\(QR\)-like algorithms for eigenvalue problems (Q1591175)

This is an excellent summary of \(GR\) algorithms, including particularly the \(QR\) algorithm, for solving matrix eigenvalue problems. Let \(A\) be an \(n\times n\) matrix, \(GR\) algorithms start with a matrix \(A_0\) similar to \(A\) and generate a sequence (\(A_m\)) of similar matrices, which converge to a block upper triangular matrix \(\left[\begin{smallmatrix} A_{11} &A_{12}\\ 0 & A_{22}\end{smallmatrix} \right]\) and transform the problem into smaller eigenvalue problems. The history, present and future of \(GR\) algorithms, their implementation and relationships to \(QR\), \(LR\) methods, shift strategies and bulge-chasing procedures are nicely discussed.