Cubic Spline Interpolation with First and Second Boundary Conditions

To implement cubic spline interpolation with first and second boundary conditions, the following MATLAB code provides a comprehensive solution: The algorithm begins by defining the input data points and boundary conditions: x = [x1 x2 x3 ... xn]; % Input x-coordinates of data points y = [y1 y2 y3 ... yn]; % Input y-coordinates of data points y1_prime = ...; % First boundary condition (derivative at first point) y2_prime = ...; % Second boundary condition (derivative at last point) The core implementation calculates the spline coefficients through these steps: h = diff(x); % Compute intervals between consecutive x-points mu = h(1:end-1) ./ (h(1:end-1) + h(2:end)); % Calculate mu values for tridiagonal system lambda = 1 - mu; % Compute lambda coefficients d = (6 ./ (h(1:end-1) + h(2:end))) .* (diff(y) ./ diff(x)); % Right-hand side vector d = [y1_prime; d; y2_prime]; % Incorporate boundary conditions into the system A key algorithmic component is constructing and solving the tridiagonal system: A = spdiags([lambda(2:end) 2*ones(n-2,1) mu(1:end-1)], [-1 0 1], n-2, n-2); % Sparse tridiagonal matrix c = A d(2:end-1); % Solve for second derivatives at interior points The interpolation process evaluates the cubic polynomials: xx = linspace(min(x),max(x),num_points); % Generate evaluation points across the domain yy = zeros(size(xx)); % Initialize output array for i = 1:length(xx) j = find(x <= xx(i), 1, 'last'); % Locate the relevant interval if j == n j = j - 1; % Handle edge case at last interval end h_j = x(j+1) - x(j); % Interval width % Cubic polynomial evaluation using calculated coefficients yy(i) = ((x(j+1)-xx(i))^3*c(j) + (xx(i)-x(j))^3*c(j+1)) / (6*h_j) + ... ((y(j)/h_j - h_j*c(j)/6)*(x(j+1)-xx(i)) + (y(j+1)/h_j - h_j*c(j+1)/6)*(xx(i)-x(j))); end This implementation efficiently handles both boundary condition types while maintaining C² continuity, producing smooth and accurate interpolation results suitable for scientific and engineering applications.