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 $7 \times 7$, 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.

1. Introduction

Parameter-dependent matrix functions arise in many problems of computational mathematics, engineering analysis, control theory, vibration modeling, finite-element discretization, and frequency-domain simulation. In these problems, the system matrix is not fixed but depends on several independent physical, geometrical, material, control, or spectral parameters. As a result, the evaluation of determinants, characteristic polynomials, inverse matrices, generalized inverses, and eigenvalue-related quantities becomes a central computational task. Such quantities are required for analyzing singularity conditions, determining stability regions, computing resonance frequencies, solving parameter-dependent linear systems, and constructing local analytical or numerical-analytical approximations of system behavior.

The computation of inverse and generalized inverse matrices for matrix-valued functions has been studied using different numerical, symbolic, and hybrid approaches. Generalized inverses and their numerical and symbolic computation are systematically treated in the monograph by Wei, Stanimirović, and Petković [1], which provides an important theoretical and computational background for this class of problems. For polynomial matrix functions, interpolation-based algorithms have been proposed for computing the inverse of two-variable polynomial matrices [2], while DFT-based algorithms have been developed for Moore–Penrose and Drazin inverses of multivariable polynomial matrices [3]. Approximation of parameter-dependent inverse operators has also been considered in connection with preconditioning and projection-based computations for large-scale parameter-dependent systems [4]. Rational approximation techniques for matrix-valued functions have been developed for nonlinear eigenvalue problems and model reduction [5], and rational pseudo-inverses of multivariable rational matrix-valued functions have been investigated in [6]. These studies demonstrate that the computation of inverse and generalized inverse matrices for multivariable matrix functions is both theoretically important and computationally demanding.

A closely related class of problems concerns the computation of determinants, characteristic polynomials, and eigenvalues of parameter-dependent matrices. In delay-differential equations, stability analysis may lead to matrix pencil, matrix polynomial, or multiparameter eigenvalue problems [7]. In numerical approximations of partial differential equations, generalized eigenvalue problems of the form $Ax=\lambda Bx$ arise, where one or both matrices may depend on physical, discretization, or stabilization parameters [8]. Parameter-dependent Lyapunov equations are used in parametric model order reduction [9]. In control theory, structural dynamics, and frequency-domain engineering problems, eigenvalues of parameter-dependent matrices determine stability, vibration frequencies, resonance behavior, and robustness properties [10,11]. These applications are particularly important when the matrices are complex-valued or rational and depend simultaneously on several physical, geometrical, material, boundary, or control parameters.

Although modern numerical and symbolic computational systems can evaluate matrix characteristics in many cases, direct computation becomes increasingly difficult when the matrix dimension, the number of independent parameters, and the algebraic complexity of the entries increase. Numerical inversion and eigenvalue computation are efficient for fixed parameter values, but they provide information only at isolated points of the parameter space and may become unreliable near singular or ill-conditioned regions. Direct symbolic computation can, in principle, produce closed-form expressions for determinants, inverses, and characteristic polynomials, but the resulting expressions may become very large, especially for rational and complex-valued matrix functions. Therefore, recurrence-based and transform-domain methods are of interest as alternatives to direct determinant–adjugate symbolic constructions.

The differential transformation method, originally introduced by Pukhov [12], provides a systematic framework for transforming differential and functional relations into algebraic recurrence relations for differential spectra. Expansion formulas and operational rules for differential transforms were further developed in [13]. The status and interpretation of the differential transformation method were analyzed by Bervillier [14], who emphasized its close connection with power-series-based semi-analytical computation. The applied theory of differential transforms was further developed in [15], where differential spectra were used as a computational tool for constructing analytical and numerical-analytical solutions. More recently, differential transformations have also been applied to matrix-related problems, including the construction of complex one-parameter generalized Moore–Penrose inverses [16]. The use of differential transforms has also been demonstrated in engineering vibration problems, such as the analysis of longitudinal vibrations of rods with variable geometry [17].

The existing literature therefore contains two complementary directions. The first direction includes numerical, symbolic, interpolation, DFT-based, rational approximation, and projection methods for inverse and generalized inverse matrices and multivariable matrix-valued functions. The second direction includes differential-transform-based methods for differential equations and one-parameter matrix problems. However, the construction of determinants, characteristic polynomials, inverse matrices, and eigenvalue-related quantities for matrices depending on several independent parameters within a multidimensional differential-transform framework remains insufficiently developed, especially for complex-valued rational matrix functions. This gap motivates the present study.

The main objective of this paper is to develop a multidimensional differential-transform-based framework for computing the principal characteristics of multiparameter complex matrix functions. The proposed approach represents a matrix-valued function by its multidimensional differential spectrum and transforms operations on matrix functions into recurrence relations for the corresponding spectra. Within this framework, D-analogues are constructed for determinants, characteristic polynomials, and inverse matrices. Special attention is given to complex-valued rational matrix functions, for which direct symbolic computation may become computationally inconvenient.

Compared with the previous D-transform-based approaches for one-parameter matrix functions and generalized inverse matrices [15,16], the present work considers complex-valued matrix functions depending on several independent parameters. This transition is not only a formal replacement of one variable by several variables. It requires the construction of multidimensional spectral recurrences, the systematic use of multidimensional convolution, the introduction of truncation vectors, and the analysis of convergence, truncation error, and residual-based verification for reconstructed matrix characteristics.

The main contributions of this paper are as follows.

• A multidimensional differential-transform framework is formulated for complex-valued multiparameter matrix functions. The matrix entries may depend simultaneously on several independent variables, and their multidimensional differential spectra are used as the basic computational objects.
• D-analogues of the Leverrier and Faddeev methods are developed in the multidimensional spectral domain. These analogues provide recurrence relations for computing the spectra of the characteristic-polynomial coefficients and determinants without direct symbolic expansion by permutations.
• A recurrence-based construction of inverse matrices is proposed through the Faddeev D-analogue. The reconstructed inverse matrix is represented by multidimensional spectra, and its correctness is verified by symbolic identities and residual-based a posteriori indicators.
• Convergence and truncation-error issues are addressed using analyticity in a polydisc, Cauchy estimates, truncation vectors, and residual norms. These tools provide practical criteria for selecting or increasing the truncation orders.
• The method is implemented in a Python-based symbolic computational framework and validated on multiparameter complex matrix examples, including an applied damped vibration system and a dense $7 \times 7$ seven-parameter matrix. The results are compared with MATLAB symbolic and numerical computations.

The remainder of the paper is organized as follows. Section 2 introduces the main definitions, operational rules, and properties of multidimensional differential transforms for scalar and matrix-valued functions. Section 3 develops D-analogues of the Leverrier and Faddeev methods for computing characteristic polynomials, determinants, and inverse matrices of multiparameter complex-valued matrices. Section 4 presents an illustrative example involving a fourth-order complex matrix depending on three independent parameters and demonstrates the reconstruction of the characteristic polynomial and inverse matrix. Section 5 concludes the paper.

