A Modified Conjugate Residual Method and Nearest Kronecker Product Preconditioner for the Generalized Coupled Sylvester Tensor Equations

This paper is devoted to proposing a modified conjugate residual method for solving the generalized coupled Sylvester tensor equations. To further improve its convergence rate, we derive a preconditioned modified conjugate residual method based on the Kronecker product approximations for solving the tensor equations. A theoretical analysis shows that the proposed method converges to an exact solution for any initial tensor at most finite steps in the absence round-off errors. Compared with a modified conjugate gradient method, the obtained numerical results illustrate that our methods perform much better in terms of the number of iteration steps and computing time.

Automated summary

This paper proposes a modified conjugate residual method for solving the generalized coupled Sylvester tensor equations, and further derives a preconditioned modified conjugate residual method based on Kronecker product approximations. Theoretical analysis and numerical results demonstrate that our methods outperform the traditional conjugate gradient method in terms of convergence rate and computational efficiency.