MATLAB Implementation of Steffensen's Iteration Method with Source Code

Steffensen's iteration method is a root-finding algorithm that accelerates convergence compared to standard fixed-point iteration methods. This implementation provides faster convergence than similar methods like Newton-Raphson, while avoiding the need for derivative calculations. The MATLAB implementation typically involves creating a function that accepts several parameters: an initial guess (x0), the target function (f), maximum iteration count (max_iter), and tolerance level (tol). The core algorithm uses Aitken's delta-squared process to accelerate convergence through the formula: x_{n+1} = x_n - (f(x_n)^2) / (f(x_n + f(x_n)) - f(x_n)). Key implementation considerations include: - Handling potential division by zero when the denominator approaches zero - Implementing convergence checks using absolute or relative error thresholds - Adding safeguards against infinite loops through iteration counters - Validating input parameters and function handles The code structure generally follows this pattern: 1. Initialize variables and validate inputs 2. Implement the main iteration loop with convergence checking 3. Apply Steffensen's acceleration formula at each step 4. Include error handling for numerical instability cases 5. Return the final approximation and convergence statistics Proper testing should verify behavior with various functions, initial guesses, and tolerance levels to ensure robustness before deployment in practical applications.