How to plot a vertical line at f(x)=0.5

Hello,

I am new to matlab and have some problems plotting the correct graphs. Here are my questions:

(1) How can I determine the x value at which it holds that my defined function f(x)=0.5? I want to draw a vertical line at this value and then colour the area to the right of the vertical line and below a function in a colour.

(2) How can I determine the y value at which it holds that the function is defined as follows f1 = f(0.5) with x=0.5. I want to draw a horizontal line at this value and color the area above and to the left of a function.

Maybe someone has an idea on how to define this in matlab to get the correct plots?

Thankful for any help!

Best AK

Answer 1

[Star Strider]

Use the xline and either the patch or fill functions:

xv = 0.5;
yv = 0.5;
figure
hold on
xline(xv)
yline(yv)
patch([[0 xv] fliplr([0 xv])], [[0 0] fliplr([yv yv])], 'g')
patch([[xv 1] fliplr([xv 1])], [[0 0] fliplr([yv yv])], 'r')
hold off
axis([0  1   0  1])

That illustrates the approach.

If you have a specific funciton and data to plot, I will help you adapt this code to that problem.

.

[AK]

Hello Star Strider,

thank you so much for the fast answer. I tried using xline and yline but for this I need to determine the specific x value at which f(x)=0.5. I can read this from the graph but is there a function that allows to get in this case phi1StarArr1 = 0.5? I attached my drawing from the ipad so that you can see how I would like it to look like.

Would be so thankful for an advice :)

p = 0.25;
thetaArr = [1.4; 2]

phi2 = 0.5:0.01:1;

for i=1:2
    theta = thetaArr(i)

    phi1StarArr1 = zeros(length(phi2), 1);
    phi1StarArr2 = zeros(length(phi2), 1);
    phi1StarArr3 = zeros(length(phi2), 1);
    phi1 = 0.5:0.01:1;
    

    for i = 1:length(phi2)
        phi1StarArr1(i) = phi1Star(phi2(i), 1-p, p, theta);
    end

    for i = 1:length(phi2)
        phi1StarArr2(i) = phi1Star(phi2(i), p, p, theta);
    end

    for i = 1:length(phi2)
        phi1StarArr3(i) = phi1Star(phi2(i), p, 1-p, theta);
    end
    
    
    figure
    plot(phi2, phi1StarArr1, 'color', 'g')
    hold on
    plot(phi2, phi1StarArr2, 'color', 'b')
    plot(phi2, phi1StarArr3, 'color', 'r')
    xlabel('\phi_2')
    ylabel('\phi_1', 'Rotation', 0)
    tit = sprintf('Focusing Areas for theta = %.1f', round(theta,1))
    title(tit)
    axis([0.5 1.0 0.5 1.0])
    
    % New functions
    plot(phi2, phi1, 'color', 'k')  

    hold off
end

function res = phi1Star(phi2, x, xPrime, theta)
    function g2_res = g_in_phi1(phi1)
           g2_res = g(phi1, phi2, x, xPrime, theta);
    end
    
    fun = @g_in_phi1;
    root = fzero(fun, 0.7);
    
    if root >= 0.5 & root <= 1.0
        res = root;
    else
        res = 0.5;
    end
end

% Funktion g
function g_res = g(phi1, phi2, x, xPrime, theta)
    g_res = (f(phi1,x) - f(1-phi1, xPrime))*theta - (f(phi2,1-x) - f(1-phi2, 1-xPrime));
end

% Funktion f
function f_res = f(y,z)
    f_res = y*z/(y*z + (1-y)*(1-z));
end

[Star Strider]

