The Modified LR Algorithm for Complex Hessenberg Matrices

  • R. S. Martin
  • J. H. Wilkinson
Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 186)


The LR algorithm of Rutishauser [3] is based on the observation that if
$$ A = LR $$
where L is unit lower-triangular and R is upper-triangular then B defined by
$$ B = {L^{ - 1}}AL = RL $$
is similar to A. Hence if we write
$$ \matrix{ {{A_s} = {L_s}{R_s},} & {{R_s}{L_s} = {A_{s + 1}},} \cr } $$
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.


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  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).MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Parlett, B. N.: The development and use of methods of LR type. SI AM Rev. 6, 275–295 (1964).MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Rutishauser, H.: Solution of eigenvalue problems with the LR transformation. Nat. Bur. Standards Appl. Math. Ser. 49, 47–81 (1958).MathSciNetGoogle Scholar
  4. 4.
    Wilkinson, J. H.: The algebraic eigenvalue problem. London: Oxford University Press 1965-zbMATHGoogle Scholar
  5. 5.
    Wilkinson, J. HConvergence of the LR ,QR and related algorithms. Comput. J. 8, 77–84 (1965).MathSciNetzbMATHCrossRefGoogle Scholar
  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/ll.MathSciNetzbMATHCrossRefGoogle Scholar
  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.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Martin, R. S., and J. H. Wilkinson, G.Peter. and J. H. Wilkinson. The QR algorithm for real Hessenberg matrices. Numer. Math. 14, 219–231 (1970). Cf. 11/14.MathSciNetzbMATHCrossRefGoogle Scholar

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© Springer-Verlag Berlin · Heidelberg 1971

Authors and Affiliations

  • R. S. Martin
  • J. H. Wilkinson

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