核心概念
Two new reorthogonalized variants of the block classical Gram-Schmidt with Pythagorean inner product (BCGS-PIP) algorithm are introduced, which feature improved bounds on the loss of orthogonality compared to the original BCGS-PIP method.
摘要
The content presents two new reorthogonalized variants of the block classical Gram-Schmidt with Pythagorean inner product (BCGS-PIP) algorithm, called BCGS-PIP+ and BCGS-PIPI+. These variants aim to improve the stability and orthogonality of the computed basis compared to the original BCGS-PIP method.
Key highlights:
BCGS-PIP is a communication-efficient variant of the classical Gram-Schmidt algorithm, but it can suffer from significant loss of orthogonality in finite-precision arithmetic.
The new BCGS-PIP+ algorithm runs BCGS-PIP twice to improve orthogonality, and has an O(ε) bound on the loss of orthogonality.
The BCGS-PIPI+ algorithm combines the two BCGS-PIP steps into a single loop, reducing the number of synchronization points, while still maintaining an O(ε) bound on the loss of orthogonality.
Detailed error analysis is provided for both new variants, including bounds on the loss of orthogonality, the standard residual, and the Cholesky residual.
The analysis also covers mixed-precision variants of the new algorithms.
Numerical experiments using the BlockStab toolbox are presented to verify the theoretical results.