Implicit QR algorithms for palindromic and even eigenvalue problems (Q1027778)

A structure preserving variant of the implicit QR algorithm is given for palindromic eigenvalue problems. The palindromic eigenvalue problem for an \((n,n)\) matrix \(A\) is to find \(\lambda\) such that \(Ax=\lambda A^* x\), \(x\neq 0\). The matrix \(A\) is said to be anti-triangular if \(a_{ij}= 0\), \(i+ j\leq n\) Given a matrix \(A\) the object of the algorithm is to reduce it to anti-triangular form. This is accomplished assuming the \(A\) is already reduced to an anti-Hessenberg matrix, \(a_{ij}= 0\), \(i+j\leq n\). A detailed description of the algorithm is presented together with application to an optimal control problem and a steel rail cooling problem. The algorithm is strongly backward stable and requires, less operations than the standard QZ algorithm.