Sharp condition-number bounds for growth factors of Higham matrices in Gaussian elimination
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917434745683968 |
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| author | Zhang, Teng |
| author_facet | Zhang, Teng |
| contents | Higham's conjecture on the growth factor of complex symmetric positive definite matrices is a longstanding problem in the stability theory of Gaussian elimination without pivoting. It asserts that every complex matrix $A=B+iC$ with $B$ and $C$ real symmetric positive definite, is called Higham matrix and has growth factor $ρ_n(A)<2$. In 2013, Drury [Linear Algebra Appl. \textbf{439} (2013), no.~10, 3129--3133] proved that $ρ_n(A)\le 2$. In fact, we will see his sectorial determinant method can be refined to give the strict bound $ρ_n(A)<2$ for each fixed Higham matrix; however, the resulting constant $1+δ_A^2$ depends on the matrix $A$. In this paper, we establish sharp condition-number-dependent lower and upper bounds for the growth factors of Higham matrices, thereby providing a quantitative refinement of Drury's result. The main ingredient is a sharp scalar Schur-complement inequality, proved via a two-dimensional domination.We also obtain corresponding sharp scalar and diagonal estimates for accretive-dissipative matrices, and an improved entrywise growth bound for that broader class. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_23024 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sharp condition-number bounds for growth factors of Higham matrices in Gaussian elimination Zhang, Teng Numerical Analysis Primary 65F05, Secondary 15A45, 65F35, 15A60 Higham's conjecture on the growth factor of complex symmetric positive definite matrices is a longstanding problem in the stability theory of Gaussian elimination without pivoting. It asserts that every complex matrix $A=B+iC$ with $B$ and $C$ real symmetric positive definite, is called Higham matrix and has growth factor $ρ_n(A)<2$. In 2013, Drury [Linear Algebra Appl. \textbf{439} (2013), no.~10, 3129--3133] proved that $ρ_n(A)\le 2$. In fact, we will see his sectorial determinant method can be refined to give the strict bound $ρ_n(A)<2$ for each fixed Higham matrix; however, the resulting constant $1+δ_A^2$ depends on the matrix $A$. In this paper, we establish sharp condition-number-dependent lower and upper bounds for the growth factors of Higham matrices, thereby providing a quantitative refinement of Drury's result. The main ingredient is a sharp scalar Schur-complement inequality, proved via a two-dimensional domination.We also obtain corresponding sharp scalar and diagonal estimates for accretive-dissipative matrices, and an improved entrywise growth bound for that broader class. |
| title | Sharp condition-number bounds for growth factors of Higham matrices in Gaussian elimination |
| topic | Numerical Analysis Primary 65F05, Secondary 15A45, 65F35, 15A60 |
| url | https://arxiv.org/abs/2604.23024 |