MATLAB Answers

## How to solve transcendental equations in matlab

Asked by alburary daniel

### alburary daniel (view profile)

on 12 Jun 2018
Latest activity Answered by Walter Roberson

### Walter Roberson (view profile)

on 30 Jul 2018
Accepted Answer by Anton Semechko

### Anton Semechko (view profile)

I am practicing solving transcendental equations in matlab I have the given equation and I want to solve for x
tan(((0.9)*pi/w)*sqrt(1.468^2-x.^2))- sqrt((x.^2-1.458^2))/(sqrt(1.468^2-x.^2))==0
plotting w varies from 1.55 to 5 I tried all day but I cannot find solution, I tried with fzero(fun,xo) without success any suggestion?

Walter Roberson

### Walter Roberson (view profile)

on 12 Jun 2018
Please do not close questions that have an Answer.
alburary daniel

### alburary daniel (view profile)

on 12 Jun 2018
A feel confuse because a will trying to solve problemas using newton rapson method
Walter Roberson

### Walter Roberson (view profile)

on 13 Jun 2018
Do not close questions that have an Answer. The volunteers spent time responding to you, and their responses may help other people with similar questions.
I know that I spent about 2 hours investigating this myself, and I am disappointed that my contribution would just disappear as if I had never done the work.

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## 2 Answers

Answer by Anton Semechko

### Anton Semechko (view profile)

on 12 Jun 2018
Accepted Answer

This is a complex-valued function. Just from visual inspection, it is apparent that the real part of this function has infinitely many solutions. The imaginary part, however, appears to have a finite number of zeros. Are those the solutions you are looking for?
x=linspace(-2*pi,2*pi,1E4);
w=1.55;
f=tan(((0.9)*pi/w)*sqrt(1.468^2-x.^2)) - sqrt((x.^2-1.458^2))./(sqrt(1.468^2-x.^2));
figure('color','w')
plot(x,real(f))
hold on
plot(x,imag(f))
set(gca,'YLim',[-10 10])

alburary daniel

### alburary daniel (view profile)

on 12 Jun 2018
but in this case I want that x-axis will be W not b, so change the result rigth in the code? by the way what do Z? can you explain that sentences
Anton Semechko

### Anton Semechko (view profile)

on 12 Jun 2018
If you want to plot w along horizontal axis, change
plot(Z(i),W(i),'.k','MarkerSize',20) to plot(W(i),Z(i),'.k','MarkerSize',20),
plot(Z,W,'-b') to plot(W,Z,'-b')
x- and y-labels would have to be modified accordingly.
Variable Z is a vector that contains solutions to f(x,w)=0, so that abs(f(Z(i),W(i)))<tol, where tol=1E-16. W is vector of w values that go from 1.5 to 5.
Anton Semechko

### Anton Semechko (view profile)

on 12 Jun 2018
Here is modified version of the code and visualization of computed results: % Equation
a=0.9;
n1=1;
n2=1.5;
f=@(x,w) tan((a*pi/w)*sqrt(n2^2-x.^2)) - sqrt((x.^2-n1^2))./(sqrt(n2^2-x.^2));
% Look for zeros for w=[1.5 5]
W=linspace(1.5,5,1E2);
X=linspace(n1+1E-12,n2-1E-12,1E3);
Z=0.5*(n1+n2)*ones(size(W));
Opt=optimset('Display','off');
figure('color','w')
for i=1:numel(W)
% Initialize search
F=f(X,W(i));
[F_min,id]=min(abs(F));
Z(i)=X(id);
% Refine solution
f_i=@(x) f(x,W(i));
[Zi,Fi]=fzero(f_i,Z(i),Opt);
if abs(Fi)<abs(F_min), Z(i)=Zi; end
plot(W(i),Z(i),'.k','MarkerSize',20)
hold on
end
plot(W,Z,'-b')
xlabel('w','FontSize',25,'FontWeight','bold')
ylabel('b','FontSize',25,'FontWeight','bold')
set(gca,'FontSize',20)
grid on

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Answer by Walter Roberson

### Walter Roberson (view profile)

on 30 Jul 2018

Stop closing your questions that people have already answered!!
We are not free private consultants: public questions and public answers are the price we "charges". We answer publicly as volunteers so that everyone can benefit. When you close questions after people have answered, then you are taking advantage of what you learned but you are grabbing it away to prevent anyone else from benefiting from the time and knowledge that other people have contributed.

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