A Study of Numerical Analysis for Multi-Linear Algebra and Differential Equations

Exploring the Advances in Numerical Multi-Linear Algebra and its Applications

Abstract

Numerical multi-linear algebra, in which instead of matrices and vectors the higher-order tensors are considered in numerical viewpoint, is a new branch of computational mathematics. Although the linear numerical algebra is an extension of it, it has many essential differences and more difficulties than the linear numerical algebra. This paper presents an incomplete survey of state-of-the-art knowledge on this issue and shows new trends in further research. Our survey also includes an important part of a detailed bibliography. A new branch of computer maths is numerical multilinear algebra. It deals with the numerical handling by replacing matrix of higher-order tensors. Various computational issues related to the higher order tensors are included, such as decomposition of the tensor, tensor range calculation, own-value computation of the tensor, lower rank tensor approximations, numerical stability, and tensor calculation disturbance analysis, etc. This new business branch has a strong practical background and broad applications in the fields of digital image restore psychometrics, chemometrics, econometrics and multi-way data analysis.