The Complex-Step Newton Method is a numerical root-finding method that combines Newton iterations with complex-step derivative approximations. By avoiding the numerical cancellation errors that appear in finite-difference methods, the approach provides highly accurate derivative estimates and improved computational reliability. The Complex step iteration can be expressed as the equation or system of equations of the form:
where the product between Jacobian matrix and the vector is approximated by the complex-step Jacobian approximation for small complex-step parameter h>0.
Therefore, the corresponding linear system can be solved using Krylov methods without forming the Jacobian but using instead this matrix-vector product approximation.
An important feature of the method is its convergence behavior. For systems of nonlinear equations, the method retains quadratic convergence even for relatively large values of the complex-step parameter. In the scalar case, the convergence is generally first order for finite values of the complex step, but approaches quadratic convergence as the complex-step parameter h tends to zero. In practice, this does not present a significant limitation, since the complex-step approximation remains numerically stable even for extremely small values of h, unlike classical finite-difference methods which suffer from subtractive cancellation errors.
My work on this method includes theoretical analysis, numerical implementation, and the development of open-source software tools for scientific computing applications. The project is motivated by the broader goal of designing accurate and efficient numerical methods for nonlinear problems arising in applied mathematics and engineering.
An open-source implementation of the method has also been developed in C++ with Python interface and can be accessed in PyPi and in GitHub
References:
D. Mitsotakis, The complex-step Newton method and its convergence, Numerische Mathematik, 157(2025), 993-1021 DOI:10.1007/s00211-025-01471-w (PDF) (Appendix)
D. Mitsotakis, Computational Mathematics: An Introduction to Numerical Analysis and Scientific Computing with Python, Chapman and Hall/CRC New York, 2023, DOI: 10.1201/9781003287292
In other words, the complex-step Newton method is the defined through the iteration