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A quaternion QR algorithm

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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.

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Bunse-Gerstner, A., Byers, R. & Mehrmann, V. A quaternion QR algorithm. Numer. Math. 55, 83–95 (1989).

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