# The QR Algorithm for Real Hessenberg Matrices

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

## Abstract

The QR algorithm of Francis [1] and Kublanovskaya [4] with shifts of origin is described by the relations
$$\matrix{ {{Q_s}({A_s} - {k_s}I) = {R_s},} & {{A_{s + 1}} = {R_s}Q_s^T + {k_s}I,} & {giving} \cr } \matrix{ {{A_{s + 1}} = } \hfill \cr } {Q_s}{A_s}Q_s^T,$$
(1)
where Q s is orthogonal, R s is upper triangular and k s is the shift of origin. When the initial matrix A 1 is of upper Hessenberg form then it is easy to show that this is true of all A s . The volume of work involved in a QR step is far less if the matrix is of Hessenberg form, and since there are several stable ways of reducing a general matrix to this form [3,5, 8], the QR algorithm is invariably used after such a reduction.

## Keywords

National Physical Laboratory Matrix Iteration Principal Submatrix Hessenberg Matrix Current Matrix
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Francis, J. C. F.: The QR transformation -a unitary analogue to the LR transformation. Comput. J. 4, 265–271 and 332–345 (1961/62).
2. 2.
Householder, A. S.: Unitary triangularization of a non-symmetric matrix. J. Assoc. Comput. Mach. 5, 339–342 (1958).
3. 3.
Householder, A. S Bauer, F. L.: On certain methods for expanding the characteristic polynomial. Numer. Math. 1, 29–37 (1959).
4. 4.
Kublanovskaya, V. N.: On some algorithms for the solution of the complete eigenvalue problem. 2. Vyisl. Mat. i Mat. Fiz. 1, 555 -570 (1961).Google Scholar
5. 5.
Martin, R. S., Wilkinson, J. H.: Similarity reduction of a general matrix to Hessenberg form. Numer. Math. 12, 349–368 (1968). Cf. 11/13.
6. 6.
Parlett, B. N.: Global convergence of the basic QR algorithm on Hessenberg matrices. Math. Comp. 22, 803 -817 (1968).
7. 7.
Parlett, B. N.: Reinsch, C.: Balancing a matrix for calculation of eigenvalues and eigenvectors. Numer. Math. 13, 293 -304 (1969). Cf. II/ll.
8. 8.
Wilkinson, J. H.: The algebraic eigenvalue problem. London: Oxford University Press 1965.

© Springer-Verlag Berlin · Heidelberg 1971

## Authors and Affiliations

• R. S. Martin
• G. Peters
• J. H. Wilkinson

There are no affiliations available