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An improved Shifted CholeskyQR based on columns

Yuwei Fan, Haoran Guan, Zhonghua Qiao

TL;DR

This work tackles the stability and applicability limitations of deterministic QR methods for ill-conditioned matrices by introducing a column-based metric [X]_{g} to compute a smaller shift s in Shifted CholeskyQR3. It defines ISCholeskyQR and ISCholeskyQR3, proves orthogonality and residual properties, and demonstrates via theory (Theorems 41–43) and experiments that reduced s improves the sufficient condition on κ_{2}(X) and broadens the method’s usable conditioning range. Numerical results on SVD-based, Hilbert, and arrowhead inputs show comparable or superior stability to HouseholderQR with competitive CPU times, confirming practical gains for moderately sized problems. The approach provides tighter error bounds through [X]_{g}, enabling more robust rounding-error analysis and suggesting further extensions to structured matrices and parallel/GPU implementations. Overall, the paper advances deterministic QR factorization for ill-conditioned data by balancing stability, accuracy, and efficiency through a geometry-aware column-norm metric.

Abstract

Among all the deterministic CholeskyQR-type algorithms, Shifted CholeskyQR3 is specifically designed to address the QR factorization of ill-conditioned matrices. This algorithm introduces a shift parameter $s$ to prevent failure during the initial Cholesky factorization step, making the choice of this parameter critical for the algorithm's effectiveness. Our goal is to identify a smaller $s$ compared to the traditional selection based on $\norm{X}_{2}$. In this research, we propose a new definition for the input matrix $X$ called $[X]_{g}$, which is based on the column properties of $X$. $[X]_{g}$ allows us to obtain a reduced shift parameter $s$ for the Shifted CholeskyQR3 algorithm, thereby improving the sufficient condition of $κ_{2}(X)$ for this method. We provide rigorous proofs of orthogonality and residuals for the improved algorithm using our proposed $s$. Numerical experiments confirm the enhanced numerical stability of orthogonality and residuals with the reduced $s$. We find that Shifted CholeskyQR3 can effectively handle ill-conditioned $X$ with a larger $κ_{2}(X)$ when using our reduced $s$ compared to the original $s$. Furthermore, we compare CPU times with other algorithms to assess performance improvements.

An improved Shifted CholeskyQR based on columns

TL;DR

This work tackles the stability and applicability limitations of deterministic QR methods for ill-conditioned matrices by introducing a column-based metric [X]_{g} to compute a smaller shift s in Shifted CholeskyQR3. It defines ISCholeskyQR and ISCholeskyQR3, proves orthogonality and residual properties, and demonstrates via theory (Theorems 41–43) and experiments that reduced s improves the sufficient condition on κ_{2}(X) and broadens the method’s usable conditioning range. Numerical results on SVD-based, Hilbert, and arrowhead inputs show comparable or superior stability to HouseholderQR with competitive CPU times, confirming practical gains for moderately sized problems. The approach provides tighter error bounds through [X]_{g}, enabling more robust rounding-error analysis and suggesting further extensions to structured matrices and parallel/GPU implementations. Overall, the paper advances deterministic QR factorization for ill-conditioned data by balancing stability, accuracy, and efficiency through a geometry-aware column-norm metric.

Abstract

Among all the deterministic CholeskyQR-type algorithms, Shifted CholeskyQR3 is specifically designed to address the QR factorization of ill-conditioned matrices. This algorithm introduces a shift parameter to prevent failure during the initial Cholesky factorization step, making the choice of this parameter critical for the algorithm's effectiveness. Our goal is to identify a smaller compared to the traditional selection based on . In this research, we propose a new definition for the input matrix called , which is based on the column properties of . allows us to obtain a reduced shift parameter for the Shifted CholeskyQR3 algorithm, thereby improving the sufficient condition of for this method. We provide rigorous proofs of orthogonality and residuals for the improved algorithm using our proposed . Numerical experiments confirm the enhanced numerical stability of orthogonality and residuals with the reduced . We find that Shifted CholeskyQR3 can effectively handle ill-conditioned with a larger when using our reduced compared to the original . Furthermore, we compare CPU times with other algorithms to assess performance improvements.
Paper Structure (24 sections, 18 theorems, 87 equations, 18 tables, 6 algorithms)

This paper contains 24 sections, 18 theorems, 87 equations, 18 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. CholeskyQR2 and its properties
  3. Shifted CholeskyQR3 and its problems
  4. Our contributions in this work
  5. Outline and notations
  6. Preliminary Lemmas of rounding error analysis
  7. Some properties of $[\cdot]_{g}$
  8. Theoretical analysis of the improved Shifted CholeskyQR3
  9. General settings and assumptions
  10. Algorithms
  11. Some lemmas for proving theorems
  12. Proof of Theorem \ref{['thm:41']}
  13. Proof of Theorem \ref{['thm:42']}
  14. Proof of Theorem \ref{['thm:43']}
  15. Experimental Results
  16. ...and 9 more sections

Key Result

lemma thmcounterlemma

For $X \in \mathbb{R}^{m\times n}$ and $[Q_{1},R_{2}]=\hbox{CholeskyQR2}(X)$, when $8\kappa_{2}(X)\sqrt{mn{\bf u}+n(n+1){\bf u}} \le 1$, we have

Theorems & Definitions (34)

  • lemma thmcounterlemma: Rounding error analysis of CholeskyQR2
  • lemma thmcounterlemma: Rounding error analysis of Shifted CholeskyQR
  • lemma thmcounterlemma: The relationship between $\kappa_{2}(X)$ and $\kappa_{2}(Q)$ for Shifted CholeskyQR
  • lemma thmcounterlemma: Rounding error analysis of Shifted CholeskyQR3
  • definition thmcounterdefinition: The definition of ${[\cdot]_{g}}$
  • theorem 1: Rounding error analysis of the improved Shifted CholeskyQR
  • theorem 2: The relationship between $\kappa_{2}(X)$ and $\kappa_{2}(Q)$ for the improved Shifted CholeskyQR
  • theorem 3: Rounding error analysis of the improved Shifted CholeskyQR3
  • lemma thmcounterlemma: Weyl's Theorem MatrixC
  • lemma thmcounterlemma: Rounding error in matrix multiplications Higham
  • ...and 24 more