Iterative Methods for Symmetric Outer Product Tensor Decompositions

We study the symmetric outer product decomposition which decomposes a fully (partially) symmetric tensor into a sum of rank-one fully (partially) symmetric tensors. We present iterative algorithms for the third-order partially symmetric tensor and fourth-order fully symmetric tensor. The numerical examples indicate a faster convergence rate for the new algorithms than the standard method of alternating least squares

nth Root extraction: Double iteration process and Newton's method

AbstractA double iteration process already used to find the nth root of a positive real number is analysed and showed to be equivalent to the Newton's method. These methods are of order two and three. Higher-order methods for finding the nth root are also mentioned

Optimization via Chebyshev Polynomials

This paper presents for the first time a robust exact line-search method based on a full pseudospectral (PS) numerical scheme employing orthogonal polynomials. The proposed method takes on an adaptive search procedure and combines the superior accuracy of Chebyshev PS approximations with the high-order approximations obtained through Chebyshev PS differentiation matrices (CPSDMs). In addition, the method exhibits quadratic convergence rate by enforcing an adaptive Newton search iterative scheme. A rigorous error analysis of the proposed method is presented along with a detailed set of pseudocodes for the established computational algorithms. Several numerical experiments are conducted on one- and multi-dimensional optimization test problems to illustrate the advantages of the proposed strategy.Comment: 26 pages, 6 figures, 2 table