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Peer-reviewedCC BY 4.0Mathematics

On the Factored Newton Iteration for Positive-Real Algebraic Riccati Equations via the Matrix Sign Function

  • Yiqin Lin1
  • Wenbo Li2
  • Liping Zhou1
  1. 1School of Science, Hunan Institute of Science and Engineering, Yongzhou 425199, China
  2. 2School of Sciences, Xi’an University of Technology, Xi’an 710054, China
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Abstract

We extend the factored Newton iteration originally developed by Benner et al. for algebraic Riccati equations with a negative quadratic term—to the positive-real case. Unlike the negative-sign setting, the positive sign introduces two well-posedness issues: each intermediate matrix Ak must remain nonsingular, and the DC-gain condition must be preserved to ensure the validity of the factored inversion formula. We prove rigorously that both properties are invariant under the Newton update. The resulting structure-exploiting solver, GENPF, reduces the per-iteration cost from 16n3 to approximately 2n3 floating-point operations when the input matrices are tall and skinny. Numerical experiments confirm that GENPF preserves the quadratic convergence and accuracy of the conventional Newton iteration, requires significantly less computational time, and maintains the DC-gain condition throughout the iteration.

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Cite

BibTeXRIS
BibTeX
@article{lin2026factored,
  title = {{On the Factored Newton Iteration for Positive-Real Algebraic Riccati Equations via the Matrix Sign Function}},
  author = {Yiqin Lin and Wenbo Li and Liping Zhou},
  journal = {Mathematics},
  year = {2026},
  month = oct,
  doi = {10.3390/math14203802},
  url = {https://www.mdpi.com/2227-7390/14/20/3802},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - On the Factored Newton Iteration for Positive-Real Algebraic Riccati Equations via the Matrix Sign Function
AU  - Yiqin Lin
AU  - Wenbo Li
AU  - Liping Zhou
PY  - 2026
DA  - 2026/10/20
JO  - Mathematics
DO  - 10.3390/math14203802
UR  - https://www.mdpi.com/2227-7390/14/20/3802
AB  - We extend the factored Newton iteration originally developed by Benner et al. for algebraic Riccati equations with a negative quadratic term—to the positive-real case. Unlike the negative-sign setting, the positive sign introduces two well-posedness issues: each intermediate matrix Ak must remain nonsingular, and the DC-gain condition must be preserved to ensure the validity of the factored inversion formula. We prove rigorously that both properties are invariant under the Newton update. The resulting structure-exploiting solver, GENPF, reduces the per-iteration cost from 16n3 to approximately 2n3 floating-point operations when the input matrices are tall and skinny. Numerical experiments confirm that GENPF preserves the quadratic convergence and accuracy of the conventional Newton iteration, requires significantly less computational time, and maintains the DC-gain condition throughout the iteration.
SN  - 2227-7390
ER  -