Improved Lower Bounds for the Maximum Growth Factor in Gaussian Elimination with Complete Pivoting

Основные понятия

The maximum growth factor under complete pivoting is larger than 1.0045n for all n > 10, and the lim sup of the ratio with n is greater than or equal to 3.317. This disproves the long-standing conjecture that the growth factor is at most n.

Аннотация

The paper combines modern numerical computation with theoretical results to improve the understanding of the growth factor problem for Gaussian elimination with complete pivoting.

Key highlights:

1. Numerical search and stability/extrapolation results provide improved lower bounds for the maximum growth factor:

   • The growth factor is at least 1.0045n for all n > 10.
   • The lim sup of the ratio of the growth factor to n is at least 3.317.
   • This disproves the long-standing conjecture that the growth factor is at most n.

2. Theoretical results:

   • The maximum growth factor over matrices with entries restricted to a subset of the reals is nearly equal to the maximum growth factor over all real matrices.
   • The growth factors under floating point arithmetic and exact arithmetic are nearly identical.

3. Extensive numerical computations:

   • Improvements over previously known maximum growth factors for small matrix sizes.
   • Observations about Hadamard matrices and their relationship to the growth factor problem.
   • Tabulated numerical results for matrix sizes up to n = 75 and n = 100.

The paper provides strong evidence that the maximum growth factor exhibits super-linear growth, contradicting the old conjecture that it might never be bigger than n.

Статистика

The paper does not contain any key metrics or important figures to support the author's key logics. The focus is on theoretical results and numerical computations of the maximum growth factor.

Цитаты

The paper does not contain any striking quotes supporting the author's key logics.

Ключевые выводы из

Some New Results on the Maximum Growth Factor in Gaussian Elimination

by Alan Edelman... в arxiv.org 04-09-2024

https://arxiv.org/pdf/2303.04892.pdf

Дополнительные вопросы

How can the theoretical results be extended to provide tighter bounds on the asymptotic behavior of the maximum growth factor under complete pivoting

To provide tighter bounds on the asymptotic behavior of the maximum growth factor under complete pivoting, one could explore more refined numerical search strategies. By optimizing the search algorithm parameters, such as the starting points, step sizes, and convergence criteria, researchers can potentially uncover matrices with even larger growth factors. Additionally, incorporating advanced optimization techniques or machine learning algorithms could help in efficiently exploring the solution space and identifying matrices that push the bounds further. Furthermore, theoretical advancements in understanding the structural properties of matrices that lead to high growth factors could aid in formulating tighter bounds based on mathematical insights.

Are there any counterarguments or limitations to the numerical approach used in the paper to compute the maximum growth factors

While the numerical approach used in the paper is effective in computing the maximum growth factors for matrices, there are some counterarguments and limitations to consider. One limitation is the computational complexity associated with searching for matrices with high growth factors, especially as the matrix size increases. The exhaustive search over a large solution space can be computationally intensive and time-consuming. Additionally, the numerical results obtained may be sensitive to the choice of starting points and optimization parameters, potentially leading to local optima rather than the global maximum. Moreover, the numerical approach may not provide a complete theoretical understanding of the underlying factors influencing the growth factor, as it primarily focuses on empirical observations rather than analytical derivations.

What are the potential implications of the super-linear growth of the maximum growth factor for the practical use of Gaussian elimination with complete pivoting in numerical linear algebra applications

The discovery of super-linear growth in the maximum growth factor has significant implications for the practical use of Gaussian elimination with complete pivoting in numerical linear algebra applications. Firstly, it suggests that as the matrix size increases, the growth factor may grow at a rate faster than linear, indicating a potential increase in computational complexity and numerical instability. This could impact the efficiency and accuracy of numerical algorithms relying on Gaussian elimination, highlighting the importance of considering alternative approaches or optimizations to mitigate the effects of high growth factors. Additionally, the super-linear growth implies that for larger matrices, the potential for numerical errors and round-off effects to amplify during Gaussian elimination with complete pivoting becomes more pronounced, emphasizing the need for robust numerical techniques and precision management in practical implementations.