21 5.57 6.30];

I am trying to fit the ydata, which are stresses denoted by P, to the analytical equivalent (see equation) where the xdata represents the corresponding strains denoted by λ.

I am attempting to use the lsqcurvefit function in the follwing script but I am encountering an error. How may I resolve this? I am unfamiliar with most of the information returned to me

clear all
close all
clc

% xdata (Stretch, Lamda)
xdata = [1 1.01 1.12 1.24 1.39 1.61 1.89 2.17 2.42 3.01 3.58 4.03 4.76 5.36 5.76 6.16 6.40 6.62 6.87 7.05 7.16 7.27 7.43 7.50 7.61];

% ydata (Nominal Stress, P)
ydata = [0 0.03 0.14 0.23 0.32 0.41 0.50 0.58 0.67 0.85 1.04 1.21 1.58 1.94 2.29 2.67 3.02 3.39 3.75 4.12 4.47 4.85 5.21 5.57 6.30];

% Proposition
% ydata = (a*b*xdata^3 - 1))/(xdata)*(xdata*b - xdata^3 + 3*xdata - 2));

% Gent Model Y-Axis
gent_function = @(x,xdata)(x(1)*x(2)*(xdata^3 - 1))/((xdata)*(xdata*x(2) - xdata^3 + 3*xdata - 2));

% Fit Model w/ Start 
x0 = [100, 0.1];
x = lsqcurvefit(gent_function,x0,xdata,ydata);

% Plot Data & Fitted Curve
times = linspace(xdata(1),xdata(end));
plot(xdata,ydata,'ko',times,gent_function(x,times),'b-')
legend('Data','Fitted Exponential')
title('Data and Fitted Curve')

--------------------------------------------------------------------------------------------------
Error using  ^  (line 51)
Incorrect dimensions for raising a matrix to a
power. Check that the matrix is square and the power
is a scalar. To perform elementwise matrix powers,
use '.^'.

Error in
check>@(x,xdata)(x(1)*x(2)*(xdata^3-1))/((xdata)*(xdata*x(2)-xdata^3+3*xdata-2))
(line 15)
gent_function = @(x,xdata)(x(1)*x(2)*(xdata^3 -
1))/((xdata)*(xdata*x(2) - xdata^3 + 3*xdata - 2));

Error in lsqcurvefit (line 225)
            initVals.F =
            feval(funfcn_x_xdata{3},xCurrent,XDATA,varargin{:});

Error in check (line 19)
x = lsqcurvefit(gent_function,x0,xdata,ydata);

Caused by:
    Failure in initial objective function
    evaluation. LSQCURVEFIT cannot continue.

Answer 1

[Walter Roberson]

% xdata (Stretch, Lamda)
xdata = [1 1.01 1.12 1.24 1.39 1.61 1.89 2.17 2.42 3.01 3.58 4.03 4.76 5.36 5.76 6.16 6.40 6.62 6.87 7.05 7.16 7.27 7.43 7.50 7.61];

% ydata (Nominal Stress, P)
ydata = [0 0.03 0.14 0.23 0.32 0.41 0.50 0.58 0.67 0.85 1.04 1.21 1.58 1.94 2.29 2.67 3.02 3.39 3.75 4.12 4.47 4.85 5.21 5.57 6.30];

% Proposition
% ydata = (a*b*xdata^3 - 1))/(xdata)*(xdata*b - xdata^3 + 3*xdata - 2));

% Gent Model Y-Axis
gent_function = @(x,xdata)(x(1)*x(2)*(xdata.^3 - 1))./((xdata).*(xdata*x(2) - xdata.^3 + 3*xdata - 2));

% Fit Model w/ Start 
x0 = [100, 0.1];
x = lsqcurvefit(gent_function,x0,xdata,ydata);

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.

% Plot Data & Fitted Curve
times = linspace(xdata(1),xdata(end));
plot(xdata,ydata,'ko',times,gent_function(x,times),'b-')
legend('Data','Fitted Exponential')
title('Data and Fitted Curve')

[Walter Roberson]

% xdata (Stretch, Lamda)
xdata = [1 1.01 1.12 1.24 1.39 1.61 1.89 2.17 2.42 3.01 3.58 4.03 4.76 5.36 5.76 6.16 6.40 6.62 6.87 7.05 7.16 7.27 7.43 7.50 7.61];

% ydata (Nominal Stress, P)
ydata = [0 0.03 0.14 0.23 0.32 0.41 0.50 0.58 0.67 0.85 1.04 1.21 1.58 1.94 2.29 2.67 3.02 3.39 3.75 4.12 4.47 4.85 5.21 5.57 6.30];

% Proposition
% ydata = (a*b*xdata^3 - 1))/(xdata)*(xdata*b - xdata^3 + 3*xdata - 2));

% Gent Model Y-Axis
gent_function = @(x,xdata)(x(1)*x(2)*(xdata.^3 - 1))./((xdata).*(xdata*x(2) - xdata.^3 + 3*xdata - 2));
residue_function = @(x) sum(((x(1)*x(2)*(xdata.^3 - 1))./((xdata).*(xdata*x(2) - xdata.^3 + 3*xdata - 2)) - ydata).^2);

% Fit Model w/ Start 
x0 = [100, 0.1];
x = fmincon(residue_function,x0);

Local minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing in 
feasible directions, to within the value of the optimality tolerance,
and constraints are satisfied to within the value of the constraint tolerance.

% Plot Data & Fitted Curve
times = linspace(xdata(1),xdata(end));
plot(xdata,ydata,'ko',times,gent_function(x,times),'b-')
legend('Data','Fitted Exponential')
title('Data and Fitted Curve')

[Walter Roberson]

% xdata (Stretch, Lamda)
xdata = [1 1.01 1.12 1.24 1.39 1.61 1.89 2.17 2.42 3.01 3.58 4.03 4.76 5.36 5.76 6.16 6.40 6.62 6.87 7.05 7.16 7.27 7.43 7.50 7.61];

% ydata (Nominal Stress, P)
ydata = [0 0.03 0.14 0.23 0.32 0.41 0.50 0.58 0.67 0.85 1.04 1.21 1.58 1.94 2.29 2.67 3.02 3.39 3.75 4.12 4.47 4.85 5.21 5.57 6.30];

% Proposition
% ydata = (a*b*xdata^3 - 1))/(xdata)*(xdata*b - xdata^3 + 3*xdata - 2));

% Gent Model Y-Axis
gent_function = @(x,xdata)(x(1)*x(2)*(xdata.^3 - 1))./((xdata).*(xdata*x(2) - xdata.^3 + 3*xdata - 2));
residue_function = @(x) sum(((x(1)*x(2)*(xdata.^3 - 1))./((xdata).*(xdata*x(2) - xdata.^3 + 3*xdata - 2)) - ydata).^2);


% Fit Model w/ Start 
x0 = [100, 0.1];

gs = GlobalSearch;
problem = createOptimProblem('fmincon', 'x0', x0, ...
    'objective', residue_function);
x = run(gs,problem)

GlobalSearch stopped because it analyzed all the trial points.

All 8 local solver runs converged with a positive local solver exit flag.

x = 1×2

0.2451 79.4367

●

% Plot Data & Fitted Curve
times = linspace(xdata(1),xdata(end));
plot(xdata,ydata,'ko',times,gent_function(x,times),'b-')
legend('Data','Fitted Exponential')
title('Data and Fitted Curve')

[Joshua Rees]

Hello,

Thanks for your time and helping out. I think the first solution you posted is correct, by running the amended script I can now produce a graph (which is similar if not the same to the graph in the third solution).