2. Preliminaries and Multidimensional Differential Transforms

2.1. Multidimensional Differential Transforms

Multidimensional differential transformations for functions of several independent variables were introduced by Pukhov [12,13]. Let $\mathrm{u}\left(\mathbf{x}\right)=\mathrm{u}\left({\mathrm{x}}_1,{\mathrm{x}}_2,\dots ,{\mathrm{x}}_{\mathrm{m}}\right)$ be a sufficiently differentiable function in a neighborhood of the point ${\mathbf{x}}_{\mathbf{0}}=\left({\mathrm{x}}_{10},{\mathrm{x}}_{20},\dots ,{\mathrm{x}}_{\mathrm{m}0}\right).$ The direct multidimensional differential transform of $\mathrm{u}\left(\mathrm{x}\right)$ is defined as

$$\mathrm{U}\left({\mathrm{k}}_1,{\mathrm{k}}_2,\dots ,{\mathrm{k}}_{\mathrm{m}}\right)={\displaystyle \frac{{\mathrm{H}}_1^{{\mathrm{k}}_1}{\mathrm{H}}_2^{{\mathrm{k}}_2}\cdots {\mathrm{H}}_{\mathrm{m}}^{{\mathrm{k}}_{\mathrm{m}}}}{{\mathrm{k}}_1!{\mathrm{k}}_2!\cdots {\mathrm{k}}_{\mathrm{m}}!}}{{\displaystyle \frac{{\partial }^{{\mathrm{k}}_1+{\mathrm{k}}_2+\cdots +{\mathrm{k}}_{\mathrm{m}}}\mathrm{u}\left({\mathrm{x}}_1,{\mathrm{x}}_2,\dots ,{\mathrm{x}}_{\mathrm{m}}\right)}{\partial {\mathrm{x}}_1^{{\mathrm{k}}_1}\partial {\mathrm{x}}_2^{{\mathrm{k}}_2}\cdots \partial {\mathrm{x}}_{\mathrm{m}}^{{\mathrm{k}}_{\mathrm{m}}}}}|}_{{\mathrm{x}}_1={\mathrm{x}}_{10},{\mathrm{x}}_2={\mathrm{x}}_{20},\dots ,{\mathrm{x}}_{\mathrm{m}}={\mathrm{x}}_{\mathrm{m}0}}$$

(1)

The inverse differential transform, or the reconstruction relation, has the form

$$\mathrm{u}\left({\mathrm{x}}_1,{\mathrm{x}}_2,\dots ,{\mathrm{x}}_{\mathrm{m}}\right)={\displaystyle \sum _{\mathrm{K}=0}^{\mathrm{\infty }}}\sum _{\begin{array}{c}\begin{array}{c}{\mathrm{k}}_1+{\mathrm{k}}_2+\cdots +{\mathrm{k}}_{\mathrm{m}}=K, \end{array}\\ {\mathrm{k}}_1,{\mathrm{k}}_2,\dots ,{\mathrm{k}}_{\mathrm{m}}\ge 0\end{array}}\mathrm{U}\left({\mathrm{k}}_1,{\mathrm{k}}_2,\dots ,{\mathrm{k}}_{\mathrm{m}}\right){\displaystyle \prod _{\mathrm{r}=1}^{\mathrm{m}}}{\left({\displaystyle \frac{{\mathrm{x}}_{\mathrm{r}}-{\mathrm{x}}_{\mathrm{r}0}}{{\mathrm{H}}_{\mathrm{r}}}}\right)}^{{\mathrm{k}}_{\mathrm{r}}}$$

(2)

Equivalently, relation (2) can be written as the multidimensional power series

$$\mathrm{u}\left(\mathbf{x}\right)={\displaystyle \sum _{{\mathrm{k}}_1=0}^{\mathrm{\infty }}}{\displaystyle \sum _{{\mathrm{k}}_2=0}^{\mathrm{\infty }}}\dots {\displaystyle \sum _{{\mathrm{k}}_{\mathrm{m}}=0}^{\mathrm{\infty }}}\mathrm{U}\left({\mathrm{k}}_1,{\mathrm{k}}_2,\dots ,{\mathrm{k}}_{\mathrm{m}}\right){\displaystyle \prod _{\mathrm{r}=1}^{\mathrm{m}}}{\left({\displaystyle \frac{{\mathrm{x}}_{\mathrm{r}}-{\mathrm{x}}_{\mathrm{r}0}}{{\mathrm{H}}_{\mathrm{r}}}}\right)}^{{\mathrm{k}}_{\mathrm{r}}}$$

(3)

Here $\mathrm{u}\left(\mathbf{x}\right)$ is the original function, $\mathrm{U}\left({\mathrm{k}}_1,{\mathrm{k}}_2,\dots ,{\mathrm{k}}_{\mathrm{m}}\right)$ is its multidimensional differential image, or differential spectrum, depending on the integer arguments $\left({\mathrm{k}}_1,{\mathrm{k}}_2,\dots ,{\mathrm{k}}_{\mathrm{m}}\right)$. The quantities $\left({\mathrm{H}}_1,{\mathrm{H}}_2,\dots ,{\mathrm{H}}_{\mathrm{m}}\right)$ are scale factors, and $\left({\mathrm{x}}_{10},{\mathrm{x}}_{20},\dots ,{\mathrm{x}}_{\mathrm{m}0}\right)$ are the coordinates of the expansion center. It is assumed that all partial derivatives required in (1) exist in a neighborhood of $\left({\mathrm{x}}_0\right)$.

For compact notation, we introduce the multi-index

$$\mathbf{k}=\left({\mathrm{k}}_1,{\mathrm{k}}_2,\dots ,{\mathrm{k}}_{\mathrm{m}}\right),\hspace{1em}{\mathrm{k}}_{\mathrm{r}}\in {\mathbb{N}}_0,\hspace{1em}\mathrm{r}=1,2,\dots ,\mathrm{m},$$

and write

$$\left|\mathbf{k}\right|={\mathrm{k}}_1+{\mathrm{k}}_2+\cdots +{\mathrm{k}}_{\mathrm{m}},$$

$$\mathbf{k}!={\mathrm{k}}_1!{\mathrm{k}}_2!\cdots {\mathrm{k}}_{\mathrm{m}}!,$$

$${\mathbf{H}}^{\mathbf{k}}={\mathrm{H}}_1^{{\mathrm{k}}_1}{\mathrm{H}}_2^{{\mathrm{k}}_2}\cdots {\mathrm{H}}_{\mathrm{m}}^{{\mathrm{k}}_{\mathrm{m}}}.$$

Using this notation, the direct transform (1) can be rewritten as

$$\mathrm{U}\left(\mathbf{k}\right)={\displaystyle \frac{{\mathrm{H}}^{\mathbf{k}}}{\mathbf{k}!}}{{\displaystyle \frac{{\partial }^{\left|\mathbf{k}\right|}\mathrm{u}\left(\mathrm{x}\right)}{\partial {\mathrm{x}}_1^{{\mathrm{k}}_1}\partial {\mathrm{x}}_2^{{\mathrm{k}}_2}\cdots \partial {\mathrm{x}}_{\mathrm{m}}^{{\mathrm{k}}_{\mathrm{m}}}}}|}_{\mathrm{x}={\mathrm{x}}_0}.$$

(4)

