Hyperbolic Householder transformations for Up- and Downdating Cholesky factorizations.

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  • License Expression: LGPL-3.0-or-later
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  • Author: Pieter P
  • Tags linear-algebra , cholesky , update , downdate , matrix
  • Requires: Python >=3.9
  • Provides-Extra: test , docs
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arXiv Preprint CI: Linux PyPI Downloads

hyhound

Hyperbolic Householder transformations for Up- ‘n’ Downdating Cholesky factorizations.

Purpose

Given a Cholesky factor \(L\) of a dense matrix \(H\), the hyhound::update_cholesky function computes the Cholesky factor \(\tilde L\) of the matrix

\begin{equation*} \tilde H = \tilde L \tilde L^\top = H + A \Sigma A^\top, \end{equation*}

where \(H,\tilde H\in\mathbb{R}^{n\times n}\) with \(H \succ 0\) and \(\tilde H \succ 0\), \(A \in \mathbb{R}^{n\times m}\), \(\Sigma \in \mathbb{R}^{m\times m}\) diagonal, and \(L, \tilde L\in\mathbb{R}^{n\times n}\) lower triangular.

Computing \(\tilde L\) in this way is done in \(mn^2 + \mathcal{O}(n^2 + mn)\) operations rather than the \(\tfrac16 n^3 + \tfrac12 mn^2 + \mathcal{O}(n^2 + mn)\) operations required for the explicit evaluation and factorization of \(\tilde H\). When \(m \ll n\), this results in a considerable speedup over full factorization, enabling efficient low-rank updates of Cholesky factorizations, for use in e.g. iterative algorithms for numerical optimization.

Additionally, hyhound includes efficient routines for updating factorizations of the Riccati recursion for optimal control problems.

Preprint

The paper describing the algorithms in this repository can be found on arXiv: https://arxiv.org/abs/2503.15372v1

@misc{pas_blocked_2025,
   title = {Blocked {Cholesky} factorization updates of the {Riccati} recursion using hyperbolic {Householder} transformations},
   url = {http://arxiv.org/abs/2503.15372},
   doi = {10.48550/arXiv.2503.15372},
   publisher = {arXiv},
   author = {Pas, Pieter and Patrinos, Panagiotis},
   month = mar,
   year = {2025},
   note = {Accepted for publication in the Proceedings of CDC 2025}
}

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  • License Expression: LGPL-3.0-or-later
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  • Author: Pieter P
  • Tags linear-algebra , cholesky , update , downdate , matrix
  • Requires: Python >=3.9
  • Provides-Extra: test , docs
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