OP-00236 Solution claimed

Preservation of real H-matrix structure by Newton square-root iterates

Let $n \ge 1$ and let $A \in \mathbb{R}^{n \times n}$ have positive diagonal entries and be a nonsingular H-matrix, that is, one whose comparison matrix is a nonsingular M-matrix. Consider Newton's iteration for the square root, in exact arithmetic, $$X_0 = A, \qquad X_{k+1} = \tfrac12\bigl(X_k + X_k^{-1}A\bigr), \quad k \ge 0.$$

Its iterates converge to the principal square root $A^{1/2}$, which Lin and Liu proved is again a nonsingular H-matrix with positive diagonal entries. Guo proved that when every diagonal entry of $A$ is at most $1$, the Newton iterates started from $X_0 = I$ are all nonsingular H-matrices with positive diagonal entries.

Problem. Is every $X_k$ a nonsingular H-matrix with positive diagonal entries?

Source: Higham (2008), Problem 6.25, a research problem (Functions of Matrices, p. 170), on the Newton iteration (6.12); stated by OpenProblemsInNLA for real H-matrices, following the book's errata to Theorem 6.10.

Imported from github.com/ajt60gaibb/OpenProblemsInNLA.

Classification

Primary: NA — Numerical Analysis
Secondary: RA
Keywords: H-matrices, matrix square roots, Newton iterations, OpenProblemsInNLA
Literature checked: Oct. 8, 2026 by clemens-admin

References

@book{higham2008,
  author    = {Higham, Nicholas J.},
  title     = {Functions of Matrices: Theory and Computation},
  publisher = {SIAM},
  address   = {Philadelphia},
  year      = {2008},
  doi       = {10.1137/1.9780898717778},
  note      = {Section 6.3, equation (6.12), Section 6.8.3 and Problem 6.25},
}
@article{lin2001,
  author  = {Lin, Lu and Liu, Zhong-Yun},
  title   = {On the square root of an {H}-matrix with positive diagonal elements},
  journal = {Ann. Oper. Res.},
  volume  = {103},
  number  = {1-4},
  year    = {2001},
  pages   = {339--350},
  doi     = {10.1023/A:1012931928589},
}
@article{guo2010,
  author  = {Guo, Chun-Hua},
  title   = {On {N}ewton's method and {H}alley's method for the principal $p$th root of a matrix},
  journal = {Linear Algebra Appl.},
  volume  = {432},
  number  = {8},
  year    = {2010},
  pages   = {1905--1922},
  doi     = {10.1016/j.laa.2009.02.030},
}
@misc{openproblemsinnla-sf01,
  author       = {Townsend, Alex},
  title        = {{SF-01}: Preservation of real H-matrix structure by Newton square-root iterates},
  howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
  year         = {2026},
  url          = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/matrix-functions-and-stability/SF-01/README.md},
  note         = {Read at commit eb7c9d6, 2026-10-07},
}

Resolution claims

Preservation of positive-diagonal H-matrices by scaled Newton and Halley square-root iterations v1 GitHub ↗

Matthew J. Colbrook

Every scaled Newton square-root iterate of a real nonsingular H-matrix with positive diagonal entries is again a real nonsingular H-matrix with positive diagonal entries, when initialized by a positive scalar multiple of the identity, a positive scalar multiple of the input, or more generally a nonzero nonnegative affine function of the input. The scaling parameter may be any positive scalar …

Full resolution · not verified · fidelity not established · posted on GitHub 11 Sep 2026

Status history

When	Change	By	Reason
2026-10-08	Needs verification → Solution claimed	clemens-admin	Listed with the record of its solution, unreviewed.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00236, "Preservation of real H-matrix structure by Newton square-root iterates", clopen.solutions.