Core Concepts
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.
Abstract
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.