A Real QZ Algorithm for Structured Companion Pencils
This work provides a more efficient eigenvalue algorithm for polynomial rootfinding, a common problem in numerical linear algebra, but the improvement is incremental as it adapts an existing algorithm to a specific structure.
The paper presents a fast implicit real QZ algorithm for computing eigenvalues of structured companion pencils from polynomial rootfinding problems, achieving O(N) flops per iteration and O(N) memory storage, with numerical experiments confirming effectiveness and stability.
We design a fast implicit real QZ algorithm for eigenvalue computation of structured companion pencils arising from linearizations of polynomial rootfinding problems. The modified QZ algorithm computes the generalized eigenvalues of an $N\times N$ structured matrix pencil using $O(N)$ flops per iteration and $O(N)$ memory storage. Numerical experiments and comparisons confirm the effectiveness and the stability of the proposed method.