Summary
This paper extends the Francis QR algorithm to quaternion and antiquaternion matrices. It calculates a quaternion version of the Schur decomposition using quaternion unitary similarity transformations. Following a finite step reduction to a Hessenberg-like condensed form, a sequence of implicit QR steps reduces the matrix to triangular form. Eigenvalues may be read off the diagonal. Eigenvectors may be obtained from simple back substitutions. For serial computation, the algorithm uses only half the work and storage of the unstructured Francis QR iteration. By preserving quaternion structure, the algorithm calculates the eigenvalues of a nearby quaternion matrix despite rounding errors.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bunse-Gerstner, A., Byers, R., Mehrmann, V.: Numerically stable structure preserving methods for eigenproblems with special structure. Fakultät für Mathematik, Universität Bielefeld. Postf. 8640, D-4800 Bielefeld 1 (1987)
Bunse-Gerstner, A.: Symplectic QR-like methods. Habilitationsschrift, Fakultät für Mathematik, Universität Bielefeld, Postf. 8640, D-4800 Bielefeld 1 (1986)
Bunse-Gerstner, A., Gragg, W.: Singular value decompositions of complex symmetric matrices. J. Comput. Appl. Math.21, 41–54 (1988)
Dongarra, J.J., Gabriel, J.R., Kolling, D.D., Wilkinson, J.H.: The eigenvalue problem for hermitian matrices with time reversal symmetry. Linear Algebra Appl.60, 27–42 (1984)
Francis, J.: The QR transformation, Part II. Comput. J.5, 332–345 (1962)
Golub, G., Van Loan, C.: Matrix computations, 1st Ed. Baltimore, Maryland: Hopkins 1983
Reference deleted
Herstein, I.N.: Topics in algebra. 2nd Ed. Toronto: Xerox College 1975
Lax, M.: Symmetry principles in solid state and molecular physics, 1st Ed. New York: Wiley 1974
Martin, R.S., Wilkinson, J.H.: Similarity reduction of a general matrix to Hessenberg form. Numer. Math.12, 349–368 (1968)
Parlett, B.N.: Global convergence of the basic QR algorithm on Hessenberg matrices. Math. Comput.22, 803–817 (1968)
Rosch, N.: Time-reversal symmetry, Kramers' degeneracy and the algebraic eigenvalue problem. Chemical Phys.80, 1–5 (1983)
Watkins, D.: Understanding the QR algorithm. SIAM Review24, 427–440 (1982)
Wigner, E.P.: Group theory and its application the quantum mechanics of atomic spectra. New York: Academic Press 1959
Wilkinson, J.H.: The algebraic eigenvalue problem. London: Oxford University Press (Clarendon) 1965