Solving systems of equations graphically and finding where they cross.

I've got the following non-linear equation that I need to solve: F(x) = 2x - 3sin(x) +5. I know how to graph it and my code is below. However, I don't know how to find the intersection points. Am I supposed to use fplot or smth?

How can I find the exact coordinates for where y and F(x) intersect?

Could I have found the x-intercepts of F(x) without considering y?

%define x, y and F(x)
x = linspace(-20,20,2000);
F = 2*x-3*sin(x)+5;
y = zeros(size(x));
% plot the graphs on the same graph
hold on 
plot(x,F)
plot(x,y)
% make the graph lookd neater
grid on 
xlabel('x')
ylabel('F(x)')
title(' plot of F(x)')

Answer 1

[Matt J]

x_intersect=fzero(@(x) 2*x-3*sin(x)+5  ,[-20,+20])

x_intersect = -2.8832

[Sumer]

Thanks very much! This is extremely practical.

Answer 2

[Fabio Freschi]

@Matt J solution is the cleanest. I have understood you want to find the graphical intersection. If you want to do it yourself you can calculate the line between the two points before and after the crossing and find the zero of the line.

clear variables, close all

%define x, y and F(x)
x = linspace(-20,20,2000);
F = @(x)2*x-3*sin(x)+5;  % <- anonymous function
y = zeros(size(x));
% plot the graphs on the same graph
hold on 
plot(x,F(x))
plot(x,y)
% make the graph lookd neater
grid on 
xlabel('x')
ylabel('F(x)')
title(' plot of F(x)')

% find where you are crossing the 0
idx = find(diff(sign(F(x))));

% find crossing of linear interpolation of these two points
x0 = x(idx)+F(x(idx))*(x(idx+1)-x(idx))/(F(x(idx+1))-F(x(idx)))

x0 = -2.8997

plot(x0,F(x0),'o');

Additional notes:

• You should also check if by chance, one of your x is a solution, that is F(x) = 0.
• If you have more intersections, idx is a vector and you can loop over the vector length to find all intersections

[Sumer]

Thanks for the solution!

Answer 3

[Walter Roberson]

How can I find the exact coordinates for where y and F(x) intersect?

You cannot. There is no known closed-form expression for the coordinates. The best you can do is get a numeric approximation. You could use fzero or fsolve for that, or vpasolve

[Sumer]

Thanks very much!