Try this —

p = 0.25;
thetaArr = [1.4; 2]

thetaArr = 2×1

1.4000 2.0000

phi2 = 0.5:0.01:1;

for i1=1:2
    theta = thetaArr(i1)

    phi1StarArr1 = zeros(length(phi2), 1);
    phi1StarArr2 = zeros(length(phi2), 1);
    phi1StarArr3 = zeros(length(phi2), 1);
    phi1 = 0.5:0.01:1;
    

    for i = 1:length(phi2)
        phi1StarArr1(i) = phi1Star(phi2(i), 1-p, p, theta);
    end

    for i = 1:length(phi2)
        phi1StarArr2(i) = phi1Star(phi2(i), p, p, theta);
    end

    for i = 1:length(phi2)
        phi1StarArr3(i) = phi1Star(phi2(i), p, 1-p, theta);
    end
    
    
    figure
    plot(phi2, phi1StarArr1, 'color', 'g')
    hold on
    plot(phi2, phi1StarArr2, 'color', 'b')
    plot(phi2, phi1StarArr3, 'color', 'r')
    yv1 = phi1StarArr3(1);                                                      % Lower Red Border
    yline(yv1)                                                                  % Added
    xvix = find(diff(sign(phi1StarArr1-0.5)));                                  % Reference Index — Green Left Border
    idxrng = (0:1)+xvix;                                                        % Index Range For Interpolation
    xv1 = interp1(phi1StarArr1(idxrng),phi2(idxrng),0.5)                        % Interpolate
    xv1 = xv1.*(i1==1) + 1.*(i1==2)                                             % Conditionally Set 'xv1'
    xline(xv1)
    xlabel('\phi_2')
    ylabel('\phi_1', 'Rotation', 0)
    tit = sprintf('Focusing Areas for theta = %.1f', round(theta,1))
    title(tit)
    axis([0.5 1.0 0.5 1.0])
    patch([phi2 fliplr(phi2)], [max([ones(size(phi2))*yv1; phi1]) ones(size(phi2))*max(ylim)], 'r', 'EdgeColor','none', 'FaceAlpha',0.25)
    phi2xv1Lv = phi2 >= xv1;                                                    % Define 'phi2' Subset For Green Patch
    patch([phi2(phi2xv1Lv) fliplr(phi2(phi2xv1Lv))], [phi1StarArr2(phi2xv1Lv); ones(size(phi1StarArr2(phi2xv1Lv)))*min(ylim)].', 'g', 'EdgeColor','none', 'FaceAlpha',0.25)
    % New functions
    plot(phi2, phi1, 'color', 'k')  

    hold off
end

theta = 1.4000

xv1 = 0.9400

xv1 = 0.9400

tit = 'Focusing Areas for theta = 1.4'

theta = 2

xv1 = 0.9900

xv1 = 1

tit = 'Focusing Areas for theta = 2.0'

function res = phi1Star(phi2, x, xPrime, theta)
    function g2_res = g_in_phi1(phi1)
           g2_res = g(phi1, phi2, x, xPrime, theta);
    end
    
    fun = @g_in_phi1;
    root = fzero(fun, 0.7);
    
    if root >= 0.5 & root <= 1.0
        res = root;
    else
        res = 0.5;
    end
end

% Funktion g
function g_res = g(phi1, phi2, x, xPrime, theta)
    g_res = (f(phi1,x) - f(1-phi1, xPrime))*theta - (f(phi2,1-x) - f(1-phi2, 1-xPrime));
end

% Funktion f
function f_res = f(y,z)
    f_res = y*z/(y*z + (1-y)*(1-z));
end

Make appropriate changes to get the result you want.

.

[AK]

Thank you so much! This is amazing :)

I just have one small problem when I change p from 0.25 to 0.2, the second graph for "focusing areas for theta 2.0" does not show the red coloured area anymore for some reason. Do I have to adjust something else then?

[Star Strider]

My pleasure! Thank you!

The problem is that in that situation, ‘phi1StarArr1’ does not change (it is always 0.5), so ‘xvix’ is empty, and the interpolation fails. This throws an error and interrupts the execution of the rest of the code. I added a test so that if ‘xvix’ is empty (there are no changes in ‘phi1StarArr1’ so there are no sign changes to test for), my code skips the interpolation entirely, and the green patch region is then not plotted.