The inverse transform takes the form

$$\mathrm{u}\left(\mathbf{x}\right)={\displaystyle \sum _{\mathrm{K}=0}^{\mathrm{\infty }}}\sum _{\begin{array}{c}\begin{array}{c}\left|\mathbf{k}\right|=\mathrm{K},\\ \mathbf{k}\in {\mathbb{N}}_0^{\mathrm{m}}\end{array}\end{array}}\mathrm{U}\left(\mathbf{k}\right){\displaystyle \prod _{\mathrm{r}=1}^{\mathrm{m}}}{\left({\displaystyle \frac{{\mathrm{x}}_{\mathrm{r}}-{\mathrm{x}}_{\mathrm{r}0}}{{\mathrm{H}}_{\mathrm{r}}}}\right)}^{{\mathrm{k}}_{\mathrm{r}}}$$

(5)

For the construction of D-analogues of matrix algorithms in the following sections, the scale factors are taken to be equal to unity, that is,

$${\mathrm{H}}_1={\mathrm{H}}_2=\cdots ={\mathrm{H}}_{\mathrm{m}}=1.$$

Under this assumption, the direct multidimensional differential transform becomes

$$\mathrm{U}\left(\mathbf{k}\right)={\displaystyle \frac{1}{\mathbf{k}!}}{{\displaystyle \frac{{\partial }^{\left|\mathbf{k}\right|}\mathrm{u}\left(\mathbf{x}\right)}{\partial {\mathrm{x}}_1^{{\mathrm{k}}_1}\partial {\mathrm{x}}_2^{{\mathrm{k}}_2}\cdots \partial {\mathrm{x}}_{\mathrm{m}}^{{\mathrm{k}}_{\mathrm{m}}}}}|}_{\mathbf{x}={\mathbf{x}}_{\mathbf{0}}}.$$

(6)

The corresponding inverse transform is written as

$$\mathrm{u}\left(\mathbf{x}\right)={\displaystyle \sum _{\mathrm{K}=0}^{\mathrm{\infty }}}\sum _{\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{c}\left|\mathbf{k}\right|=\mathbf{K},\\ \mathbf{k}\in {\mathbb{N}}_0^{\mathrm{m}}\end{array}\end{array}\end{array}\end{array}}\mathrm{U}\left(\mathbf{k}\right){\left(\mathbf{x}-{\mathbf{x}}_{\mathbf{0}}\right)}^{\mathbf{k}},$$

(7)

where

$${\left(\mathbf{x}-{\mathbf{x}}_{\mathbf{0}}\right)}^{\mathbf{k}}={\left({\mathrm{x}}_1-{\mathrm{x}}_{10}\right)}^{{\mathrm{k}}_1}{\left({\mathrm{x}}_2-{\mathrm{x}}_{20}\right)}^{{\mathrm{k}}_2}\cdots {\left({\mathrm{x}}_{\mathrm{m}}-{\mathrm{x}}_{\mathrm{m}0}\right)}^{{\mathrm{k}}_{\mathrm{m}}}.$$

(8)

2.2. Operational Rules of Multidimensional Differential Transforms

The multidimensional differential transform satisfies a number of operational rules [12,13], that make it possible to transform algebraic and differential relations in the original domain into recurrence relations in the domain of differential spectra. In the following, let

$$\mathrm{u}\left(\mathbf{x}\right)⟷\mathrm{U}\left(\mathbf{k}\right),\hspace{1em}\hspace{1em}\mathrm{v}\left(\mathbf{x}\right)⟷\mathrm{V}\left(\mathbf{k}\right).$$

where the symbol $⟷$ denotes the correspondence between the direct and inverse differential transforms.

For constants $\mathsf{\alpha }$ and $\mathsf{\beta },$ the linear combination

$$\mathrm{w}\left(\mathbf{x}\right)=\mathsf{\alpha }\mathrm{u}\left(\mathbf{x}\right)+\mathsf{\beta }\mathrm{v}\left(\mathbf{x}\right)$$

has the differential spectrum

$$\mathrm{W}\left(\mathbf{k}\right)=\mathsf{\alpha }\mathrm{U}\left(\mathbf{k}\right)+\mathsf{\beta }\mathrm{V}\left(\mathbf{k}\right).$$

Thus,

$$\mathsf{\alpha }\mathrm{u}\left(\mathbf{x}\right)+\mathsf{\beta }\mathrm{v}\left(\mathbf{x}\right)⟷\mathsf{\alpha }\mathrm{U}\left(\mathbf{k}\right)+\mathsf{\beta }\mathrm{V}\left(\mathbf{k}\right).$$

(9)

The product of two functions,

$$\mathrm{w}\left(\mathbf{x}\right)=\mathrm{u}\left(\mathbf{x}\right)\mathrm{v}\left(\mathbf{x}\right),$$

is transformed into a multidimensional convolution of their spectra:

$$\mathrm{W}\left(\mathbf{k}\right)=\left(\mathrm{U}\ast \mathrm{V}\right)\left(\mathbf{k}\right).$$

The convolution operation is defined by

$$\left(\mathrm{U}\ast \mathrm{V}\right)\left(\mathbf{k}\right)=\sum _{\mathbf{s}\le \mathbf{k}}\mathrm{U}\left(\mathbf{s}\right)\mathrm{V}\left(\mathbf{k}-\mathbf{s}\right),$$

(10)

where

$$\mathbf{s}=\left({\mathrm{s}}_1,{\mathrm{s}}_2,\dots ,{\mathrm{s}}_{\mathrm{m}}\right),\mathbf{s}\le \mathbf{k}\hspace{1em}⟺\hspace{1em}{\mathrm{s}}_{\mathrm{r}}\le {\mathrm{k}}_{\mathrm{r}},\mathrm{r}=1,2,\dots ,\mathrm{m},$$

and

$$\mathbf{k}-\mathbf{s}=\left({\mathrm{k}}_1-{\mathrm{s}}_1,{\mathrm{k}}_2-{\mathrm{s}}_2,\dots ,{\mathrm{k}}_{\mathrm{m}}-{\mathrm{s}}_{\mathrm{m}}\right).$$

In expanded form, (10) is written as

$$\left(\mathrm{U}\ast \mathrm{V}\right)\left({\mathrm{k}}_1,{\mathrm{k}}_2,\dots ,{\mathrm{k}}_{\mathrm{m}}\right)={\displaystyle \sum _{{\mathrm{s}}_1=0}^{{\mathrm{k}}_1}}{\displaystyle \sum _{{\mathrm{s}}_2=0}^{{\mathrm{k}}_2}}\dots {\displaystyle \sum _{{\mathrm{s}}_{\mathrm{m}}=0}^{{\mathrm{k}}_{\mathrm{m}}}}\mathrm{U}\left({\mathrm{s}}_1,{\mathrm{s}}_2,\dots ,{\mathrm{s}}_{\mathrm{m}}\right)\mathrm{V}\left({\mathrm{k}}_1-{\mathrm{s}}_1,{\mathrm{k}}_2-{\mathrm{s}}_2,\dots ,{\mathrm{k}}_{\mathrm{m}}-{\mathrm{s}}_{\mathrm{m}}\right)$$

(11)

Therefore,

$$\mathrm{u}\left(\mathbf{x}\right)\mathrm{v}\left(\mathbf{x}\right)⟷\sum _{\mathbf{s}\le \mathbf{k}}\mathrm{U}\left(\mathbf{s}\right)\mathrm{V}\left(\mathbf{k}-\mathbf{s}\right).$$

(12)

These operational rules are the basis for constructing differential analogues of matrix algorithms.

