ZND-ZeaD Models and Theoretics Including Proofs for Takagi Factorization of Complex Time-Dependent Symmetric Matrix

Takagi factorization (TF) is a kind of matrix factorization (also termed, decomposition) method. In this paper, the problem of TF for complex time-dependent symmetric matrix (simply termed, time-dependent Takagi factorization) is presented and theoretically studied. To simplify the tough problem, we first consider converting it into an equivalent equation system. Then, a continuous-time (CT) TF (CTTF) model is presented and derived theoretically, by adopting ZND (Zhang neural dynamics, or termed, Zhang neural network) method and a dimensional reduction technique. In order to facilitate the implementation on digital hardware platforms, the corresponding discrete-time (DT) ZND solution models (or termed, algorithms) with different discretization precisions of the CTTF model are presented and taken into theoretical consideration. Specifically, we present and investigate a discrete-time TF (DTTF) model, termed eight-instant DTTF (8IDTTF) model, by using a new eight-instant Zhang et al discretization (ZeaD, also termed, Zhang time discretization, ZTD) formula to discretize the CTTF model. Besides, two other DTTF models, termed four-instant DTTF (4IDTTF) and six-instant DTTF (6IDTTF) models, are presented respectively, on the basis of two corresponding multiinstant ZeaD formulas. Moreover, the effective solution step-size intervals of the three DTTF models are further presented and proved.