I have no idea how to make it more robust since I do not know what you want to change. Clearly, not all values of ‘p’ are going to work with my changes to the original code, because the intepolation to calculate the ‘x’ axis location for the xline call and the left border of the green patch region, will fail in some instances. It seems to work for , however I cannot guarantee that it will work for all values of ‘p’.

p = 0.2;
thetaArr = [1.4; 2]

thetaArr = 2×1

1.4000 2.0000

phi2 = 0.5:0.01:1;

for i1=1:2
    theta = thetaArr(i1)

    phi1StarArr1 = zeros(length(phi2), 1);
    phi1StarArr2 = zeros(length(phi2), 1);
    phi1StarArr3 = zeros(length(phi2), 1);
    phi1 = 0.5:0.01:1;
    

    for i = 1:length(phi2)
        phi1StarArr1(i) = phi1Star(phi2(i), 1-p, p, theta);
    end
% phi1StarArr1 
% CheckLimits = [min(phi1StarArr1), max(phi1StarArr1)]
    for i = 1:length(phi2)
        phi1StarArr2(i) = phi1Star(phi2(i), p, p, theta);
    end
% phi1StarArr2
    for i = 1:length(phi2)
        phi1StarArr3(i) = phi1Star(phi2(i), p, 1-p, theta);
    end
% phi1StarArr3
    
    figure
    plot(phi2, phi1StarArr1, 'color', 'g')
    hold on
    plot(phi2, phi1StarArr2, 'color', 'b')
    plot(phi2, phi1StarArr3, 'color', 'r')
    yv1 = phi1StarArr3(1);                                                      % Lower Red Border
    yline(yv1)                                                                  % Added
    xvix = find(diff(sign(0.5-phi1StarArr1)));                                  % Reference Index — Green Left Border
    if ~isempty(xvix)
        idxrng = (0:1)+xvix;                                                    % Index Range For Interpolation
        xv1 = interp1(phi1StarArr1(idxrng),phi2(idxrng),0.5, 'pchip','extrap'); % Interpolate
        xline(xv1)
    end
    xlabel('\phi_2')
    ylabel('\phi_1', 'Rotation', 0)
    tit = sprintf('Focusing Areas for theta = %.1f', round(theta,1))
    title(tit)
    axis([0.5 1.0 0.5 1.0])
    patch([phi2 fliplr(phi2)], [max([ones(size(phi2))*yv1; phi1]) ones(size(phi2))*max(ylim)], 'r', 'EdgeColor','none', 'FaceAlpha',0.25)
    phi2xv1Lv = phi2 >= xv1;                                                    % Define 'phi2' Subset For Green Patch
    if ~isempty(xvix)
        patch([phi2(phi2xv1Lv) fliplr(phi2(phi2xv1Lv))], [phi1StarArr2(phi2xv1Lv); ones(size(phi1StarArr2(phi2xv1Lv)))*min(ylim)].', 'g', 'EdgeColor','none', 'FaceAlpha',0.25)
    end
        % New functions
    plot(phi2, phi1, 'color', 'k')  

    hold off
end

theta = 1.4000

tit = 'Focusing Areas for theta = 1.4'

theta = 2

tit = 'Focusing Areas for theta = 2.0'

function res = phi1Star(phi2, x, xPrime, theta)
    function g2_res = g_in_phi1(phi1)
           g2_res = g(phi1, phi2, x, xPrime, theta);
    end
    
    fun = @g_in_phi1;
    root = fzero(fun, 0.7);
    
    if root >= 0.5 & root <= 1.0
        res = root;
    else
        res = 0.5;
    end
end

% Funktion g
function g_res = g(phi1, phi2, x, xPrime, theta)
    g_res = (f(phi1,x) - f(1-phi1, xPrime))*theta - (f(phi2,1-x) - f(1-phi2, 1-xPrime));
end

% Funktion f
function f_res = f(y,z)
    f_res = y*z/(y*z + (1-y)*(1-z));
end

.

[AK]

Dear Star Strider,

I have a change that I want to make to the graphs but I am not sure how.

