The Modified LR 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

$${A_s} = {L_s}{R_s},\quad {R_s}{L_s} = {A_{s + 1}}$$

((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 A _s 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.

Prepublished in Numer. Math. 12, 369–376 (1968).