Linear Algebra pp 396-403

# The Modified LR Algorithm for Complex Hessenberg Matrices

• R. S. Martin
• J. H. Wilkinson
Part of the Handbook for Automatic Computation book series (HDBKAUCO, volume 2)

## 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.

## References

1. 1.
Francis, J. G. F.: The QR transformation — a unitary analogue to the LR transformation. Comput. J. 4, 265–271 and 332–345 (1961/62).
2. 2.
Parlett, B. N.: The development and use of methods of LR type. SIAM Rev. 6, 275–295 (1964).
3. 3.
Rutishauser, H.: Solution of eigenvalue problems with the LR transformation. Nat. Bur. Standards Appl. Math. Ser. 49, 47–81 (1958).
4. 4.
Wilkinson, J. H.: The algebraic eigenvalue problem. London: Oxford University Press 1965.
5. 5.
Wilkinson, J. H.: Convergence of the LR, QR and related algorithms. Comput. J. 8, 77–84 (1965).
6. 6.
Parlett, B. N., and C. Reinsch: Balancing a matrix for calculation of eigenvalues and eigenvectors. Numer. Math. 13, 293–304 (1969). Cf. II/11.
7. 7.
Martin, R. S., and J. H. Wilkinson: Similarity reduction of a general matrix to Hessenberg form. Numer. Math. 12, 349–368 (1968). Cf. II/13-
8. 8.
Martin, R. S., G. Peters and J. H. Wilkinson: The QR algorithm for real Hessenberg matrices. Numer. Math. 14, 219–231 (1970). Cf. II/14.