Update: Removing the dotted lines and instead, add an additional line flu(φ2). For this, I need to define

h(φ1 , φ2 , x, x′ ) := [f (φ1 , x) − f (1 − φ1 , 1 − x′ )]θ + [f (φ2 , 1 − x) − f (1 − φ2 , x′ )] Like above, for a fixed φ2 ∈ [0.5,1] define φ1∗∗(x,x′) ∈ [0.5,1] as the unique root of h(·, φ2, x, x′). Shades are adjusted as indicated in the figure.

flu(φ2)=φ1∗∗(p,1−p)

This is how I would change the code but I am not sure how to plot the purple line and change the patches.

p = 0.25;
thetaArr = [1.4; 2]
phi2 = 0.5:0.01:1;

for i1=1:2
    theta = thetaArr(i1)
    phi1StarArr1 = zeros(length(phi2), 1);
    phi1StarArr2 = zeros(length(phi2), 1);
    phi1StarArr3 = zeros(length(phi2), 1);
    phi1StarArr4 = zeros(length(phi2), 1); %Update for line fl^u
    phi1 = 0.5:0.01:1;
    
    for i = 1:length(phi2)
        phi1StarArr1(i) = phi1Star(phi2(i), 1-p, p, theta); %Line fl^s2
    end
    for i = 1:length(phi2)
        phi1StarArr2(i) = phi1Star(phi2(i), p, p, theta); %Line fl^2
    end
    for i = 1:length(phi2)
        phi1StarArr3(i) = phi1Star(phi2(i), p, 1-p, theta); %Line fl^s1
    end
    for i = 1:length(phi2)
        phi1StarArr4(i) = phi1StarStar(phi2(i), p, 1-p, theta); %Line fl^u
    end
    
    
    figure
    plot(phi2, phi1StarArr1, 'color', 'g')
    hold on
    plot(phi2, phi1StarArr2, 'color', 'b')
    plot(phi2, phi1StarArr3, 'color', 'r')
    yv1 = phi1StarArr3(1);                                                      % Lower Red Border                                                                 
    xvix = find(diff(sign(phi1StarArr1-0.5)));                                  % Reference Index — Green Left Border
    idxrng = (0:1)+xvix;                                                        % Index Range For Interpolation
    xv1 = interp1(phi1StarArr1(idxrng),phi2(idxrng),0.5)                        % Interpolate
    xv1 = xv1.*(i1==1) + 1.*(i1==2)                                             % Conditionally Set 'xv1'
    plot(xv1, yv1, 'color', 'k')
    
    %Axis labels
    xlabel('\phi_2')
    ylabel('\phi_1', 'Rotation', 0)
    tit = sprintf('Focusing Areas for theta = %.1f', round(theta,1))
    title(tit)
    axis([0.5 1.0 0.5 1.0])
    
    %Colored patches
    patch([phi2 fliplr(phi2)], [max([ones(size(phi2))*yv1; phi1]) ones(size(phi2))*max(ylim)], 'r', 'EdgeColor','none', 'FaceAlpha',0.25)
    phi2xv1Lv = phi2 >= xv1;                                                    % Define 'phi2' Subset For Green Patch
    patch([phi2(phi2xv1Lv) fliplr(phi2(phi2xv1Lv))], [phi1StarArr2(phi2xv1Lv); ones(size(phi1StarArr2(phi2xv1Lv)))*min(ylim)].', 'g', 'EdgeColor','none', 'FaceAlpha',0.25)
    
    % Function von (0.5/0.5) bis (1/1)
    plot(phi2, phi1, 'color', 'k')  
    hold off
end

%Unique root of g
function res = phi1Star(phi2, x, xPrime, theta)
    function g2_res = g_in_phi1(phi1)
           g2_res = g(phi1, phi2, x, xPrime, theta);
    end
    
    fun = @g_in_phi1;
    root = fzero(fun, 0.7);
    
    if root >= 0.5 & root <= 1.0
        res = root;
    else
        res = 0.5;
    end
end

