Find all possible roots of transcendental function

Please can help with how find all the possible roots of q for xi and u and use the results to compute tau?

clear 
clc
close all
%% Model parameters
alpha3 = -0.001104; eta1 = 0.0240; K1 = 6*10^(-12);              
gamma1 = 0.1093; chi_a = 1.219*10^(-6); d = 0.0002;
H_c  =   pi*sqrt(K1/chi_a)/d;                    
alpha = 1-((alpha3).^2/(gamma1*eta1));
xi =linspace(-0.3, 0.3, 81);              
u = linspace(0, 3, 81);                   
for iu=1:length(u)
    for  ixi=1:length(xi)
        q(iu, ixi) - (1-alpha)*tan(q) + (1/gamma1)*((alpha3*ixi/eta1)*tan(q) + chi_a*(iu*H_c)^2*q)*(alpha*gamma1*  (4*K1*q(iu, ixi).^2/d.^2  - (alpha3*ixi)/eta1  - chi_a*(iu*H_c).^2)^(-1));
        tau(iu, ixi) = (alpha*gamma1*(4*K1*q(iu, ixi).^2/d.^2  - (alpha3*ixi)/eta1  - chi_a*(iu*H_c).^2)^(-1)
    end 
end

[James Tursa]

Please post an image of the function you are trying to find the roots of, rather than have us trying to decipher your code.

Answer 1

[Torsten]

Really all zeros ? Then you cannot save them in any array since for each u and xi, you get an infinite number of them. Is the function correct ? There were some strange settings therein: the "q(iu, ixi) -" at the beginning and the use of ixi and iu instead of xi(ixi) and u(iu) in the function definition.

%% Model parameters
alpha3 = -0.001104; eta1 = 0.0240; K1 = 6*10^(-12);              
gamma1 = 0.1093; chi_a = 1.219*10^(-6); d = 0.0002;
H_c  =   pi*sqrt(K1/chi_a)/d;                    
alpha = 1-((alpha3).^2/(gamma1*eta1));
fun = @(q,xi,u)(1-alpha).*tan(q) + (1./gamma1).*((alpha3.*xi./eta1).*tan(q) + chi_a.*(u*H_c).^2.*q).*(alpha.*gamma1.*(4.*K1.*q.^2/d.^2-(alpha3.*xi)./eta1-chi_a.*(u.*H_c).^2).^(-1));
xi = 0.2;
u = 1.5;
hold on
q = 0:0.01:pi/2-0.1;
plot(q,fun(q,xi,u),'b')
q = pi/2+0.1:0.01:3*pi/2-0.1;
plot(q,fun(q,xi,u),'b')
q = 3*pi/2+0.1:0.01:5*pi/2-0.1;
plot(q,fun(q,xi,u),'b')
q = 5*pi/2+0.1:0.01:7*pi/2-0.1;
plot(q,fun(q,xi,u),'b')
hold off
grid on

[University]

@Torsten Thanks for you response. I have explore the function in Maple. I had to find the odd asymptotes using seq(evalf(1/2*(2*j + 1)*pi), j = 0 .. n) for each mode n, and then find the zeros between each asymptotes.

[Sam Chak]

Hi @University,

The problem looks interesting. From your description, it seems that is a bivariate function.

Can you plot the surface within the intervals and ? It may help to visualize the regions of zero-crossing so that the initial guesses can be narrow down.

However, I cannot interpret this line into a valid mathematical expression or function.

q(iu, ixi) - (1-alpha)*tan(q) + (1/gamma1)*((alpha3*ixi/eta1)*tan(q) + chi_a*(iu*H_c)^2*q)*(alpha*gamma1*  (4*K1*q(iu, ixi).^2/d.^2  - (alpha3*ixi)/eta1  - chi_a*(iu*H_c).^2)^(-1));

Is q an implicit function?

Meanwhile, you mentioned that the roots of q will be used to compute another function called tau, , which also depends on q. But I don't see any indication of roots in tau. Hope you can clarify that.

tau(iu, ixi) = (alpha*gamma1*(4*K1*q(iu, ixi).^2/d.^2  - (alpha3*ixi)/eta1  - chi_a*(iu*H_c).^2)^(-1)

By the way, could you explain this Maple code?

seq(evalf(1/2*(2*j + 1)*pi), j = 0 .. n)

[University]

Thank you Sam. The maple code is a bit long. Unless, you want the complete code. I have attached the contour plot of tau with respect to xi and u

[Sam Chak]

@University, Based on the contour plot, it is most likely that the circled region represents the zero-crossing region where the roots are located. Therefore, the solution should appear as a curved line. However, if we examine the presumed implicit function , which has yet to be clarified, it is possible that this region corresponds to where discontinuities occur due to the tangent function in .