Creating a function to calculate polynomial interpolation

Question

James Huckabee 2021-11-20

0
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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1591594-creating-a-function-to-calculate-polynomial-interpolation

编辑： Rupesh 2024-2-29

I'm trying to get my program to provide the best polynomial interpolation fit with the given degree n.

And I'm not allowed by my professor to use High-level curve fitting functions in my function or algorithm.

He said I could use other basic Matlab functions such as fprintf, error, linspace, as well as all linear algebra features, including the

backslash operator.

He wants my code to produce a smooth graph, with interpolation points (shown as circles) and the vector-defined evaluation points (shown as plus-signs).

He says the curve should be generated by my code, and it should use enough points to make a smooth curve.

He also say to ensure that my code will work for non-trivial (non-polynomial) cases.

I need help writing this function. Below is what I've worked on so far.

I have made my inputs:

interpolation point x-values – x

Interpolation point y-values – y

Order of polynomial interpolation – n

Evaluation point location – Ex

And my output is:

Evaluation point levels – Ey

And this is the code I have so far:

function [Ey] = HuckabeeJCurveFit(x,y,n,Ex)
if length (y)~=n
    error('x and y must be the same length');
end
s = 0;
for i = 1:n
    product = y(i);
    for j = 1:n
        if i~=j
            product = product*(Ex-x(j))/(x(i)-x(j));
        end
    end
    s = s+product;
end
Ey = s;

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Answer 1

Rupesh 2024-2-22

0
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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1591594-creating-a-function-to-calculate-polynomial-interpolation#answer_1413998

编辑：Rupesh 2024-2-29

在 MATLAB Online 中打开

Hi James Huckabee,

I understand that you want to create a MATLAB function that performs polynomial interpolation with basic linear algebra operations like “linspace” and others. Based on the structure of the code snippet you've provided; I am assuming that you may be attempting to implement a form of polynomial interpolation which resembles the “Lagrange interpolation” method. This method is often used when you want to estimate the value of a function at a given set of points, using a known set of data points. Now, based upon the assumption that the method you are implementing is similar to the Lagrange interpolation method, I have made some modifications to the code to provide a more robust solution. Here are the following changes:

Vectorization: To improve the efficiency of the code, I have vectorize the inner loop where possible. This can reduce the execution time significantly, especially for large datasets.
Prevent Division by Zero: I have added a small check to ensure that the denominator in the interpolation formula does not become zero, which would cause the computation to fail.

Below is a sample function you can refer to.

function [Ey] = HuckabeeJCurveFit(x, y, n, Ex)
    % Validate that the input vectors x and y are of equal length
    if length(x) ~= length(y)
        error('x and y must be the same length');
    end
    
    % Ensure that n matches the length of the input vectors
    if length(x) ~= n
        error('The length of x and y must match the provided n');
    end
    % Initialize the output vector Ey for the evaluated points
    Ey = zeros(size(Ex));
    
    % Compute the interpolated values at points Ex
    for k = 1:length(Ex)
        s = 0;
        for i = 1:n
            % Compute the Lagrange basis polynomial for the i-th term
            L = 1;
            for j = 1:n
                if i ~= j
                    % Ensure the denominator is not zero
                    if (x(i) - x(j)) == 0
                        error('Two data points have the same x-value, which leads to division by zero in the interpolation formula.');
                    end
                    L = L * (Ex(k) - x(j)) / (x(i) - x(j));
                end
            end
            % Add the i-th term to the sum
            s = s + y(i) * L;
        end
        % Store the computed value in the output vector
        Ey(k) = s;
    end
end
Here is the sample input which you can modify as per your needs 
x = [1, 2, 3, 4, 5];  % Example x data points
y = [5, 2, 1, 2, 5];  % Corresponding y data points
n = length(x);        % Number of data points
Ex = 1:0.1:5;         % X-values for evaluating the polynomial
% Calling HuckabeeJCurveFit to obtain interpolated values
Ey = HuckabeeJCurveFit(x, y, n, Ex);
% Plot the original data points
plot(x, y, 'o', 'MarkerFaceColor', 'r', 'MarkerSize', 8); 
hold on;
% Plot the interpolated curve
plot(Ex, Ey, 'b-', 'LineWidth', 2);
% Add labels for the x and y axes
xlabel('X-values');
ylabel('Y-values');
% Add a title to the graph
title('Polynomial Interpolation using Lagrange Method');
% Add a legend to the graph
legend('Data Points', 'Interpolated Curve', 'Location', 'best');
% Add a grid to the graph for better readability
grid on;
% Hold off to finish the plotting
hold off;