%Unique root of h
function res2 = phi1StarStar(phi2, x, xPrime, theta)
    function h2_res2 = h_in_phi1(phi1)
           h2_res2 = h(phi1, phi2, x, xPrime, theta);
    end
    
    fun = @h_in_phi1;
    root = fzero(fun, 0.7);
    
    if root >= 0.5 & root <= 1.0
        res2 = root;
    end
end

% Funktion g
function g_res = g(phi1, phi2, x, xPrime, theta)
    g_res = (f(phi1,x) - f(1-phi1, xPrime))*theta - (f(phi2,1-x) - f(1-phi2, 1-xPrime));
end

% Funktion h
function h_res = h(phi1, phi2, x, xPrime, theta)
    h_res = (f(phi1,x) - f(1-phi1, 1-xPrime))*theta + (f(phi2,1-x) - f(1-phi2, xPrime));
end

% Funktion f
function f_res = f(y,z)
    f_res = y.*z ./(y.*z + (1-y).*(1-z));
end

Do you maybe have an idea on the changes I need to make?

Best AS

[Star Strider]

The code is generating an error because in the ‘phi1StarStar’ function, ‘res2’ is not always assigned. I worked around that by assigning it a NaN default value so something is always assigned to it, and added a comment on that line.

However, the plots I am getting do not look the same as the one you posted, so I cannot go farther until that problem is resolved. The second straight line is missing, and the blue (or violet) line does not appear to be in the correct location.

p = 0.25;
thetaArr = [1.4; 2]

thetaArr = 2×1

1.4000 2.0000

phi2 = 0.5:0.01:1;

for i1=1:2
    theta = thetaArr(i1)
    phi1StarArr1 = zeros(length(phi2), 1);
    phi1StarArr2 = zeros(length(phi2), 1);
    phi1StarArr3 = zeros(length(phi2), 1);
    phi1StarArr4 = zeros(length(phi2), 1); %Update for line fl^u
    phi1 = 0.5:0.01:1;
    
    for i = 1:length(phi2)
        phi1StarArr1(i) = phi1Star(phi2(i), 1-p, p, theta); %Line fl^s2
    end
    for i = 1:length(phi2)
        phi1StarArr2(i) = phi1Star(phi2(i), p, p, theta); %Line fl^2
    end
    for i = 1:length(phi2)
        phi1StarArr3(i) = phi1Star(phi2(i), p, 1-p, theta); %Line fl^s1
    end
    for i = 1:length(phi2)
        phi1StarArr4(i) = phi1StarStar(phi2(i), p, 1-p, theta); %Line fl^u
    end
    
    
    figure
    plot(phi2, phi1StarArr1, 'color', 'g')
    hold on
    plot(phi2, phi1StarArr2, 'color', 'b')
    plot(phi2, phi1StarArr3, 'color', 'r')
    yv1 = phi1StarArr3(1);                                                      % Lower Red Border                                                                 
    xvix = find(diff(sign(phi1StarArr1-0.5)));                                  % Reference Index — Green Left Border
    idxrng = (0:1)+xvix;                                                        % Index Range For Interpolation
    xv1 = interp1(phi1StarArr1(idxrng),phi2(idxrng),0.5)                        % Interpolate
    xv1 = xv1.*(i1==1) + 1.*(i1==2)                                             % Conditionally Set 'xv1'
    plot(xv1, yv1, 'color', 'k')
    
    %Axis labels
    xlabel('\phi_2')
    ylabel('\phi_1', 'Rotation', 0)
    tit = sprintf('Focusing Areas for theta = %.1f', round(theta,1))
    title(tit)
    axis([0.5 1.0 0.5 1.0])
    
    %Colored patches
    patch([phi2 fliplr(phi2)], [max([ones(size(phi2))*yv1; phi1]) ones(size(phi2))*max(ylim)], 'r', 'EdgeColor','none', 'FaceAlpha',0.25)
    phi2xv1Lv = phi2 >= xv1;                                                    % Define 'phi2' Subset For Green Patch
    patch([phi2(phi2xv1Lv) fliplr(phi2(phi2xv1Lv))], [phi1StarArr2(phi2xv1Lv); ones(size(phi1StarArr2(phi2xv1Lv)))*min(ylim)].', 'g', 'EdgeColor','none', 'FaceAlpha',0.25)
    
