Conjugate gradient-like methods for solving general tensor equation with Einstein product

Abstract Tensors have a wide application in data mining, chemistry, information sciences, documents analysis and medical engineering. In this work, we study the general tensor equation ∑ i = 1 l F i * P X * Q G i = H with Einstein product where F i , G i , H , for i = 1 , 2 , . . , l , are known tensors and X is an unknown tensor to be determined. The main motivation for this study is the investigation of conjugate gradient-like methods for solving this tensor equation. We show that the conjugate gradient-like methods converge to tensor solutions in a finite number of steps in the absence of round-off errors. Numerical examples confirm the theoretical results and demonstrate the accuracy and computational efficiency of the methods.