ShaneLogic/chebyshev-barycentric-interpolation

Chebyshev Barycentric Interpolation

Barycentric Formula

Use the barycentric formula (Theorem 4) to evaluate the polynomial p(x) at any point.

Formula Structure

p(x) = (Sum'' [ (-1)^k * f(x_k) / (x - x_k) ]) / (Sum'' [ (-1)^k / (x - x_k) ])

Notation Rules

• Sum'': Summation from k=0 to k=N
• Double prime: Terms for k=0 and k=N (endpoints) must be multiplied by 1/2
• x_k: Chebyshev points
• f(x_k): Function values at Chebyshev points

Special Case

If x is exactly equal to a node x_k, return f(x_k) directly to avoid division by zero.

Implementation

    % Check if x is exactly a node
    [found, idx] = ismembertol(x, x_nodes, 1e-14);
    if found
        p_val = f_values(idx);
        return;
    end
    
    % Compute numerator and denominator
    N = length(x_nodes) - 1;
    numerator = 0;
    denominator = 0;
    
    for k = 0:N
        weight = (-1)^k;
        if k == 0 || k == N
            weight = weight / 2;  % Double prime notation
        end
        
        term = weight / (x - x_nodes(k+1));
        numerator = numerator + term * f_values(k+1);
        denominator = denominator + term;
    end
    
    p_val = numerator / denominator;
end

Variables

• x: Evaluation point (scalar)
• x_k: Array of Chebyshev points
• f(x_k): Array of function values at Chebyshev points

When to Use

• Evaluating Chebyshev polynomial interpolants
• Understanding Chebfun's underlying interpolation method
• Implementing custom polynomial interpolation
• Analyzing interpolation error and stability

Notes

• The barycentric formula is numerically stable
• Avoids explicit polynomial coefficient computation
• Handles the endpoint weights correctly via the double prime notation
• Direct evaluation at nodes prevents division by zero