    % Function von (0.5/0.5) bis (1/1)
    plot(phi2, phi1, 'color', 'k')  
    hold off
end

theta = 1.4000

xv1 = 0.9400

xv1 = 0.9400

tit = 'Focusing Areas for theta = 1.4'

theta = 2

xv1 = 0.9900

xv1 = 1

tit = 'Focusing Areas for theta = 2.0'

%Unique root of g
function res = phi1Star(phi2, x, xPrime, theta)
    function g2_res = g_in_phi1(phi1)
           g2_res = g(phi1, phi2, x, xPrime, theta);
    end
    
    fun = @g_in_phi1;
    root = fzero(fun, 0.7);
    
    if root >= 0.5 & root <= 1.0
        res = root;
    else
        res = 0.5;
    end
end

%Unique root of h
function res2 = phi1StarStar(phi2, x, xPrime, theta)
    function h2_res2 = h_in_phi1(phi1)
           h2_res2 = h(phi1, phi2, x, xPrime, theta);
    end
    
    fun = @h_in_phi1;
    root = fzero(fun, 0.7);
    res2 = NaN;                                         % Added To Avoid An Error When 'res2' Is Not Assigned
    if root >= 0.5 & root <= 1.0
        res2 = root;
    end
end

% Funktion g
function g_res = g(phi1, phi2, x, xPrime, theta)
    g_res = (f(phi1,x) - f(1-phi1, xPrime))*theta - (f(phi2,1-x) - f(1-phi2, 1-xPrime));
end

% Funktion h
function h_res = h(phi1, phi2, x, xPrime, theta)
    h_res = (f(phi1,x) - f(1-phi1, 1-xPrime))*theta + (f(phi2,1-x) - f(1-phi2, xPrime));
end

% Funktion f
function f_res = f(y,z)
    f_res = y.*z ./(y.*z + (1-y).*(1-z));
end

.

[AK]

Hello Star Strider,

thanks a lot for discovering the problem with the ‘phi1StarStar’ function. I do get the same graphs:

Now I just need to figure out how to plot the purple line defined by function fl^u by having defined function h as above.

for i = 1:length(phi2)
        phi1StarArr4(i) = phi1StarStar(phi2(i), p, 1-p, theta); %Line fl^u
    end

Do you have an idea how I can make the graph look like this?

Thanks so much already!

Best AS

[Star Strider]

I am lost.

Where does this go:

for i = 1:length(phi2)

phi1StarArr4(i) = phi1StarStar(phi2(i), p, 1-p, theta); %Line fl^u

end

in the code and in the plot calls?

I assume that the plot call using it would be:

plot(phi2, phi1StarArr4, '-m')

however I do not see that anywhere.

Perhaps —

p = 0.25;
thetaArr = [1.4; 2]

thetaArr = 2×1

1.4000 2.0000

phi2 = 0.5:0.01:1;

for i1=1:2
    theta = thetaArr(i1)
    phi1StarArr1 = zeros(length(phi2), 1);
    phi1StarArr2 = zeros(length(phi2), 1);
    phi1StarArr3 = zeros(length(phi2), 1);
    phi1StarArr4 = zeros(length(phi2), 1); %Update for line fl^u
    phi1 = 0.5:0.01:1;
    
    for i = 1:length(phi2)
        phi1StarArr1(i) = phi1Star(phi2(i), 1-p, p, theta); %Line fl^s2
    end
    for i = 1:length(phi2)
        phi1StarArr2(i) = phi1Star(phi2(i), p, p, theta); %Line fl^2
    end
    for i = 1:length(phi2)
        phi1StarArr3(i) = phi1Star(phi2(i), p, 1-p, theta); %Line fl^s1
    end
    for i = 1:length(phi2)
        phi1StarArr4(i) = phi1StarStar(phi2(i), p, 1-p, theta); %Line fl^u
    end
    