We now extend the multidimensional differential transform to matrix-valued functions. Let

$$\mathrm{A}\left(\mathbf{x}\right)={\left[{\mathrm{a}}_{\mathrm{i}\mathrm{j}}\left(\mathbf{x}\right)\right]}_{\mathrm{i},\mathrm{j}=1}^{\mathrm{n}}$$

be an $\mathrm{n}\times \mathrm{n}$ matrix-valued function whose entries are sufficiently differentiable in a neighborhood of ${\mathrm{x}}_0$. The multidimensional differential spectrum of $\mathrm{A}\left(\mathrm{x}\right)$ is defined as the matrix sequence

$$\mathrm{A}\left(\mathbf{k}\right)={\left[{\mathrm{A}}_{\mathrm{i}\mathrm{j}}\left(\mathbf{k}\right)\right]}_{\mathrm{i},\mathrm{j}=1}^{\mathrm{n}},$$

where

$${\mathrm{A}}_{\mathrm{i}\mathrm{j}}\left(\mathbf{k}\right)={\displaystyle \frac{1}{\mathbf{k}!}}{{\displaystyle \frac{{\partial }^{\left|\mathbf{k}\right|}{\mathrm{a}}_{\mathrm{i}\mathrm{j}}\left(\mathrm{x}\right)}{\partial {\mathrm{x}}_1^{{\mathrm{k}}_1}\partial {\mathrm{x}}_2^{{\mathrm{k}}_2}\cdots \partial {\mathrm{x}}_{\mathrm{m}}^{{\mathrm{k}}_{\mathrm{m}}}}}|}_{\mathbf{x}={\mathbf{x}}_{\mathbf{0}}}.$$

(13)

Thus,

$$\mathrm{A}\left(\mathbf{x}\right)⟷\mathrm{A}\left(\mathbf{k}\right).$$

The inverse transform gives the representation

$$\mathrm{A}\left(\mathbf{x}\right)={\displaystyle \sum _{\mathrm{K}=0}^{\mathrm{\infty }}}\sum _{\begin{array}{c}\begin{array}{c}\begin{array}{c}\left|\mathbf{k}\right|=\mathrm{K},\\ \mathbf{k}\in {\mathbb{N}}_0^{\mathrm{m}}\end{array}\end{array}\end{array}}\mathrm{A}\left(\mathbf{k}\right){\left(\mathbf{x}-{\mathbf{x}}_{\mathbf{0}}\right)}^{\mathbf{k}}.$$

(14)

If

$$A\left(\mathbf{x}\right)⟷A\left(\mathbf{k}\right),\hspace{1em}\hspace{1em}B\left(\mathbf{x}\right)⟷B\left(\mathbf{k}\right),$$

and

$$C\left(\mathbf{x}\right)=\alpha A\left(\mathbf{x}\right)+\beta B\left(\mathbf{x}\right),$$

where $\alpha$ and $\beta$ are constants, then the differential spectrum of $C\left(\mathbf{x}\right)$ is given by

$$C\left(\mathbf{k}\right)=\alpha A\left(\mathbf{k}\right)+\beta B\left(\mathbf{k}\right).$$

(15)

Componentwise, this relation can be written as

$$C_{ij}\left(\mathbf{k}\right)=\alpha A_{ij}\left(\mathbf{k}\right)+\beta B_{ij}\left(\mathbf{k}\right), i,j=1, 2,\dots ,n.$$

(16)

If

$$\mathrm{C}\left(\mathbf{x}\right)=\mathrm{A}\left(\mathbf{x}\right)\mathrm{B}\left(\mathbf{x}\right),$$

then the differential spectrum of $\mathrm{C}\left(\mathbf{x}\right)$ is given by

$$\mathrm{C}\left(\mathbf{k}\right)=\sum _{\mathbf{s}\le \mathbf{k}}\mathrm{A}\left(\mathbf{s}\right)\mathrm{B}\left(\mathbf{k}-\mathbf{s}\right).$$

(17)

In this relation, the multiplication inside the summation is ordinary matrix multiplication.

In expanded form, the matrix convolution (16) can be written as

$$C(k_1,k_2,\dots ,k_m)={\displaystyle \sum _{s_1=0}^{k_1}}{\displaystyle \sum _{s_2=0}^{k_2}}\dots {\displaystyle \sum _{s_m=0}^{k_m}}A(s_1,s_2,\dots ,s_m)B(k_1-s_1,k_2-s_2,\dots ,k_m-s_m)$$

(18)

Expanded componentwise, this relation has the form

$${\mathrm{C}}_{\mathrm{i}\mathrm{j}}\left(\mathbf{k}\right)=\sum _{\mathbf{s}\le \mathbf{k}}{\displaystyle \sum _{\mathcal{l}=1}^{\mathrm{n}}}{\mathrm{A}}_{\mathrm{i}\mathcal{l}}\left(\mathbf{s}\right){\mathrm{B}}_{\mathcal{l}\mathrm{j}}\left(\mathbf{k}-\mathbf{s}\right),\hspace{1em}\hspace{1em}\mathrm{i},\mathrm{j}=1,2,\dots ,\mathrm{n}.$$

(19)

For the identity matrix ${\mathrm{I}}_{\mathrm{n}}$, the corresponding differential spectrum is

$${\mathrm{I}}_{\mathrm{n}}\left(\mathbf{k}\right)=\{\begin{array}{cc}{\mathbf{I}}_{\mathbf{n}},& \mathbf{k}=\mathbf{0},\\ {\mathbf{0}}_{\mathbf{n}\times \mathbf{n}},& \mathbf{k}\ne \mathbf{0}.\end{array}$$

(20)

Here $\mathbf{0}=\left(0,0,\dots ,0\right).$

The framework is directly applicable to complex-valued matrix functions. If

$$\mathrm{A}\left(\mathbf{x}\right)={\mathrm{A}}_{\mathrm{R}}\left(\mathbf{x}\right)+i{\mathrm{A}}_{\mathrm{I}}\left(\mathbf{x}\right),\hspace{1em}{i}^2=-1,$$

(21)

where ${\mathrm{A}}_{\mathrm{R}}\left(\mathbf{x}\right)\mathrm{a}\mathrm{n}\mathrm{d} {\mathrm{A}}_{\mathrm{I}}\left(\mathbf{x}\right)$ are real-valued matrix functions, and $i$ denotes the imaginary unit, then

$$\mathrm{A}\left(\mathbf{k}\right)={\mathrm{A}}_{\mathrm{R}}\left(\mathbf{k}\right)+i{\mathrm{A}}_{\mathrm{I}}\left(\mathbf{k}\right).$$

(22)

Therefore, the operational rules of multidimensional differential transforms remain valid for complex-valued matrix functions, provided that all algebraic operations are performed over the field of complex numbers. The definitions and operational properties introduced in this section will be used below to derive recurrence procedures for the determinant, characteristic polynomial, and inverse matrix of multiparameter matrix-valued functions.

2.3. Convergence and Truncation Error

The multidimensional differential transform used in this paper is equivalent to the multidimensional Taylor expansion of the considered function at the expansion center [12,13,14]. Therefore, the convergence of the inverse transform is determined by the analyticity properties of the original function in a neighborhood of the expansion point.

