Newton's Method Graphing

Hey everyone, I need some assitance in plotting my code. I have the computational work done, I just need help plotting the values. My goal is to plot the original function, then the approximated points along the original curve in a different color to be able to see the accuracy. I was thinking of creating the plot in the while loop, but ran into issues where multiple graphs would appear, or the variable I wanted to be the y axis would be zero. I was also having problems trying to store the value in the while loop without being constantly overwritten.

Thank you for your time and help!

Below is the code, feel free to edit or use it for whatever you may need.

clc; clear all; close all;
format long

functn = @(x) 2*sin(x) - x %Starting function
interval = 0.001; %Small x-step used to approximate the function derivative
x = 0.01:0.01:1; %Range of x
dydx = (functn(x+interval) - functn(x))/interval; %Definition of a derivative
Root_value = 0.069; % Function value
ytolerance = 1e-6; % The convergence tolerance
initial_guess = 2; % The initial guess for the location of the root x0

x = initial_guess; %Setting x to be the initial guess, using initial guess to compute the function
yerror = Root_value-functn(x); %Computing error
counter = 1; %A basic loop counter
previous_x = 0; %Setting previous x value to be empty, "0"
t = 0; %Iteration counter

fprintf('Iteration#\t\t x\t\tf(x)\t\terror\n') %Output headers

while abs(yerror)>ytolerance %While the absolute value of the error > error tolerance, the loop continues
    
    dydx = (functn(x+interval)-functn(x))/interval; %Definition of a derivative
    fx=functn(x); %Renaming function to something simpler
    dx = fx/dydx; %Original function divided by the derivative
    previous_x=x; %Setting the previous x value to the current x value to progress the loop
    x=x-dx; %Definition of Newton's Method
    yerror = Root_value - functn(x); %Recalculating error
    counter = counter+1; %A basic loop counter
    
        if previous_x==x %Once the previous x value equals the current x value, the loop breaks, and the program ends
            break
        end
        
    t = t+1; %Iteration counter
    fprintf('%d\t\t\t%f\t%f\t%f\n',t,x,fx,yerror) %Outputting all values at each iteration
    
end

[darova]

Can you make a drawing or something?

[Nicholas Fuller]

So this plot is very close to what I am trying to do. Basically, at each iteration, a new point would appear on the graph with the other points still on the graph. I made a slight edit to my code where I am able to get plots and points; however, the plots are overwritten at each iteration of the loop.

The plot was taken from: https://www.mathworks.com/matlabcentral/fileexchange/44487-newton-s-method-animation

For the example where f(x) = 2*sin(x) - x, I know that the plotted root approximation points, the red *, will be very close to each other and almost indistinguishable.

Here is the new code:

clc; 
clear all; 
close all;

format long

functn = @(x) 2*sin(x) -x %Starting function %Starting function
interval = 0.001; %Small x-step used to approximate the function derivative
x = 0.01:0.01:1; %Range of x
dydx = (functn(x+interval) - functn(x))/interval; %Definition of a derivative
Root_value = 0.069; % Function value
ytolerance = 1e-6; % The convergence tolerance
initial_guess = 2; % The initial guess for the location of the root x0

x = initial_guess; %Setting x to be the initial guess, using initial guess to compute the function
yerror = Root_value-functn(x); %Computing error
counter = 1; %A basic loop counter
previous_x = 0; %Setting previous x value to be empty, "0"
t = 0; %Iteration counter


fprintf('Iteration#\t\t x\t\tf(x)\t\terror\n') %Output headers

while abs(yerror)>ytolerance %While the absolute value of the error > error tolerance, the loop continues
    
    dydx = (functn(x+interval)-functn(x))/interval; %Definition of a derivative
    fx=functn(x); %Renaming function to something simpler
    dx = fx/dydx; %Original function divided by the derivative
    previous_x=x; %Setting the previous x value to the current x value to progress the loop
    x=x-dx; %Definition of Newton's Method
    yerror = Root_value - functn(x); %Recalculating error
    counter = counter+1; %A basic loop counter
    
        if previous_x==x %Once the previous x value equals the current x value, the loop breaks, and the program ends
            break
        end
        
    t = t+1; %Iteration counter
    fprintf('%d\t\t\t%f\t%f\t%f\n',t,x,fx,yerror) %Outputting all values at each iteration
    
    %Plotting
    figure
    fplot(functn) %Plotting original funtion
    hold on
    plot(x, 'r', 'Marker', "*") %Plotting approximated root value
    plot(initial_guess, 'k', "Marker","o") %Plotting the initial guess value
    hline = refline(0, 0); %Refernce line to act as x-axis
    hline.Color = 'k';
    
end

hold off

Answer 1

[darova]

Try this way

%Plotting
xx = 0:0.1:1;
figure
plot(xx,functn(xx),'k') %Plotting original funtion
hold on
plot(initial_guess,functn(initial_guess), 'k', "Marker","o") %Plotting the initial guess value
while % condition
    % some code
    plot(x,functn(x), 'r', 'Marker', "*") %Plotting approximated root value
end
hold off

[Nicholas Fuller]

Yep that did thank you so much!