A General Matrix Eigenvalue Algorithm

A general QR-type process called the VZ algorithm is presented for the solution of the general matrix eigenvalue problem ACx = λ BDx. The matrices involved may be rectangular. For appropriate choices of A, B, C and D, we have some of the more familiar types of eigenproblems, and this is reflected in the fact that the QR, QZ and SVD algorithms are all special cases of the VZ algorithm. The main emphasis is upon the algorithm's generality as well as its bearing upon the generalized singular value problem AT Ax = μ2BTBx.