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<p>We introduce two-parameter and matrix-weighted block-splitting preconditioners for the normal equations of indefinite least squares (ILS). The schemes are <i>minimax-optimal</i> over the spectral interval of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\((P^{-1}Q)\)</EquationSource> </InlineEquation>, ensuring smaller worst-case contraction factors than one-parameter counterparts. We derive a spectral representation, characterize the minimax solution via equioscillation, and provide efficient computation through semidefinite programming or a closed-form two-point formula. The approach is robust to spectral uncertainty and readily integrates with (F)GMRES. Numerical tests on commuting and nearly commuting systems confirm faster convergence and reduced time-to-solution.</p>

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Minimax-Optimal Two-Parameter and Matrix-Weighted Block-Splitting Preconditioners for Indefinite Least Squares

  • 期刊论文

  • 发表日期：

    2026年1月5日

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  • Saeed Hashemi Sababe

摘要

We introduce two-parameter and matrix-weighted block-splitting preconditioners for the normal equations of indefinite least squares (ILS). The schemes are minimax-optimal over the spectral interval of \((P^{-1}Q)\) , ensuring smaller worst-case contraction factors than one-parameter counterparts. We derive a spectral representation, characterize the minimax solution via equioscillation, and provide efficient computation through semidefinite programming or a closed-form two-point formula. The approach is robust to spectral uncertainty and readily integrates with (F)GMRES. Numerical tests on commuting and nearly commuting systems confirm faster convergence and reduced time-to-solution.