Algorithm 1019: A Task-based Multi-shift QR/QZ Algorithm with Aggressive Early Deflation

The QR algorithm is one of the three phases in the process of computing the eigenvalues and the eigenvectors of a dense nonsymmetric matrix. This paper describes a task-based QR algorithm for reducing an upper Hessenberg matrix to real Schur form. The task-based algorithm also supports generalized eigenvalue problems (QZ algorithm) but this paper concentrates on the standard case. The task-based algorithm adopts previous algorithmic improvements, such as tightly-coupled multi-shifts and Aggressive Early Deflation (AED) , and also incorporates several new ideas that significantly improve the performance. This includes, but is not limited to, the elimination of several synchronization points, the dynamic merging of previously separate computational steps, the shortening and the prioritization of the critical path, and experimental GPU support. The task-based implementation is demonstrated to be multiple times faster than multi-threaded LAPACK and ScaLAPACK in both single-node and multi-node configurations on two different machines based on Intel and AMD CPUs. The implementation is built on top of the StarPU runtime system and is part of the open-source StarNEig library.

Representations for the image-kernel (p,q)-inverses of block matrices in rings

We present the conditions for a block matrix of a ring to have the image-kernel{(p,q)}-inverse in the generalized Banachiewicz–Schur form. We give representations for the image-kernel inverses of the sum and the product of two block matrices. Some characterizations of the image-kernel{(p,q)}-inverse in a ring with involution are investigated too.