finding all roots of a trignometric equation

Can we find all roots of a trignometric equation using matlab?

for e.g., tan(x)-x=0.

Answer 1

[Ameer Hamza]

range of tan(x) is (-inf inf), so this equation has an infinite number of solutions. Also, the solutions to this equation cannot be represented analytically. There is no general way to find multiple solutions to such equations. One solution is to start the numerical solver with several starting points and choose unique values. For example

eq = @(x) tan(x)-x;

x_range = 0:0.1:20;
sols = ones(size(x_range));
for i = 1:numel(sols)
    sols(i) = fsolve(eq, x_range(i));
end
sols = uniquetol(sols, 1e-2);

[Walter Roberson]

The solutions to tan(x)-x tend towards being close to 2*pi apart, so it is possible to generate reasonable starting points for as many points as desired until you start losing too much floating point precision.

However, you will never have enough memory to return them "all".

[Ameer Hamza]

Yes, thats a good observation to give the initial points. The solutions are pi apart from each. Following will also give same result as one in the answer.

eq = @(x) tan(x)-x;

x_range = 0.1:pi:20;
sols = ones(size(x_range));
for i = 1:numel(sols)
    sols(i) = fsolve(eq, x_range(i));
end

Answer 2

[Image Analyst]

Well here's one way. Use fminbnd() to find out where the equation is closest to zero:

x = linspace(-1, 1, 1000);
y = tan(x);
plot(x, x, 'b-', 'LineWidth', 2);
hold on;
plot(x, y, 'r-', 'LineWidth', 2);
grid on;
axis equal

% Use fminbnd() to find where d = 0, which is where tan(x) = x.
xIntersection = fminbnd(@myFunc, -1.5,1.5)
% Put a line there
xline(xIntersection, 'Color', 'g', 'LineWidth', 3);
caption = sprintf('xIntersection = %f', xIntersection);
title(caption, 'FontSize', 20)

function d = myFunc(x)
d = abs(tan(x) - x);
end

You get

xIntersection =

-1.66533453693773e-16