Theoretical Extensions and Numerical Analysis of Parameter-Manipulated HSS-Type Iterations for Sparse Linear Systems

Abstract

Background: Large, sparse linear systems of the form Ax = b is ubiquitous in Management Information Systems and business analytics, appearing in supply-chain optimization, PageRank and web-scale ranking, portfolio optimization, and networked stochastic models. The Hermitian/Skew-Hermitian Splitting (HSS) method is a flexible iterative framework for such problems, but its practical performance is highly sensitive to the choice of the acceleration parameter and can be slow or unstable without careful tuning.

Materials and Methods: We develop two parameter-manipulation extensions of HSS: (i) Adaptive Parameter HSS (AP-HSS), which dynamically updates the scalar acceleration parameter using a low-cost residual-minimization heuristic and safeguarded quadratic interpolation among candidate values; and (ii) Cyclic Parameter HSS (CP-HSS), which applies a pre-defined geometric cycle of parameters with the possibility of precomputing factorizations for the cycle. Both methods are compared against standard HSS and NHSS using representative test matrices drawn from synthetic convection–diffusion discretizations, Markov transition systems, and portfolio/KKT problems; prototype implementations employ sparse direct solvers and measure iteration counts, CPU time, and relative residual norms.

Results: Numerical experiments on the representative suite show that both AP-HSS and CP-HSS consistently outperform fixed-parameter HSS in iterationwise residual reduction and final residuals for convection–diffusion and PSD portfolio tests; AP-HSS delivers the most robust improvement across heterogeneous problems by adapting to local spectral characteristics, while CP-HSS is highly efficient when a small cycle of parameters can be precomputed and amortized. Stochastic/Markov examples highlight the need for additional stabilization: naive parameter choices can destabilize HSS, whereas AP-HSS’s guarded evaluation improves robustness though additional preconditioning may still be required to reach strict tolerances.

Conclusion: Structured parameter tuning for HSS-type iterations provides a useful approach for enhanced speed and robustness for linear solvers in Management Information Systems (MIS) and business analytics. AP-HSS offers a black-box, automated solution, thus minimizing computationally expensive spectral precomputation as part of an analysis pipeline. CP-HSS offers an alternative with improved performance when precomputation is not prohibitively expensive. These techniques enable efficient large-scale simulations and analytics processes to be implemented in business intelligence; future efforts should include developing formal convergence proofs, exploring broader generalizations of the methodology, predicting parameters using machine learning, and deploying these techniques onto very large scales.