Let $u\left(\mathbf{x}\right)=u\left(x_1,\dots ,x_m\right)$ be analytic in the polydisc

$${\mathcal{D}}_{\mathsf{\rho }}\left({\mathbf{x}}_0\right)=\left\{\mathbf{x}\in {\mathbb{C}}^m:\left|x_r-x_{r0}\right|<{\rho }_r,\hspace{1em}r=1,\dots ,m\right\},$$

where

$$\mathsf{\rho }=\left({\rho }_1,\dots ,{\rho }_m\right),\hspace{1em}\hspace{1em}{\rho }_r>0.$$

Then $u\left(\mathbf{x}\right)$ admits an absolutely and uniformly convergent multidimensional Taylor expansion in every closed polydisc [18]

$${\overline{\mathcal{D}}}_{\mathbf{r}}\left({\mathbf{x}}_0\right)=\left\{\mathbf{x}\in {\mathbb{C}}^m:\left|x_r-x_{r0}\right|\le r_r<{\rho }_r,\hspace{1em}r=1,\dots ,m\right\}.$$

For $H_1=\cdots =H_m=1$, this expansion coincides with the inverse multidimensional differential transform (7). Assume that

$$\left|u\left(\mathbf{x}\right)\right|\le M$$

in the polydisc ${\mathcal{D}}_{\mathsf{\rho }}\left({\mathbf{x}}_0\right)$. By the multidimensional Cauchy estimates [18], the differential spectrum satisfies

$$\left|U\left(\mathbf{k}\right)\right|\le {\displaystyle \frac{M}{{\rho }_1^{k_1}{\rho }_2^{k_2}\cdots {\rho }_m^{k_m}}}.$$

Let the series be truncated with respect to the rectangular truncation vector

$$\mathbf{K}=\left(K_1,\dots ,K_m\right).$$

The truncated reconstruction is

$$u_{\mathbf{K}}\left(\mathbf{x}\right)={\displaystyle \sum _{k_1=0}^{K_1}}\dots {\displaystyle \sum _{k_m=0}^{K_m}}U\left(k_1,\dots ,k_m\right){\left(x_1-x_{10}\right)}^{k_1}\cdots {\left(x_m-x_{m0}\right)}^{k_m}.$$

For

$$\left|x_r-x_{r0}\right|\le r_r<{\rho }_r,$$

introduce

$${\eta }_r={\displaystyle \frac{r_r}{{\rho }_r}},\hspace{1em}\hspace{1em}0\le {\eta }_r<1.$$

Then, using the above Cauchy estimate and the product representation of the geometric series, the truncation error satisfies

$$\left|u\left(\mathbf{x}\right)-u_{\mathbf{K}}\left(\mathbf{x}\right)\right|\le M\left[{\displaystyle \prod _{r=1}^m}{\displaystyle \frac{1}{1-{\eta }_r}}-{\displaystyle \prod _{r=1}^m}{\displaystyle \frac{1-{\eta }_r^{K_r+1}}{1-{\eta }_r}}\right].$$

This bound shows that the truncation error decreases as the truncation orders $K_r$ increase and as the evaluation point remains farther from the boundary of the analyticity polydisc.

The same argument applies componentwise to matrix-valued functions. Let

$$A\left(\mathbf{x}\right)={\left[a_{ij}\left(\mathbf{x}\right)\right]}_{i,j=1}^n$$

be analytic in ${\mathcal{D}}_{\mathsf{\rho }}\left({\mathbf{x}}_0\right)$, and suppose that all entries satisfy

$$\left|a_{ij}\left(\mathbf{x}\right)\right|\le M_A$$

in this polydisc. If $A_{\mathbf{K}}\left(\mathbf{x}\right)$ is the truncated reconstruction of $A\left(\mathbf{x}\right)$, then each entry satisfies the above scalar error bound. Consequently, for the matrix infinity norm,

$$\parallel A\left(\mathbf{x}\right)-A_{\mathbf{K}}\left(\mathbf{x}\right){\parallel }_{\mathrm{\infty }}\le n{M}_A\left[{\displaystyle \prod _{r=1}^m}{\displaystyle \frac{1}{1-{\eta }_r}}-{\displaystyle \prod _{r=1}^m}{\displaystyle \frac{1-{\eta }_r^{K_r+1}}{1-{\eta }_r}}\right].$$

For inverse matrices, an additional local nonsingularity condition is required. If $\mathrm{d}\mathrm{e}\mathrm{t}A\left({\mathbf{x}}_0\right)\ne 0,$ then, by continuity, there exists a neighborhood of ${\mathbf{x}}_0$ in which $\mathrm{d}\mathrm{e}\mathrm{t}A\left(\mathbf{x}\right)\ne 0.$ In this neighborhood, $A^{-1}\left(\mathbf{x}\right)$ is analytic, and its multidimensional differential spectrum is well defined. The radius of convergence of the inverse-matrix expansion is limited either by the nearest singularity of the entries of $A\left(\mathbf{x}\right)$ or by the nearest zero of $\det A\left(\mathbf{x}\right)$. Therefore, the reconstructed inverse matrix is a local parametric representation around the chosen expansion point. If $\det A\left(\mathbf{x}\right)=0$ at some parameter values, then the inverse matrix is not defined at those values, and the reconstructed determinant indicates the loss of invertibility. A local representation in another nonsingular parameter region can be obtained by choosing a new expansion point ${\tilde{\mathbf{x}}}_0$ satisfying $\det A\left({\tilde{\mathbf{x}}}_0\right)\ne 0$ and applying the recurrence procedure again.

In the polynomial case, the spectra of the matrix entries contain only finitely many nonzero terms. If all required nonzero spectra are retained, the reconstruction of the matrix, the determinant, and the characteristic polynomial is exact. For non-polynomial analytic entries, the method produces a local analytical or numerical-analytical approximation whose accuracy is controlled by the truncation orders and the distance from the boundary of the analyticity domain.

In addition to the a priori truncation estimates given above, the accuracy of the reconstructed inverse matrix can be assessed by residual-based a posteriori indicators. Let $B_{\mathbf{K}}\left(\mathbf{x}\right)$ denote the inverse matrix reconstructed from the truncated multidimensional differential spectra. In the exact case,

$$B_{\mathbf{K}}\left(\mathbf{x}\right)=A^{-1}\left(\mathbf{x}\right).$$

For truncated computations, the following residual matrices are introduced:

$$E_{1,\mathbf{K}}\left(\mathbf{x}\right)=A\left(\mathbf{x}\right)B_{\mathbf{K}}\left(\mathbf{x}\right)-I_n,$$

and

$$E_{2,\mathbf{K}}\left(\mathbf{x}\right)=B_{\mathbf{K}}\left(\mathbf{x}\right)A\left(\mathbf{x}\right)-I_n.$$

If $A\left(\mathbf{x}\right)$ is nonsingular at the considered parameter value, then

$$A\left(\mathbf{x}\right)B_{\mathbf{K}}\left(\mathbf{x}\right)-I_n=A\left(\mathbf{x}\right)\left(B_{\mathbf{K}}\left(\mathbf{x}\right)-A^{-1}\left(\mathbf{x}\right)\right).$$

Therefore,

$$B_{\mathbf{K}}\left(\mathbf{x}\right)-A^{-1}\left(\mathbf{x}\right)=A^{-1}\left(\mathbf{x}\right)E_{1,\mathbf{K}}\left(\mathbf{x}\right).$$

