How can I best compare the max error of b-spline interpolation?

Hello, I am preparing a project for an exam. In a particular script, I am using various methods of cubic b-spline interpolation to best approximate a given function, say .

So i decided that, for each of the 5 methods, I'll use 20 sets of an increasing number of points, from say 50 to 500.

For every method, and for every set of points, I'll extract the maximum distance of the points in the partition of the interval (hmax) and the maximum distance of the spline curve to the exact curve (errmax).

This is the result I get:

Is it a good idea to plot (1/hmax, errmax), so five error curves, each of which contains 20 points? Is there a function in matlab that lets me visualize this better? Any suggestion? (The legend will be added later)

[Mathieu NOE]

Can you share your code ?

[Salvatore Bianco]

%func:

% 1: sin(x)-sin(2x); [-pi, pi]

% 2: exp(x/2)*x; [-2, 2]

% 3: |x| + x/2 - x^2; [-2, 2]

% 4: 1/(1+x^2); [-5,5]

% 5: 1/(1+exp(x)); [-2 2]

func = 1;

k = 20;

h = zeros(5, k);

err = zeros(5, k);

for i=1:5

for j=1:k

l = (5+j)^2;

[h(i, j), err(i, j), ~] = main_spline_interp(l, i, func);

end

end

% error plot

hold on

for i=1:5

semilogy(1./h(i, :), err(i, :));

end

hold on

semilogy(1./h, h.^4);

xlabel('1/h');

ylabel('max absolute error')

[Salvatore Bianco]

Here, the i in the loop indexes the cubic spline interpolation method.

[Mathieu NOE]

you are calling this sub function : main_spline_interp

can you provide it ?

[Salvatore Bianco]

It is a function that, given a number of points, and a flag i =1, ..., 5, returns the max distance of the nodal points and the max distance of the spline to the given function in a sample of points. What I am looking for is a way to plot the errors in a way that can outline their convergence to zero, i. e. I want to visualize that graph better, and I don't know how.

[Mathieu NOE]

I don't understand what your code is doing and also what is the meaning of

extract the maximum distance of the points in the partition of the interval (hmax)

I believe this should be a simple example of spline approximation of your fist test case, vs N points of sampling for the required x range.

you can build your solution on that :

%func: 
% 1: sin(x)-sin(2x); [-pi, pi]
% 2: exp(x/2)*x; [-2, 2]
% 3: |x| + x/2 - x^2; [-2, 2]
% 4: 1/(1+x^2); [-5,5]
% 5: 1/(1+exp(x)); [-2 2]

N_tested = (50:50:500);

for ck = 1:numel(N_tested)
    N = N_tested(ck);
    x = linspace(-pi,pi,N);
    y = sin(x)-sin(2*x);

    % cubic spline interpolation
    yi = interp1(x,y,x,'spline');
    % add your own other secret spline interpolation function here (do a loop ?)

    %figure(ck),plot(x,y,'-*',x,yi,'-*'); % for me

    errmax(ck) = max(abs(y-yi)); % max error
    
end

% plot the rms error vs N
figure
plot(N_tested,errmax,'-*')

Answer 1

[Sandeep]

Hi Salvatore Bianco,

It is my understanding that you expect opinion on plotting five error curves each containing 20 points and suggestion of MATLAB function to visualize in a better way.

Yes, plotting (1/hmax, errmax) is a good approach to visualize the error curves for the different methods and sets of points. This plot will allow you to compare the performance of the methods and observe how the error changes with different numbers of points.

In MATLAB, you can use the plot function to create the error curves. To enhance the visualization, you may consider using a logarithmic scale for the x-axis (1/hmax) and the y-axis (errmax). This can be achieved using the semilogx function, which plots data with a logarithmic x-axis.

A Sample implementation to plot the error curves using semilogx in MATLAB is given below,

% Create a matrix to store the data points for each method
data = zeros(20, 5);

% Loop through each method and set of points
for method = 1:5
    for numPoints = 50:50:500
        % Calculate hmax and errmax for the current method and set of points
        hmax = calculateHmax(method, numPoints);
        errmax = calculateErrmax(method, numPoints);
        
        % Store the values in the data matrix
        data(numPoints/50, method) = errmax;
    end
end

% Plot the error curves using semilogx
figure;
semilogx(1./data(:, 1), data(:, 1), 'r');
hold on;
semilogx(1./data(:, 2), data(:, 2), 'g');
semilogx(1./data(:, 3), data(:, 3), 'b');
semilogx(1./data(:, 4), data(:, 4), 'm');
semilogx(1./data(:, 5), data(:, 5), 'c');
hold off;

% Add labels and title to the plot
xlabel('1/hmax');
ylabel('errmax');
title('Error Curves for Cubic B-spline Interpolation Methods');

% Add a legend to identify the methods
legend('Method 1', 'Method 2', 'Method 3', 'Method 4', 'Method 5');

The Demonstration snippet assumes that you have already implemented the functions calculateHmax and calculateErrmax to compute the maximum distances. You can replace these functions with your own calculations based on the specific methods you are using.

The resulting plot will show the error curves for each method, with the x-axis representing 1/hmax and the y-axis representing errmax. The legend will help differentiate between the different methods.