    figure
    plot(phi2, phi1StarArr1, 'color', 'g')
    hold on
    plot(phi2, phi1StarArr2, 'color', 'b')
    plot(phi2, phi1StarArr3, 'color', 'r')
    plot(phi2, phi1StarArr4, '-m')                                          % ADDED
    yv1 = phi1StarArr3(1);                                                      % Lower Red Border                                                                 
    xvix = find(diff(sign(phi1StarArr1-0.5)));                                  % Reference Index — Green Left Border
    idxrng = (0:1)+xvix;                                                        % Index Range For Interpolation
    xv1 = interp1(phi1StarArr1(idxrng),phi2(idxrng),0.5)                        % Interpolate
    xv1 = xv1.*(i1==1) + 1.*(i1==2)                                             % Conditionally Set 'xv1'
    plot(xv1, yv1, 'color', 'k')
    
    %Axis labels
    xlabel('\phi_2')
    ylabel('\phi_1', 'Rotation', 0)
    tit = sprintf('Focusing Areas for theta = %.1f', round(theta,1))
    title(tit)
    axis([0.5 1.0 0.5 1.0])
    
    %Colored patches
    patch([phi2 fliplr(phi2)], [max([ones(size(phi2))*yv1; phi1]) ones(size(phi2))*max(ylim)], 'r', 'EdgeColor','none', 'FaceAlpha',0.25)
    phi2xv1Lv = phi2 >= xv1;                                                    % Define 'phi2' Subset For Green Patch
    patch([phi2(phi2xv1Lv) fliplr(phi2(phi2xv1Lv))], [phi1StarArr2(phi2xv1Lv); ones(size(phi1StarArr2(phi2xv1Lv)))*min(ylim)].', 'g', 'EdgeColor','none', 'FaceAlpha',0.25)
    
    % Function von (0.5/0.5) bis (1/1)
    plot(phi2, phi1, 'color', 'k')  
    hold off
end

theta = 1.4000

phi1StarArr4 = 51×1

0.5000 NaN NaN NaN NaN NaN NaN NaN NaN NaN

xv1 = 0.9400

xv1 = 0.9400

tit = 'Focusing Areas for theta = 1.4'

theta = 2

phi1StarArr4 = 51×1

0.5000 NaN NaN NaN NaN NaN NaN NaN NaN NaN

xv1 = 0.9900

xv1 = 1

tit = 'Focusing Areas for theta = 2.0'

%Unique root of g
function res = phi1Star(phi2, x, xPrime, theta)
    function g2_res = g_in_phi1(phi1)
           g2_res = g(phi1, phi2, x, xPrime, theta);
    end
    
    fun = @g_in_phi1;
    root = fzero(fun, 0.7);
    
    if root >= 0.5 & root <= 1.0
        res = root;
    else
        res = 0.5;
    end
end

%Unique root of h
function res2 = phi1StarStar(phi2, x, xPrime, theta)
    function h2_res2 = h_in_phi1(phi1)
           h2_res2 = h(phi1, phi2, x, xPrime, theta);
    end
    
    fun = @h_in_phi1;
    root = fzero(fun, 0.7);
    res2 = NaN;                                         % Added To Avoid An Error When 'res2' Is Not Assigned
    if root >= 0.5 & root <= 1.0
        res2 = root;
    end
end

% Funktion g
function g_res = g(phi1, phi2, x, xPrime, theta)
    g_res = (f(phi1,x) - f(1-phi1, xPrime))*theta - (f(phi2,1-x) - f(1-phi2, 1-xPrime));
end

% Funktion h
function h_res = h(phi1, phi2, x, xPrime, theta)
    h_res = (f(phi1,x) - f(1-phi1, 1-xPrime))*theta + (f(phi2,1-x) - f(1-phi2, xPrime));
end

% Funktion f
function f_res = f(y,z)
    f_res = y.*z ./(y.*z + (1-y).*(1-z));
end

It appears that ‘phi1StarArr4’ is almost uniformly NaN because the value of ‘root’ are almost all outside the limits set for it. I have no idea what you are doing, so you need to fix that problem.

.