A modified CG algorithm for solving generalized coupled Sylvester tensor equations

Preconditioned TBiCOR and TCORS Algorithms for Solving the Sylvester Tensor Equation

“…From Figure 1, we can see that our preconditioned Algorithms 5-7 present better convergence than Algorithms 2-4 without preconditioning, PGMRES in [9], PBiCG and PBiCR in [30]. While Algorithms 2-4 can compare with PGMRES, PBiCG and PBiCR, and are better than CGLS [16] and MCG [23]. Algorithm 6 converges fastest among all algorithms.…”

“…In this section, we show several numerical examples to illustrate Algorithms 2-7 and compare them with CGLS in [16], MCG in [23], preconditioned GMRES (PGMRES) in [9], preconditioned BiCG (PBiCG) and preconditioned BiCR (PBiCR) in [30]. All experiments are implemented on a computer with macOS Big Sur 11.1 and 8G memory.…”

“…Algorithms 2-6 are used to solve (1.1) with the matrices A i in (6.1). These methods are compared with CGLS in [16] and MCG in [23], respectively.…”

“…The perturbation bounds and backward error are presented in [27] for solving (1.1). For more methods on other linear systems in tensor form we refer to [4,16,18,23]. Using the nearest Kronecker product (NKP) in [28], Chen and Lu [9] presented an efficient preconditioner for solving Eq.…”

Numerical solution of a class of third order tensor linear equations

“…In most cases, the authors have been interested in the presence of many summands and many Kronecker products, for which iterative methods appear to be mandatory. In this context, most approaches try to take into account the Kronecker structure and the possible low rank of the involved iteration matrices, see, e.g., [2,3,6,13,[19][20][21]. However, little has been said on "direct" dense methods for low order tensor equations, without the explicit use of the Kronecker form.…”