Search

A randomized quaternion singular value decomposition algorithm based on block Krylov iteration (RQSVD-BKI) is presented to solve the low-rank quaternion matrix approximation problem. The upper bounds of deterministic approximation error and expected approximation error for the RQSVD-BKI algorithm are also given. It is shown by numerical experiments that the running time of the RQSVD-BKI algorithm is smaller than that of the quaternion singular value decomposition, and the relative errors of the RQSVD-BKI algorithm are smaller than those of the randomized quaternion singular value decomposition algorithm in Liu et al. (SIAM J. Sci. Comput., 44(2): A870-A900 (2022)) in some cases. In order to further illustrate the feasibility and effectiveness of the RQSVD-BKI algorithm, we use it to deal with the problem of color image inpainting. [ABSTRACT FROM AUTHOR]

The Cauchy radius of a scalar polynomial is an upper bound on the magnitude of its zeros, and it is optimal among all bounds depending only on the moduli of the coefficients. It has the disadvantage of being implicit because it requires the solution of a nonlinear equation. In this note, simple and explicit upper bounds are derived that are useful approximations to the Cauchy radius. The general approach to obtain the aforementioned bounds is to embed scalar polynomials into the larger framework of their generalization to matrix polynomials and then use bounds on the eigenvalues of the latter. [ABSTRACT FROM AUTHOR]