A fast-converging Newton-based iterative scheme for the algebraic Riccati equation with step-size optimization
Bulugu Ndulu Batume, Chacha Stephen Chacha · Results in Control and Optimization · 2025
We present a matrix-free Newton–Krylov solver with exact line search (Algorithm 3) for algebraic Riccati equations and benchmark it against standard Newton variants and common baselines. Using the enhancement percentage metric, EP ( % ) = 100 ( 1 − Proposed / Other ) , the method delivers consistent and often dramatic gains across problem sizes and settings. On large problems ( n = 100 ), wall-clock time improves by 99.8–99.9% relative to classical Newton methods while achieving up to 97% EP in accuracy (final/relative residuals). In an aircraft control instance ( n = 70 , m = 35 ), Algorithm 3 attains 99.6% EP in time, reduces iterations by 50–57%, and improves residuals by 82–98%. For large diagonal families ( n = 500 , 1000 , 2000 ), Algorithm 3 converges in approximately 5 Newton steps with predictable scaling (about 0.30 s, 3.92 s, and 36.72 s, respectively), remaining well-competitive with direct solvers (e.g., dare() ) while avoiding Kronecker products and explicit Jacobians. Overall, the results indicate a robust, low-iteration, and near-instant approach that is attractive for real-time and embedded control contexts where both speed and solution quality are paramount.