Preconditioned structure-preserving Jacobi algorithms for computing the SVD of a quaternion matrix

Abstract The singular value decomposition (SVD) of quaternion matrices is a cornerstone of color image compression. However, existing computational methods for quaternion SVD remain inefficient, often lacking the speed and scalability required for real-world applications. To address this challenge, we introduce two novel preconditioned structure-preserving Jacobi algorithms designed to compute the SVD of quaternion matrices efficiently. Central to our approach is the use of quaternion rank-revealing QR (QRRQR) factorization as a preprocessing step, which serves as a robust dimension reduction strategy for both one-sided and two-sided Jacobi methods. Notably, we rigorously analyze the off-diagonal structure of the upper triangular matrix generated by QRRQR, a key innovation that enables significant acceleration of the two-sided Jacobi algorithm. By exploiting this structural property, we effectively reduce computational overhead while preserving numerical stability. Extensive numerical experiments validate the superior efficiency, accuracy, and scalability of our algorithms compared to some conventional approaches, demonstrating their potential for practical deployment in large-scale color image processing tasks.