| Author: | Guan, Haoran |
| Title: | CholeskyQR-type algorithms : development and analysis |
| Advisors: | Qiao, Zhonghua (AMA) |
| Degree: | Ph.D. |
| Year: | 2025 |
| Department: | Department of Applied Mathematics |
| Pages: | xiv, 97 pages : illustrations |
| Language: | English |
| Abstract: | This thesis focuses on the development of CholeskyQR-type algorithms, which are very popular in recent years due to their efficiency and accuracy. Compared to the traditional algorithms for QR factorization, such as HouseholderQR and MGS, CholeskyQR-type algorithms have special advantages and have raised much attention from both academia and industry. In this thesis, We present some progress we have made in CholeskyQR-type algorithms in the past several years. Though with good efficiency and accuracy, CholeskyQR is seldom used alone due to its lack of orthogonality. In order to receive numerical stability in orthogonality, CholeskyQR2 has been developed by repeating CholeskyQR twice. In recent years, researchers has proposed Shifted CholeskyQR3 to deal with QR factorization of ill-conditioned matrices, with a shifted item s in the step of Cholesky factorization to avoid numerical breakdown in ill-conditioned cases. Moreover, some other CholeskyQR-type algorithms have occurred, such as LU-CholeskyQR2 and some randomized algorithms. The development of CholeskyQR-type algorithms aims for improving the applicability of the algorithms. In this thesis, we show our improvements on the applicability of CholeskyQR-type algorithms, especially for Shifted CholeskyQR3. Some cases based on real-world problems are also considered. Shifted CholeskyQR3 avoids the problem of encountering numerical breakdown in ill-conditioned cases which belongs to CholeskyQR2. With the structure of CholeskyQR2 after Shifted CholeskyQR, Shifted CholeskyQR3 can keep numerical stability and replace CholeskyQR2. However, the original shifted item s = 11(mnu + (n + 1)nu)∥X∥22 for the input matrix X ∈ Rm×n is relatively conservative due to overestimation in rounding error analysis. We introduce a new matrix norm ∥X∥c and propose an improved shifted item s = 11(mu +(n + 1)u)∥X∥2c for Shifted CholeskyQR3. Our theoretical analysis and numerical experiments demonstrate that our new s can enhance the applicability of Shifted CholeskyQR3, while maintaining numerical stability and efficiency. In fact, in many real-world applications, the input matrix X ∈ Rm×n is often sparse, especially when m and n are large. Due to the structure of the algorithm, the sparsity of the input matrix will influence rounding error analysis of CholeskyQR and exhibit different properties compared to those of dense matrices. For sparse matrices, we build a new model and divide them into two types, T1 matrices with the dense columns and T2 matrices whose columns are all sparse. Therefore, an alternative choice of the shifted item s is proposed for Shifted CholeskyQR3 based on the structure and the key element of the input X. We prove that such an alternative s are optimal compared to the original s we propose in the previous part with certain element-norm conditions(ENCs). It can improve the applicability of Shifted CholeskyQR3 for T1 matrices and maintain numerical stability of the algorithm in this way. Numerical experiments demonstrate our findings and show that shifted CholeskyQR3 with the alternative s can also deal with more ill-conditioned cases for T2 matrices because of the potential sparsity of the orthogonal factor after Shifted CholeskyQR. The algorithm with such an s is also as efficient as the case with the original s. ∥·∥g, a definition connected to ∥·∥c, is utilized in the theoretical analysis. In recent years, probabilistic rounding error analysis has become a hot topic in numerical linear algebra. We can receive tighter error bounds compared to the deterministic ones. Based on the theoretical analysis of CholeskyQR-type algorithms, probabilistic error analysis can improve the sufficient condition of κ2(X) for X ∈ Rm×n and bring more accurate error analysis. Therefore, we do probabilistic error analysis of CholeskyQR-type algorithms. We receive tighter upper bounds of both orthogonality and residual for CholeskyQR-type algorithms, together with looser sufficient conditions of κ2(X) with the corresponding probabilities. Additionally, a probabilistic s with ∥X∥c is proposed for Shifted CholeskyQR3. Numerical experiments show that such a probabilistic s can improve the applicability of the algorithm further. Shifted CholeskyQR3 with such a probabilistic s is also numerical stable and robust enough after numerous experiments. Generally speaking, we propose and utilize new tools for more accurate rounding error analysis of CholeskyQR-type algorithms theoretically, which also helps to improve the properties of the algorithm. Our improvements on the applicability of the algorithm are effective according to numerical experiments, which correspond to the new theoretical results in this work. |
| Rights: | All rights reserved |
| Access: | open access |
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