Shifted Kronecker product systems

A fast method for solving a linear system of the form (A((p)) x ... x A((1)) - lambda I)x = b is given where each A((i)) is an n(i)-by-n(i) matrix. The first step is to convert the problem to triangular form (T-(p) x ... x T-(1) - lambda I)y = c by computing the (complex) Schur decompositions of the A((i)). This is followed by a recursive back-substitution process that fully exploits the Kronecker structure and requires just O(N(n(1) +...+ n(p))) flops where N = n(1) ... n(p). A similar method is employed when the real Schur decomposition is used to convert each A((i)) to quasi-triangular form. The numerical properties of these new methods are the same as if we explicitly formed (T-(p) x ... x T-(1) - lambda I) and used conventional back-substitution to solve for y.