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Shifted CholeskyQR for sparse matrices

Haoran Guan, Yuwei Fan

TL;DR

This work targets efficient and accurate QR factorization for sparse matrices via Shifted CholeskyQR (SCholeskyQR). It introduces an alternative shifted item $s$ that leverages sparsity through the number of non-zeros nnze and the largest input entry, and it develops a new norm $\|\cdot\|_{g}$ for sharper rounding error analysis. Theoretical results bound key error terms and relate the condition numbers of the input $X$ and the computed $W$, establishing accuracy even for ill-conditioned sparse data. Numerical experiments on sparse matrices demonstrate improved applicability and residuals with the new $s$ while preserving computational efficiency. These findings connect sparsity structure with numerical stability in CholeskyQR-type algorithms, expanding their practical viability for large sparse problems.

Abstract

In this work, we focus on Shifted CholeskyQR (SCholeskyQR) for sparse matrices. We provide a new shifted item $s$ for Shifted CholeskyQR3 (SCholeskyQR3) based on the number of non-zero elements (nnze) and the element with the largest absolute value of the input sparse $X \in \mathbb{R}^{m\times n}$ with $m \ge n$. We do rounding error analysis of SCholeskyQR3 with such an $s$ and show that SCholeskyQR3 is accurate in this case. Therefore, an alternative choice of $s$ can be taken for SCholeskyQR3 with the comparison between our new $s$ and the $s$ shown in the previous work when the input $X$ is sparse, improving the applicability and residual of the algorithm for the ill-conditioned cases. Numerical experiments demonstrate the advantage of SCholeskyQR3 with our alternative choice of $s$ in both applicablity and accuracy over the case with the original $s$, together with the same level of efficiency. This work is also the first to build connections between sparsity and numerical algorithms with detailed rounding error analysis to the best of our knowledge.

Shifted CholeskyQR for sparse matrices

TL;DR

This work targets efficient and accurate QR factorization for sparse matrices via Shifted CholeskyQR (SCholeskyQR). It introduces an alternative shifted item that leverages sparsity through the number of non-zeros nnze and the largest input entry, and it develops a new norm for sharper rounding error analysis. Theoretical results bound key error terms and relate the condition numbers of the input and the computed , establishing accuracy even for ill-conditioned sparse data. Numerical experiments on sparse matrices demonstrate improved applicability and residuals with the new while preserving computational efficiency. These findings connect sparsity structure with numerical stability in CholeskyQR-type algorithms, expanding their practical viability for large sparse problems.

Abstract

In this work, we focus on Shifted CholeskyQR (SCholeskyQR) for sparse matrices. We provide a new shifted item for Shifted CholeskyQR3 (SCholeskyQR3) based on the number of non-zero elements (nnze) and the element with the largest absolute value of the input sparse with . We do rounding error analysis of SCholeskyQR3 with such an and show that SCholeskyQR3 is accurate in this case. Therefore, an alternative choice of can be taken for SCholeskyQR3 with the comparison between our new and the shown in the previous work when the input is sparse, improving the applicability and residual of the algorithm for the ill-conditioned cases. Numerical experiments demonstrate the advantage of SCholeskyQR3 with our alternative choice of in both applicablity and accuracy over the case with the original , together with the same level of efficiency. This work is also the first to build connections between sparsity and numerical algorithms with detailed rounding error analysis to the best of our knowledge.
Paper Structure (22 sections, 13 theorems, 74 equations, 12 tables, 6 algorithms)

This paper contains 22 sections, 13 theorems, 74 equations, 12 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Summary of CholeskyQR-type algorithms
  3. New considerations
  4. Our contributions in this work
  5. The improved SCholeskyQR3
  6. SCholeskyQR3 for sparse matrices
  7. Outline of this work
  8. Some lemmas regarding rounding error analysis
  9. Proof of Theorem \ref{['THM 4.2']}-Theorem \ref{['THM 4.1']}
  10. Lemmas to prove Theorem \ref{['THM 4.2']}-Theorem \ref{['THM 4.1']}
  11. Proof of Theorem \ref{['THM 4.2']}
  12. Estimating $\|W^{\top}W-I\|_{2}$
  13. Estimating $\|\Delta X\|_{F}$
  14. The relationship between $\kappa_{2}(X)$ and $\kappa_{2}(W)$
  15. Proof of Theorem \ref{['THM 4.3']}
  16. ...and 7 more sections

Key Result

Lemma 1.2

For $X \in \mathbb{R}^{m\times n}$ and $[W,Y]=\hbox{ISCholeskyQR}(X)$, with $mn{\bf u} \le \frac{1}{64}$, $n(n+1){\bf u} \le \frac{1}{64}$, $11(mn{\bf u}+n(n+1){\bf u})\|X\|_{g}^{2} \le s \le \frac{1}{100}\|X\|_{g}^{2}$ and $\kappa_{2}(X) \le \frac{1}{4.89pn^{2}{\bf u}}$, we have Here, we have $t=\frac{s}{\|X\|_{2}^{2}} \le \frac{1}{100}$. When $[Q,R]=\hbox{ISCholeskyQR3}(X)$, if we take $s=11(mn

Theorems & Definitions (24)

  • Definition 1.1: The definition of ${\|\cdot\|_{g}}$
  • Lemma 1.2: The relationship between $\kappa_{2}(X)$ and $\kappa_{2}(W)$ for ISCholeskyQR
  • Lemma 1.3: Rounding error analysis of ISCholeskyQR3
  • Theorem 1.4: The relationship between $\kappa_{2}(X)$ and $\kappa_{2}(W)$ for the sparse $X \in \mathbb{R}^{m\times n}$
  • Theorem 1.5: Rounding error analysis of SCholeskyQR3 for the sparse $X \in \mathbb{R}^{m\times n}$
  • Theorem 1.6: A choice of the shifted item $s$ for the optimal one
  • Remark 1
  • Lemma 2.1: Weyl's Theorem for singular values
  • Lemma 2.2: Rounding error in matrix multiplications
  • Lemma 2.3: Rounding error in Cholesky factorization
  • ...and 14 more