The modifiedLR algorithm for complex Hessenberg matrices

Abstract

The LR algorithm of Rutishauser [3] is based on the observation that if $$ A = LR $$ (1) where L is unit lower-triangular and R is upper-triangular then B defined by $$ B = {L^{ - 1}}AL = RL $$ (2) is similar to A. Hence if we write $$ \matrix{ {{A_s} = {L_s}{R_s},} & {{R_s}{L_s} = {A_{s + 1}},} \cr } $$ (3) a sequence of matrices is obtained each of which is similar to A 1. Rutishauser showed [3, 4, 5] if A 1 has roots of distinct moduli then, in general As tends to upper triangular form, the diagonal elements tending to the roots arranged in order of decreasing modulus. If A 1 has some roots of equal modulus then A s does not tend to strictly triangular form but rather to block-triangular form. Corresponding to a block of k roots of the same modulus there is a diagonal block of order k which does not tend to a limit, but its roots tend to the k eigenvalues.