How do I plot multiple intersections between two functions?

I have been tasked with plotting two functions as well as marking where the two intersect. I also have to use a for or while loop to automatically find all of the intersections in the given domain and print the results of those intersections.

if true
  % code

    %%Find all of the intersections of two functions in the domain of x
     % Variable 3 ≤ x ≤ 8
    % Functions
    F1 = @(x) 90.*exp(-x) - 1;
    F2 = @(x) sin(2.*pi.*x);
    fplot(F1,[3,8]);
    grid on;
    hold on;
    fplot(F2,[3,8]);
for k = 1:7
    int = fzero(@(x) F1(x)-F2(x),k); % The x-coord
    plot(int,F1(int),'*k'); % The coordinates.

    fprintf('Intersection %0.0f is at x = %0.4f\n', k, int);
end
  end

My for loop is not working the way I need it to, so any help would be excellent. Here are a couple hints I have been given:

Hint 1: How do you make an anonymous function that takes one variable (x) and returns zero when the two functions intersect?

Hint 2: You need to call fzero a bunch of times with a reasonable set of guesses, enough to make sure that you actually get all of the intersections. Each time you calculate a new intersection, compare it to ALL of the intersections that you have already calculated. If the difference between the new intersection and any of the old ones is very small (<0.00006), do not add it to your list and just move on to calculating the next one. Don’t forget to check that the intersection is in the given domain.

Answer 1

[Star Strider]

This works:

F1 = @(x) 90.*exp(-x) - 1;
F2 = @(x) sin(2.*pi.*x);

x = linspace(3, 8, 500);                                % Domain ‘x’

fcndif = @(x) F1(x) - F2(x);                            % Function Differences

zx = fcndif(x) .* circshift(fcndif(x), [0 -1]);         % Detect Zero-Crossings
gues = find(zx <= 0);                                   % Find Indices Of Zero Crossings

for k1 = 1:length(gues)
    intsct(k1) = fzero(fcndif, x(gues(k1)));            % Find ‘x’ at Zero Crossings
end

figure(1)
plot(x, F1(x),  x, F2(x))
hold on
plot(intsct, F1(intsct), 'bp')
hold off
grid

[Mark Dillon]

Thanks Star Strider, you have been incredibly helpful for me!

[Mark Dillon]

Also what exactly does circshift do?

[Star Strider]

My pleasure!

The circshift function shifts an array circularly by a specific number of steps in a specific direction. Here, I had it shift the row to the left by one index value. Multiplying that by the unshifted vector creates negative numbers at the zero-crossings. (Experiment with it to see how it works.) The indices of those, referencing values of ‘x’, create the initial guesses for fzero.

[Mark Dillon]

Excellent!

Final question, I am having trouble getting it to print out the proper x values for each of the 9 intersections, the number are not matching up properly.

I also noticed that intersection 7 and 9 are not showing on the plot, is there reason for this?

[Star Strider]

This can be a problem with fzero if similar guesses are too close, since it will converge on the same value. (I thought it had detected all of them when I looked. It’s been a long day.) The solution is to alter the shift value. Here, I changed it to -2:

F1 = @(x) 90.*exp(-x) - 1;
F2 = @(x) sin(2.*pi.*x);

x = linspace(3, 8, 150);                                % Domain ‘x’

fcndif = @(x) F1(x) - F2(x);                            % Function Differences

zx = fcndif(x) .* circshift(fcndif(x), [0 -2]);         % Detect Zero-Crossings
gues = find(zx <= 0);                                   % Find Indices Of Zero Crossings

for k1 = 1:length(gues)
    intsct(k1) = fzero(fcndif, x(gues(k1)));            % Find ‘x’ at Zero Crossings
end
intsct = unique(round(intsct*10^6)./10^6);

figure(1)
plot(x, F1(x),  x, F2(x))
hold on
plot(intsct, F1(intsct), 'bp')
hold off
grid

That caught all of them. I added the rounding and unique call for aesthetics. It simply eliminates duplicate values within the specified tolerance.

[Mark Dillon]

Thank you,

I am still having trouble printing the x values for intersection. Instead of 9 I get 20, a few of them are repeats. All I did was

if true
  % 
for k = 1:length(f)
  int(k) = fzero(D, x(f(k))); % Find ‘x’ at intersections

    % Printing
    fprintf('Intersection Number %0.0f, x = %0.0f\n',k, x(k));
end
  end

I have tried playing around with it, and in it's current form it isn't even outputting the right numbers now. I messed up somewhere...

[Star Strider]

My pleasure.

As for printing, using my code, and putting this just before (or just after) the figure(1) block:

intsnr = [1:length(intsct); intsct;  F1(intsct)];
fprintf(1, 'Intersection Number %.0f: x = %.4f, y = %+.4f\n', intsnr)

Intersection Number 1: x = 4.0857, y = +0.5130
Intersection Number 2: x = 4.5000, y = -0.0002
Intersection Number 3: x = 4.9419, y = -0.3573
Intersection Number 4: x = 5.6175, y = -0.6730
Intersection Number 5: x = 5.8662, y = -0.7450
Intersection Number 6: x = 6.6733, y = -0.8862
Intersection Number 7: x = 6.8211, y = -0.9019
Intersection Number 8: x = 7.7045, y = -0.9594
Intersection Number 9: x = 7.7935, y = -0.9629

If you don’t want the ‘+’ sign but you still want the columns to be aligned, use this:

fprintf(1, 'Intersection Number %.0f: x = %.4f, y = % .4f\n', intsnr)

Note that the space in: % .4f is significant, and leaves a space instead of printing the sign if the value for that descriptor is positive.

Also, you don’t need a loop to print it. Take advantage of MATLAB’s matrix properties with fprintf. (I just realised that I specified 1 or stdout in fprintf out of habit. That used to be required.)