Consequently, for any consistent matrix norm,

$$\parallel B_{\mathbf{K}}\left(\mathbf{x}\right)-A^{-1}\left(\mathbf{x}\right)\parallel \le \parallel A^{-1}\left(\mathbf{x}\right)\parallel \parallel E_{1,\mathbf{K}}\left(\mathbf{x}\right)\parallel .$$

Similarly, from

$$B_{\mathbf{K}}\left(\mathbf{x}\right)A\left(\mathbf{x}\right)-I_n=\left(B_{\mathbf{K}}\left(\mathbf{x}\right)-A^{-1}\left(\mathbf{x}\right)\right)A\left(\mathbf{x}\right),$$

we obtain

$$B_{\mathbf{K}}\left(\mathbf{x}\right)-A^{-1}\left(\mathbf{x}\right)=E_{2,\mathbf{K}}\left(\mathbf{x}\right)A^{-1}\left(\mathbf{x}\right),$$

and hence

$$\parallel B_{\mathbf{K}}\left(\mathbf{x}\right)-A^{-1}\left(\mathbf{x}\right)\parallel \le \parallel E_{2,\mathbf{K}}\left(\mathbf{x}\right)\parallel \parallel A^{-1}\left(\mathbf{x}\right)\parallel .$$

Thus, the residual norms $\parallel E_{1,\mathbf{K}}\left(\mathbf{x}\right)\parallel$ and $\parallel E_{2,\mathbf{K}}\left(\mathbf{x}\right)\parallel$ provide practical indicators of the accuracy of the reconstructed inverse matrix. In particular, if these residuals decrease as the truncation vector $\mathbf{K}$ increases, then the reconstructed inverse matrix converges to the exact inverse in the considered parameter domain. When the exact inverse norm is not available, the residual still provides a useful computable accuracy indicator. Moreover, if

$$\parallel E_{1,\mathbf{K}}\left(\mathbf{x}\right)\parallel <1,$$

then the following bound can be used:

$$\parallel B_{\mathbf{K}}\left(\mathbf{x}\right)-A^{-1}\left(\mathbf{x}\right)\parallel \le {\displaystyle \frac{\parallel B_{\mathbf{K}}\left(\mathbf{x}\right)\parallel \parallel E_{1,\mathbf{K}}\left(\mathbf{x}\right)\parallel }{1-\parallel E_{1,\mathbf{K}}\left(\mathbf{x}\right)\parallel }}.$$

An analogous estimate can be written in terms of $E_{2,\mathbf{K}}\left(\mathbf{x}\right)$. Therefore, in practical computations, the residual matrices $E_{1,\mathbf{K}}$ and $E_{2,\mathbf{K}}$ can be used both as verification tools and as a posteriori indicator for selecting or increasing the truncation vector.

For polynomial matrix functions, if all nonzero spectra required by the recurrence relations are included, the residual matrices vanish identically. For truncated reconstructions, the residual norms quantify the influence of omitted higher-order spectral terms. This is the criterion used in the computational examples below to assess the accuracy of the reconstructed inverse matrices.

3. D-Analogues for Multiparameter Matrices

In this section, we present D-analogues of numerical methods for determining the characteristics of multiparameter matrices by using multidimensional differential transforms. The proposed approach is based on transferring the corresponding matrix algorithms from the original domain to the domain of differential spectra. For a given multiparameter matrix, the multidimensional differential spectra of its entries are first determined. Then, the required algebraic operations are performed in the differential-transform domain according to the operational rules introduced above. Finally, the original parameter-dependent characteristics of the matrix are reconstructed by means of the inverse multidimensional differential transform given by relation (2).

In what follows, D-analogues of the Leverrier and Faddeev methods are constructed for computing characteristic polynomials and related matrix characteristics. These constructions provide recurrence-based procedures for computing the differential spectra of the corresponding matrix characteristics and are applicable to real and complex matrix-valued functions depending on several independent parameters.

3.1. D-Analogue of the Leverrier Method for Multiparameter Matrices

Let

$$\mathrm{A}\left(\mathbf{x}\right)=\mathrm{A}\left({\mathrm{x}}_1,{\mathrm{x}}_2,\dots ,{\mathrm{x}}_{\mathrm{m}}\right)={\left[{\mathrm{a}}_{\mathrm{i}\mathrm{j}}\left({\mathrm{x}}_1,{\mathrm{x}}_2,\dots ,{\mathrm{x}}_{\mathrm{m}}\right)\right]}_{\mathrm{i},\mathrm{j}=1}^{\mathrm{n}}\in {\mathbb{C}}^{\mathrm{n}\times \mathrm{n}}$$

be an $n\times n$ matrix whose entries are sufficiently differentiable functions of the independent parameters ($x_1,x_2,\dots ,x_m$) in a neighborhood of the expansion center ${\mathbf{x}}_0$. In what follows, the entries of $\mathrm{A}\left(\mathbf{x}\right)$ may be real-valued or complex-valued functions.

• For the matrix $\mathrm{A}\left(\mathbf{x}\right)$, consider the characteristic polynomial

$$\mathrm{P}\left(\mathsf{\lambda },\mathbf{x}\right)=\det \left(\mathsf{\lambda }{\mathrm{I}}_{\mathrm{n}}-\mathrm{A}\left(\mathbf{x}\right)\right).$$

(23)

It can be written in the form

$$\mathrm{P}\left(\mathsf{\lambda },\mathbf{x}\right)={\mathsf{\lambda }}^{\mathrm{n}}+{\mathrm{p}}_1\left(\mathbf{x}\right){\mathsf{\lambda }}^{\mathrm{n}-1}+{\mathrm{p}}_2\left(\mathbf{x}\right){\mathsf{\lambda }}^{\mathrm{n}-2}+\cdots +{\mathrm{p}}_{\mathrm{n}}\left(\mathbf{x}\right),$$

(24)

where the coefficients ${\mathrm{p}}_1\left(\mathbf{x}\right),{\mathrm{p}}_2\left(\mathbf{x}\right),\dots ,{\mathrm{p}}_{\mathrm{n}}\left(\mathbf{x}\right)$ are functions of the parameters ${\mathrm{x}}_1,{\mathrm{x}}_2,\dots ,{\mathrm{x}}_{\mathrm{m}}$. Thus, for a multiparameter matrix, the coefficients of the characteristic polynomial are also multiparameter functions.

The D-analogue of the Leverrier method for non-autonomous one-parameter matrices was considered in [15]. Here, we extend this idea to the case of multiparameter matrices by using the multidimensional differential transforms introduced in Section 2.

