fzero of MATLAB and FindRoot of Mathematica give different results

Dear all, I am writing a small script to solve the temperature dependence of magnetization using a toy model. Mathematically, the problem is to determine the relation between reduced magnetization and reduced temperature . The equation is given by , where is the Brillouin function.

To solve this problem, I transform it into a root-finding problem, specifically . I used MATLAB's fzero and Mathematica's FindRoot to solve this equation. However, I noticed that the results from the two methods are not consistent. The result from Mathematica matches the reference, while the MATLAB result also appears reasonable. In both cases, m as a function of τ tends to zero from two different directions as τ increases.

Is this correct? Could you please comment or share your opinions on this matter?

Here is the MATLAB code:

clearvars; clc; close all; fclose all; format compact; format short;

B = @(J, x) ((2*J+1)/(2*J))*coth(((2*J+1)/(2*J))*x)-(1/(2*J)).*coth((1/(2*J))*x);

J = 1/2;

n = 101;

tau_array = linspace(0, 1.2, n);

m_array = zeros(1, n);

for i = 1:n

tau = tau_array(i);

fun = @(m) B(J, ((3*J)/(J+1))*m*(1/tau)) - m;

x = fzero(fun,[-1, 1]);

m_array(i) = x;

end

fig = figure();

ax = axes(fig);

plot(ax, tau_array, m_array,"LineWidth",2,"Color","red");

xlabel('Reudced temperature, \tau');

ylabel('Redueced magnetization, m')

title('m as a function of \tau')

ax.TickDir = 'out';

Here is the Mathematica code:

Here is the reference result, which is taken from this paper [DOI: 10.1109/ISSE.2008.5276604; https://ieeexplore.ieee.org/document/5276604](See Equation 20 and Figure 3)

Thank you.

Answer 1

[Malay Agarwal]

Hi @qilin guo,

You're seeing a different graph since you're finding the root around the initial value [-1, 1] in MATLAB while the root is around the initial value 1 in Mathematica. If you change your call to "fzero" from:

x = fzero(fun, [-1, 1]);

To:

x = fzero(fun, 1);

The graph is as expected. Here's the complete code for your reference:

clearvars; clc; close all; fclose all; format compact; format short;
B = @(J, x) ((2*J+1)/(2*J))*coth(((2*J+1)/(2*J))*x)-(1/(2*J)).*coth((1/(2*J))*x);
J = 1/2;
n = 101;
tau_array = linspace(0, 1.2, n);
m_array = zeros(1, n);
for i = 1:n
    tau = tau_array(i);
    fun = @(m) (B(J, ((3*J)/(J+1))*m*(1/tau)) - m);
    x = fzero(fun, 1); % Note the change from [-1, 1] to 1
    m_array(i) = x;
end
fig = figure();
ax = axes(fig);
plot(ax, tau_array, m_array,"LineWidth",2,"Color","red");
xlabel('Reudced temperature, \tau');
ylabel('Redueced magnetization, m')
title('m as a function of \tau')
ax.TickDir = 'out';

Please note that only the y-axis limits are different from the expected graph. You can change it easily using the "ylim" function.

Please refer to the following links for more information:

• "fzero" documentation - https://www.mathworks.com/help/releases/R2023a/matlab/ref/fzero.html.
• "ylim" documentation - https://www.mathworks.com/help/releases/R2023a/matlab/ref/ylim.html.

Hope this helps!

[qilin guo]

Dear Malay Agarwal, it helps me a lot. Thank you!