Exponential Growth Factors of Orthogonal Matrices under Gaussian Elimination with Partial and Complete Pivoting

المفاهيم الأساسية

The growth factors of orthogonal matrices under Gaussian elimination with partial and complete pivoting can exhibit exponential growth, with the partial pivoting strategy potentially leading to much larger growth factors compared to complete pivoting.

الملخص

The content explores the growth factors of orthogonal matrices under Gaussian elimination (GE) with partial pivoting (GEPP) and complete pivoting (GECP). It provides the following key insights:

The author establishes an explicit construction of an orthogonal matrix, denoted as Qn, that attains exponential GEPP growth, with a growth factor of 2^(n-1)√3(1+o(1)). This improves upon previous bounds on the maximum GEPP growth for orthogonal matrices.

The author shows that every orthogonal matrix with maximal GEPP growth is sign-equivalent to the Qn matrix, and that the set of such orthogonal matrices has a finite size of 2^(2n-1).

The author explores the relationship between the GEPP and GECP growth factors on the same linear systems, establishing lower bounds on the maximum difference between the two. This difference can be exponentially large, with the GEPP growth potentially much larger than the GECP growth.

The author studies the local behavior of GEPP and GECP growth factors around matrices that exhibit large differences in growth between the two pivoting strategies. The growth remains stable under the pivoting strategy that has minimal initial growth, while the larger growth model has local behavior that progressively concentrates near the smaller initial growth.

Overall, the content provides new insights into the complex behavior of Gaussian elimination with different pivoting strategies, especially for structured orthogonal matrices that can exhibit extreme growth factor differences between the partial and complete pivoting approaches.

الإحصائيات

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اقتباسات

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الرؤى الأساسية المستخلصة من

Growth factors of orthogonal matrices and local behavior of Gaussian elimination with partial and complete pivoting

by John Peca-Me... في arxiv.org 04-17-2024

https://arxiv.org/pdf/2308.16146.pdf

استفسارات أعمق

How can the precise value of the constant c in the exponential GEPP growth bound for orthogonal matrices be determined

To determine the precise value of the constant c in the exponential GEPP growth bound for orthogonal matrices, one approach is to focus on maintaining more of the orthogonality constraints when optimizing the problem. This involves considering the specific properties of orthogonal matrices and how they interact with the GEPP algorithm. By delving deeper into the orthogonality constraints and their impact on the growth factors, it may be possible to refine the optimization process and narrow down the range for the constant c. Additionally, conducting further theoretical analysis and possibly numerical experiments can help in honing in on the exact value of c.

What are the implications of the large potential differences between GEPP and GECP growth factors on the practical performance and numerical stability of these linear solvers

The large potential differences between GEPP and GECP growth factors have significant implications for the practical performance and numerical stability of these linear solvers. When there are substantial discrepancies in growth factors between GEPP and GECP, it can lead to varying levels of accuracy and efficiency in solving linear systems. In practice, this means that the choice of pivoting strategy can greatly impact the outcome of the Gaussian elimination process. Systems with high growth factors may experience more numerical instability and error accumulation, potentially leading to inaccuracies in the solutions obtained. Understanding and managing these differences is crucial for ensuring the reliability and robustness of linear solvers in numerical computations.

Are there other classes of structured matrices, beyond orthogonal matrices, that can exhibit extreme differences in growth factors between GEPP and GECP

Beyond orthogonal matrices, there are other classes of structured matrices that can exhibit extreme differences in growth factors between GEPP and GECP. For example, matrices with specific patterns or properties that affect the pivoting process can result in varying growth behaviors under different pivoting strategies. Structured matrices such as banded matrices, sparse matrices, or matrices with certain symmetry properties may showcase pronounced differences in growth factors when subjected to GEPP and GECP. Exploring these classes of matrices and analyzing their behavior under Gaussian elimination with different pivoting strategies can provide valuable insights into the impact of matrix structure on the efficiency and stability of numerical solvers.