On Efficient Iterative Algorithms and Conditioning for the Nonlinear Sylvester Equation

Nonlinear Sylvester‐type matrix equations arise in applications such as control theory, observer design, model reduction, and data‐driven systems, where they model higher order interactions in matrix transformations. Unlike the classical Sylvester equation, the presence of nonlinear terms introduces additional analytical challenges and increased numerical complexity, particularly for large‐scale problems. This work explores a nonlinear Sylvester equation with a polynomial matrix power and develops Newton‐based iterative methods tailored to its structure. These include the classical Newton method, line‐search stabilized variants, an overrelaxed (OSOR) scheme, and a matrix‐free Newton–Krylov approach exploiting structured matrix–vector products. We also present a conditioning and sensitivity analysis based on Fréchet derivatives, yielding explicit normwise, mixed, and componentwise condition numbers and corresponding perturbation bounds. Numerical experiments demonstrate the effectiveness of stabilization and matrix‐free techniques, particularly for ill‐conditioned or strongly nonlinear problems.