According to the classical Leverrier method, the coefficients of the characteristic polynomial are expressed through the traces of powers of the matrix. We introduce the notation

$${\mathrm{s}}_{\mathrm{q}}\left(\mathbf{x}\right)=\mathrm{tr}\left({\mathrm{A}}^{\mathrm{q}}\left(\mathbf{x}\right)\right),\hspace{1em}\hspace{1em}\mathrm{q}=1,2,\dots ,\mathrm{n}.$$

(25)

The coefficients ${\mathrm{p}}_{\mathrm{q}}\left(\mathbf{x}\right)$ satisfy the recurrence relations

$$\mathrm{q}·{\mathrm{p}}_{\mathrm{q}}\left(\mathbf{x}\right)+{\displaystyle \sum _{\mathrm{r}=1}^{\mathrm{q}}}{\mathrm{p}}_{\mathrm{q}-\mathrm{r}}\left(\mathbf{x}\right){\mathrm{s}}_{\mathrm{r}}\left(\mathbf{x}\right)=0,\mathrm{q}=1,2,\dots ,\mathrm{n},\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} {\mathrm{p}}_0\left(\mathbf{x}\right)=1.$$

(26)

Equivalently,

$${\mathrm{p}}_{\mathrm{q}}\left(\mathbf{x}\right)=-{\displaystyle \frac{1}{\mathrm{q}}}{\displaystyle \sum _{\mathrm{r}=1}^{\mathrm{q}}}{\mathrm{p}}_{\mathrm{q}-\mathrm{r}}\left(\mathbf{x}\right){\mathrm{s}}_{\mathrm{r}}\left(\mathbf{x}\right),\hspace{1em}\hspace{1em}\mathrm{q}=\mathrm{1,2},\dots ,\mathrm{n}.$$

(27)

Let the multidimensional differential spectra of $p_q\left(\mathit{x}\right)$ and $s_q\left(\mathit{x}\right)$ be denoted by

$${\mathrm{p}}_{\mathrm{q}}\left(\mathbf{x}\right)⟷{\mathrm{P}}_{\mathrm{q}}\left(\mathbf{k}\right),\hspace{1em}{\mathrm{s}}_{\mathrm{q}}\left(\mathbf{x}\right)⟷{\mathrm{S}}_{\mathrm{q}}\left(\mathbf{k}\right),$$

where

$$\mathbf{k}=\left({\mathrm{k}}_1,{\mathrm{k}}_2,\dots ,{\mathrm{k}}_{\mathrm{m}}\right).$$

Since $p_0\left(\mathit{x}\right)=1$, its differential spectrum is

$${\mathrm{P}}_0\left(\mathbf{k}\right)=\{\begin{array}{cc}1,& \mathbf{k}=\mathbf{0},\\ 0,& \mathbf{k}\ne \mathbf{0}.\end{array}$$

Applying the multidimensional differential transform to (27), and using the convolution rule for products, we obtain the D-analogue of the Leverrier recurrence:

$${\mathrm{P}}_{\mathrm{q}}\left(\mathbf{k}\right)=-{\displaystyle \frac{1}{\mathrm{q}}}{\displaystyle {\sum }_{\mathrm{r}=1}^{\mathrm{q}}}\left({\mathrm{P}}_{\mathrm{q}-\mathrm{r}}\ast {\mathrm{S}}_{\mathrm{r}}\right)\left(\mathbf{k}\right),\hspace{1em}\hspace{1em}\mathrm{q}=1,2,\dots ,\mathrm{n}.$$

(28)

Here (∗) denotes the multidimensional convolution, defined by

$$\left({\mathrm{P}}_{\mathrm{q}-\mathrm{r}}\ast {\mathrm{S}}_{\mathrm{r}}\right)\left(\mathbf{k}\right)={\sum }_{\mathbf{s}\le \mathbf{k}}{\mathrm{P}}_{\mathrm{q}-\mathrm{r}}\left(\mathbf{s}\right){\mathrm{S}}_{\mathrm{r}}\left(\mathbf{k}-\mathbf{s}\right).$$

(29)

In expanded form,

$$\begin{array}{l}\left({\mathrm{P}}_{\mathrm{q}-\mathrm{r}}\ast {\mathrm{S}}_{\mathrm{r}}\right)\left({\mathrm{k}}_1,{\mathrm{k}}_2,\dots ,{\mathrm{k}}_{\mathrm{m}}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\displaystyle \sum _{{\mathrm{s}}_1=0}^{{\mathrm{k}}_1}}{\displaystyle \sum _{{\mathrm{s}}_2=0}^{{\mathrm{k}}_2}}\dots {\displaystyle \sum _{{\mathrm{s}}_{\mathrm{m}}=0}^{{\mathrm{k}}_{\mathrm{m}}}}{\mathrm{P}}_{\mathrm{q}-\mathrm{r}}\left({\mathrm{s}}_1,{\mathrm{s}}_2,\dots ,{\mathrm{s}}_{\mathrm{m}}\right){\mathrm{S}}_{\mathrm{r}}\left({\mathrm{k}}_1-{\mathrm{s}}_1,{\mathrm{k}}_2-{\mathrm{s}}_2,\dots ,{\mathrm{k}}_{\mathrm{m}}-{\mathrm{s}}_{\mathrm{m}}\right).\end{array}$$

(30)

Therefore, the first recurrence relations are:

$${\mathrm{P}}_1\left(\mathbf{k}\right)=-{\mathrm{S}}_1\left(\mathbf{k}\right),$$

$${\mathrm{P}}_2\left(\mathbf{k}\right)=-{\displaystyle \frac{1}{2}}\left[{\mathrm{S}}_2\left(\mathbf{k}\right)+\left({\mathrm{P}}_1\ast {\mathrm{S}}_1\right)\left(\mathbf{k}\right)\right],$$

$${\mathrm{P}}_3\left(\mathbf{k}\right)=-{\displaystyle \frac{1}{3}}\left[{\mathrm{S}}_3\left(\mathbf{k}\right)+\left({\mathrm{P}}_1\ast {\mathrm{S}}_2\right)\left(\mathbf{k}\right)+\left({\mathrm{P}}_2\ast {\mathrm{S}}_1\right)\left(\mathbf{k}\right)\right],$$

and, in general,

$${\mathrm{P}}_{\mathrm{q}}\left(\mathbf{k}\right)=-{\displaystyle \frac{1}{\mathrm{q}}}\left[{\mathrm{S}}_{\mathrm{q}}\left(\mathbf{k}\right)+{\displaystyle {\sum }_{\mathrm{r}=1}^{\mathrm{q}-1}}\left({\mathrm{P}}_{\mathrm{q}-\mathrm{r}}\ast {\mathrm{S}}_{\mathrm{r}}\right)\left(\mathbf{k}\right)\right],\hspace{1em}\hspace{1em}\mathrm{q}=1,2,\dots ,\mathrm{n}.$$

(31)

It remains to compute the spectra ${\mathrm{S}}_{\mathrm{q}}\left(\mathbf{k}\right)$ of the trace functions ${\mathrm{s}}_{\mathrm{q}}\left(\mathbf{x}\right)$. For this purpose, define

$${\mathrm{B}}_{\mathrm{q}}\left(\mathbf{x}\right)={\mathrm{A}}^{\mathrm{q}}\left(\mathbf{x}\right),\mathrm{q}=1,2,\dots ,\mathrm{n}.$$

(32)

Let

$${\mathrm{B}}_{\mathrm{q}}\left(\mathbf{x}\right)⟷{\mathrm{B}}_{\mathrm{q}}\left(\mathbf{k}\right).$$

Then

$${\mathrm{B}}_1\left(\mathbf{k}\right)=\mathrm{A}\left(\mathbf{k}\right),$$

and, for ($\mathrm{q}\ge 2$), the spectra of the powers of the matrix are computed by the recurrence

$${\mathrm{B}}_{\mathrm{q}}\left(\mathbf{k}\right)={\sum }_{\mathbf{s}\le \mathbf{k}}\mathrm{A}\left(\mathbf{s}\right){\mathrm{B}}_{\mathrm{q}-1}\left(\mathbf{k}-\mathbf{s}\right).$$

(33)

The differential spectra of the trace functions are then obtained as

$${\mathrm{S}}_{\mathrm{q}}\left(\mathbf{k}\right)=\mathrm{tr}\left({\mathrm{B}}_{\mathrm{q}}\left(\mathbf{k}\right)\right),\hspace{1em}\mathrm{q}=1,2,\dots ,\mathrm{n}.$$

(34)

Thus, having computed ${\mathrm{S}}_1(\mathbf{k}$), ${\mathrm{S}}_2\left(\mathbf{k}\right),\dots ,{\mathrm{S}}_{\mathrm{n}}\left(\mathbf{k}\right)$, the spectra of the characteristic polynomial coefficients are obtained from (31).

