Core-Chasing Algorithms for the Eigenvalue Problem

Eigenvalue algorithms are often introduced through the matrices they transform. This monograph offers a different way to follow the computation: it recasts Francis’s familiar implicitly shifted QR method as a process of moving core transformations through structured matrix factorizations. That perspective leads into both the algorithm’s mathematical foundations and practical variants for important classes of eigenvalue problems.

See Francis’s QR algorithm from a new angle

The opening chapters introduce core transformations and QR decomposition, then explain Francis’s algorithm and how its bulge-chasing procedure can be understood and implemented as core chasing. Convergence, deflation, singular cases, and backward stability are treated alongside the algorithm itself, giving readers a way to connect the mechanics of an iteration with the reasons it works.

Follow the method into structured problems

Subsequent chapters examine how the approach changes when a matrix has useful structure. Topics include unitary matrices, companion matrices, symmetric matrices, and symmetric-plus-rank-one matrices. The discussion also reaches generalized eigenvalue problems, companion pencils, and matrix polynomial eigenvalue problems—settings where structure can shape the design of an efficient solver.

From mathematical ideas to computation

Beyond describing transformations, the book considers implementation concerns such as storage, computational cost, cache use, and parallelism. It also develops algorithms for generalized forms of Hessenberg structure. The result is a focused treatment of how numerical methods move from a mathematical description toward working computational procedures.

Who will find it useful?

Readers should already be comfortable with linear algebra and matrix computations. Researchers and practitioners in numerical linear algebra, as well as advanced students in mathematics, engineering, and computational science, will find a detailed account of algorithms for eigenvalues and related quantities such as eigenvectors and invariant subspaces.

For readers interested in the design, analysis, and implementation of modern eigenvalue methods, Core-Chasing Algorithms for the Eigenvalue Problem offers a sustained look at what changes when the computation is organized around cores rather than bulges.