Multidimensional Differential-Transform-Based Computation of Characteristics of Multiparameter Complex Matrix

Abstract

This paper develops a multidimensional differential-transform-based framework for computing matrix characteristics of complex-valued multiparameter matrix functions. The proposed approach extends differential-transform techniques from one-parameter matrix functions to functions depending on several independent variables and constructs multidimensional D-analogues of the Leverrier and Faddeev methods. In the spectral domain, products of parameter-dependent scalar and matrix functions are replaced by multidimensional convolutions, which makes it possible to compute the spectra of characteristic-polynomial coefficients, determinants, and inverse-matrix entries by recurrence relations. The convergence of the inverse multidimensional transform is discussed in terms of analyticity in a polydisc, and truncation-error estimates and residual-based a posteriori indicators are introduced for controlling the accuracy of reconstructed inverse matrices in locally nonsingular parameter regions. The method is implemented in a Python 3.14-based computational framework that stores and manipulates multidimensional differential spectra. The approach is verified on multiparameter complex matrix examples, including a three-degree-of-freedom damped vibration system and a dense 7 × 7, seven-parameter complex matrix. Numerical residuals confirm the consistency of the reconstructed inverse matrices with MATLAB R2025b-based verification. The results show that the proposed recurrence-based method is especially useful for sparse multidimensional spectra, bounded-order local reconstructions, and repeated evaluations of matrix characteristics near a fixed expansion point.