A New Real Structure-preserving Quaternion QR Algorithm
This work provides faster and more robust algorithms for computing eigenvalues of quaternion matrices, which is important for applications in quantum mechanics and computer graphics.
The paper introduces new real structure-preserving decompositions and algorithms for the right eigenproblem of general quaternion matrices, including a JRS-QR algorithm and an implicit double shift quaternion QR algorithm. Numerical experiments demonstrate efficiency and accuracy.
New real structure-preserving decompositions are introduced to develop fast and robust algorithms for the (right) eigenproblem of general quaternion matrices. Under the orthogonally JRS-symplectic transformations, the Francis JRS-QR step and the JRS-QR algorithm are firstly proposed for JRS-symmetric matrices and then applied to calculate the Schur forms of quaternion matrices. A novel quaternion Givens matrix is defined and utilized to compute the QR factorization of quaternion Hessenberg matrices. An implicit double shift quaternion QR algorithm is presented with a technique for automatically choosing shifts and within real operations. Numerical experiments are provided to demonstrate the efficiency and accuracy of newly proposed algorithms.