how to find the root of function?

Hi , I want to find the root of nonlinear function, without giving initial point.

here is my .m file for parameters

alpha=0.74
g =0.5
gamma=1.2
sigma=1
phi=0.8
epsilon_1=1.5
w_r= (epsilon_1-1)/epsilon_1

then I made a function like this

function a=n_h_fun(n_h)
a= n_h-(alpha*(w_r/ ((n_h-1)/(g*(1-gamma)))^sigma)^(1/phi))
end

and saved that file as n_h_fun.m

I want to find the value of n_h where a gets the value 0. I used fzero command but it shows errors.

[Rik]

Why don't you want to provide an initial estimate?

[Ani Asoyan]

I don't know the exact point

Answer 1

[Alan Stevens]

Like this?

n_h0 = 0.1;  %initial guess
n_h = fzero(@n_h_fun, n_h0);
 
disp(n_h)

    0.0107

function a=n_h_fun(n_h)
    alpha=0.74;
    g =0.5;
    gamma = 1.2;
    sigma=1;
    phi=0.8;
    epsilon_1=1.5;
    w_r= (epsilon_1-1)/epsilon_1;
    a= n_h-(alpha*(w_r/ ((n_h-1)/(g*(1-gamma)))^sigma)^(1/phi));
end

[Ani Asoyan]

thank you so much

Answer 2

[John D'Errico]

Or do this:

alpha=0.74;
g =0.5;
gamma=1.2;
sigma=1;
phi=0.8;
epsilon_1=1.5;
w_r= (epsilon_1-1)/epsilon_1;
syms n_h

a = n_h-(alpha*(w_r/ ((n_h-1)/(g*(1-gamma)))^sigma)^(1/phi));
pretty(a)

      /          1        \5/4
      | - --------------- |    37
      \   (10 n_h - 10) 3 /
n_h - ---------------------------
                   50

So a is a simple nonlinear function of n_h.

% first, plot it. ALWAYS PLOT EVERYTHING
fplot(a,[-20,20])
grid on

vpasolve(a,n_h)

ans = 

0.010682191712647594755165531568215

So vpasolve found the root near zero for n_h. There may be other solutions, but when n_h is negative, it looks like that curve goes to -inf almost linearly. We can probably show that to be true. And for positive values of n_h, above some point, it looks like a becomes complex. So the only real solution is probably the one found by vpasolve.