Index exceeds the number of array elements. Index must not exceed 1.

Hy. I'm trying to code a Hermite interpolation function in MATLAB. However , i'm getting an error of "Index exceeds the number of array elements. Index must not exceed 1." Below is my code . I need to run it using the codes commented at the end of the function. Your suggestions are highly appreciated.

%%-----------------------------------------------------------------------------------------
% Example usage. Please run in command window or new editor
Np=10;
x_j = 0.5 + (1./pi).*asin((2.*(0:Np)./Np)-1);
f_j = 1./ (1+ (x_j.*x_j));
f_prime_j =-(2*x_j)/(x_j.^2 + 1).^2;
% Generate points for interpolation
x_values = linspace(min(x_j), max(x_j), 1000);
% Perform Hermite interpolation
interpolated_values = hermite_interpolation2(x_values, x_j, f_j, f_prime_j);

Index exceeds the number of array elements. Index must not exceed 1.

Error in solution>hermite_interpolation2 (line 44)
interpolated_values = interpolated_values + f_j(j) * H + f_prime_j(j) * tilde_H;

% Plot the results
figure;
plot(x_j, f_j, 'o', 'MarkerSize', 10, 'DisplayName', 'Data Points');
hold on;
plot(x_values, interpolated_values, '-', 'LineWidth', 2, 'DisplayName', 'Hermite Interpolation');
xlabel('x');
ylabel('f(x)');
legend('Location', 'Best');
title('Hermite Interpolation');
grid on;

function interpolated_values = hermite_interpolation2(x, x_j, f_j, f_prime_j)
n = length(f_j) - 1; % Degree of the polynomial

% Initialize the interpolated values
interpolated_values = zeros(size(x));

for j = 1:n+1
    % Compute Lagrange polynomial and its derivative
    L = 1;
    L_prime = 0;
    for i = 1:n+1
        if i ~= j
            L = L .* (x - x_j(i)) / (x_j(j) - x_j(i));
            L_prime = L_prime + 1 / (x_j(j) - x_j(i));
        end
    end
    
    % Compute Hermite polynomials
    H = (1 - 2 * L_prime * (x - x_j(j))) .* L.^2;
    tilde_H = (x - x_j(j)) .* L.^2;
    
    % Add contributions to the interpolated values
    interpolated_values = interpolated_values + f_j(j) * H + f_prime_j(j) * tilde_H;
end
end

Answer 1

[Dyuman Joshi]

There was a missing element-wise division sign in the definition of f_prime_j, causing the said variable to be defined as a scalar.

When you called the function, it tried to access the data in f_prime_j via indices for which values did not exist, thus you got the error.

See the correction below -

% Example usage. Please run in command window or new editor
Np=10;
x_j = 0.5 + (1./pi).*asin((2.*(0:Np)./Np)-1);
f_j = 1./ (1+ (x_j.*x_j));


%% Missing sign 
%%                 v
f_prime_j =-(2*x_j)./(x_j.^2 + 1).^2;


% Generate points for interpolation
x_values = linspace(min(x_j), max(x_j), 1000);
% Perform Hermite interpolation
interpolated_values = hermite_interpolation2(x_values, x_j, f_j, f_prime_j);
% Plot the results
figure;
plot(x_j, f_j, 'o', 'MarkerSize', 10, 'DisplayName', 'Data Points');
hold on;
plot(x_values, interpolated_values, '-', 'LineWidth', 2, 'DisplayName', 'Hermite Interpolation');
xlabel('x');
ylabel('f(x)');
legend('Location', 'Best');
title('Hermite Interpolation');
grid on;

function interpolated_values = hermite_interpolation2(x, x_j, f_j, f_prime_j)
n = length(f_j) - 1; % Degree of the polynomial

% Initialize the interpolated values
interpolated_values = zeros(size(x));

for j = 1:n+1
    % Compute Lagrange polynomial and its derivative
    L = 1;
    L_prime = 0;
    for i = 1:n+1
        if i ~= j
            L = L .* (x - x_j(i)) / (x_j(j) - x_j(i));
            L_prime = L_prime + 1 / (x_j(j) - x_j(i));
        end
    end
    
    % Compute Hermite polynomials
    H = (1 - 2 * L_prime * (x - x_j(j))) .* L.^2;
    tilde_H = (x - x_j(j)) .* L.^2;
    
    % Add contributions to the interpolated values
    interpolated_values = interpolated_values + f_j(j) * H + f_prime_j(j) * tilde_H;
end
end

[Panda05]

Thank you so much Joshi. May God bless you ! I spent 2 hours looking for a simple ''.''

[Dyuman Joshi]

Thank you and you're welcome!