Numerical Methods for Quaternion and Matrix Computations

The field of numerical linear algebra is moving towards the development of more efficient and stable algorithms for computing eigenvalues and singular value decompositions of matrices, particularly for quaternion and dual quaternion matrices. Researchers are exploring new methods for evaluating eigenvalues, such as using trigonometric formulas and Lie theory, and are also investigating the application of these methods to navigation systems and data analysis. Notable papers include:

  • A paper on numerically stable evaluation of closed-form expressions for eigenvalues of 3x3 matrices, which presents a new algorithm that is approximately ten times faster than the highly optimized LAPACK library.
  • A paper on generalized singular value decompositions of dual quaternion matrices, which introduces several types of GSVD for dual quaternion data matrices and presents artificial examples to illustrate the principle of the DQGSVD.

Sources

Numerically stable evaluation of closed-form expressions for eigenvalues of $3 \times 3$ matrices

Runge-Kutta Approximations for Direct Coning Compensation Applying Lie Theory

Generalized singular value decompositions of dual quaternion matrices and beyond

On Eigenvector Computation and Eigenvalue Reordering for the Non-Hermitian Quaternion Eigenvalue Problem

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