Theoretical analysis and numerical solution to a vector equation $Ax-\|x\|_1x=b$
Provides theoretical and numerical methods for a specific vector equation, which is incremental for researchers in matrix equations and numerical analysis.
The paper proves existence and uniqueness of a nonnegative solution to a vector equation with an M-matrix and nonnegative right-hand side, and proposes fixed-point and Newton iterations, with a structure-preserving doubling algorithm achieving at least linear convergence with rate 1/2.
Theoretical and computational properties of a vector equation $Ax-\|x\|_1x=b$ are investigated, where $A$ is an invertible $M$-matrix and $b$ is a nonnegative vector. Existence and uniqueness of a nonnegative solution is proved. Fixed-point iterations, including a relaxed fixed-point iteration and Newton iteration, are proposed and analyzed. A structure-preserving doubling algorithm is proved to be applicable in computing the required solution, the convergence is at least linear with rate 1/2. Numerical experiments are performed to demonstrate the effectiveness of the proposed algorithms.