Iterative Methods Compared: Newton, Secant, Modified Newton & Steffensen

1. Newton's Method

Newton's method is an iterative technique based on derivative information. It uses the tangent line at the current iterate to approximate the root, derived from the Taylor expansion of the function.

Iteration formula:

Newton's method is suitable when the derivative is easy to compute and the initial guess is close to the solution, especially in engineering optimization and physical simulation.

2. Secant Method

The Secant method is a derivative‑free variant of Newton's method. It constructs a secant line through two initial points and uses its intersection with the axis as the next iterate, avoiding derivative calculations.

Iteration formula:

The Secant method works well when derivatives are difficult or impossible to compute and is less sensitive to the choice of initial values.

3. Modified Newton Method

The Modified Newton method (also called Simplified Newton) fixes the derivative at the initial point, computing it only once to reduce per‑iteration cost.

Iteration formula:

This approach is advantageous when derivative evaluations are expensive, such as in large‑scale simulations or optimization problems.

4. Steffensen Method

Steffensen's method accelerates a linearly convergent iteration using Aitken's Δ² process, achieving near‑quadratic convergence without requiring derivatives.

Iteration formula:

It is useful for speeding up simple iterative schemes.

Method Comparison

Newton's Method : requires derivative, quadratic convergence, high sensitivity to initial guess, fast and accurate when applicable.

Secant Method : derivative‑free, linear convergence, moderate sensitivity, slower but broadly applicable.

Modified Newton Method : derivative computed once, convergence between linear and quadratic, moderate sensitivity, saves computation when derivatives are costly.

Steffensen Method : derivative‑free, near‑quadratic convergence, moderate sensitivity, fast for simple iterations but limited applicability.

Case Study

A sample nonlinear function combining exponential and quadratic terms was solved using all four methods. Convergence speed and accuracy were plotted.

The results show that Newton's method (identical to the Modified Newton plot) and Steffensen's method converge fastest when a good initial guess is provided, while the Secant method converges more slowly despite not requiring derivative information. Selecting the appropriate method depends on the problem's complexity and the availability of derivative information.