Two connections between the SR and HR eigenvalue algorithms

The SR and HR algorithms are members of the family of GR algorithms for calculating eigenvalues and invariant subspaces of matrices. This paper makes two connections between the SR and HR algorithms: (1) An iteration of the SR algorithm on a 2n × 2n symplectic butterfly matrix using shifts μi, μ−1i, i = 1,…, k, is equivalent to an iteration of the HR algorithm on an n × n tridiagonal sign-symmetric matrix using shifts μi + μ−1i, i = 1,…, k. (2) An iteration of the SR algorithm on a 2n × 2n J-tridagonal Hamiltonian matrix using shifts μi, −μi, i = 1,…, k, is equivalent to an iteration of the HR algorithm on an n × n tridiagonal sign-symmetric matrix using shifts μ2i, i = 1,…, k.