According to the inverse multidimensional differential transform, the coefficient functions of the characteristic polynomial are reconstructed as

$${\mathrm{p}}_{\mathrm{q}}\left(\mathbf{x}\right)={\displaystyle \sum _{\mathrm{K}=0}^{\mathrm{\infty }}}\sum _{\begin{array}{c}\left|\mathbf{k}\right|=\mathrm{K} \mathbf{k}\in {\mathbb{N}}_0^{\mathrm{m}}\end{array}}{\mathrm{P}}_{\mathrm{q}}\left(\mathbf{k}\right){\left(\mathbf{x}-{\mathbf{x}}_0\right)}^{\mathbf{k}}, \mathrm{q}=1,2,\dots ,\mathrm{n}.$$

(35)

In practical computations, the infinite expansion (35) is replaced by the truncated form

$${\mathrm{p}}_{\mathrm{q},\mathbf{K}}\left(\mathbf{x}\right)={\displaystyle \sum _{{\mathrm{k}}_1=0}^{{\mathrm{K}}_1}}{\displaystyle \sum _{{\mathrm{k}}_2=0}^{{\mathrm{K}}_2}}\dots {\displaystyle \sum _{{\mathrm{k}}_{\mathrm{m}}=0}^{{\mathrm{K}}_{\mathrm{m}}}}{\mathrm{P}}_{\mathrm{q}}\left({\mathrm{k}}_1,{\mathrm{k}}_2,\dots ,{\mathrm{k}}_{\mathrm{m}}\right){\left(\mathbf{x}-{\mathbf{x}}_0\right)}^{\mathbf{k}},$$

(36)

where

$$\mathbf{K}=\left({\mathrm{K}}_1,{\mathrm{K}}_2,\dots ,{\mathrm{K}}_{\mathrm{m}}\right)$$

is the truncation vector.

• Consequently, the characteristic polynomial of the multiparameter matrix $\mathrm{A}\left(\mathbf{x}\right)$ is reconstructed in the form

$$\mathrm{P}\left(\mathsf{\lambda },\mathbf{x}\right)={\mathsf{\lambda }}^{\mathrm{n}}+{\displaystyle \sum _{\mathrm{q}=1}^{\mathrm{n}}}{\mathrm{p}}_{\mathrm{q}}\left(\mathbf{x}\right){\mathsf{\lambda }}^{\mathrm{n}-\mathrm{q}}.$$

(37)

Using the truncated coefficient functions, we obtain the approximate characteristic polynomial

$${\mathrm{P}}_{\mathbf{K}}\left(\mathsf{\lambda },\mathbf{x}\right){=\mathsf{\lambda }}^{\mathrm{n}}+{\displaystyle \sum _{\mathrm{q}=1}^{\mathrm{n}}}{\mathrm{p}}_{\mathrm{q},\mathbf{K}}\left(\mathbf{x}\right){\mathsf{\lambda }}^{\mathrm{n}-\mathrm{q}}.$$

(38)

Therefore, the D-analogue of the Leverrier method reduces the computation of the characteristic polynomial of a multiparameter matrix to the computation of multidimensional differential spectra of trace functions and scalar convolutions. This approach avoids direct symbolic expansion of the determinant $\det \left(\mathsf{\lambda }{\mathrm{I}}_{\mathrm{n}}-\mathrm{A}\left(\mathbf{x}\right)\right)$ and provides a local analytical or numerical-analytical representation of the characteristic polynomial coefficients.

3.2. D-Analogue of the Faddeev Method for Multiparameter Matrices

Let

$$\mathrm{A}\left(\mathbf{x}\right)=\mathrm{A}\left({\mathrm{x}}_1,{\mathrm{x}}_2,\dots ,{\mathrm{x}}_{\mathrm{m}}\right)\in {\mathbb{C}}^{\mathrm{n}\times \mathrm{n}}$$

be a multiparameter matrix whose entries are sufficiently differentiable functions of the independent parameters ($x_1,x_2,\dots ,x_m$) in a neighborhood of the expansion center ${\mathbf{x}}_0$. As in the previous section, we consider the characteristic polynomial

$$\mathrm{P}\left(\mathsf{\lambda },\mathbf{x}\right)=\det \left(\mathsf{\lambda }{\mathrm{I}}_{\mathrm{n}}-\mathrm{A}\left(\mathbf{x}\right)\right),$$

which is written in the form

$$\mathrm{P}\left(\mathsf{\lambda },\mathbf{x}\right)={\mathsf{\lambda }}^{\mathrm{n}}+{\mathrm{p}}_1\left(\mathbf{x}\right){\mathsf{\lambda }}^{\mathrm{n}-1}+{\mathrm{p}}_2\left(\mathbf{x}\right){\mathsf{\lambda }}^{\mathrm{n}-2}+\cdots +{\mathrm{p}}_{\mathrm{n}}\left(\mathbf{x}\right).$$

The Faddeev method provides a recursive procedure for computing the coefficient functions $p_q\left(\mathbf{x}\right)$, $q=1, 2,\dots ,n$, together with a sequence of auxiliary matrices. This method is closely related to the Leverrier method, but it is more suitable when the inverse matrix is also required.

Introduce the auxiliary matrix functions

$$R_0\left(\mathbf{x}\right)=I_n,$$

and, for ($q=1, 2,\dots ,n$),

$$p_q\left(\mathbf{x}\right)=-{\displaystyle \frac{1}{q}}\mathrm{tr}\left(A\left(\mathbf{x}\right)R_{q-1}\left(\mathbf{x}\right)\right),$$

(39)

$$R_q\left(\mathbf{x}\right)=A\left(\mathbf{x}\right)R_{q-1}\left(\mathbf{x}\right)+p_q\left(\mathbf{x}\right)I_n$$

(40)

Relations (39) and (40) constitute the classical Faddeev recurrence for the characteristic polynomial coefficients. The last auxiliary matrix satisfies

$$R_n\left(\mathbf{x}\right)=0_{n\times n},$$

which follows from the Cayley–Hamilton theorem and can be used as a check of the computations.

If $p_n\left(\mathbf{x}\right)\ne 0$, then the inverse matrix can be represented as

$$A^{-1}\left(\mathbf{x}\right)=-{\displaystyle \frac{1}{p_n\left(\mathbf{x}\right)}}{R}_{n-1}\left(\mathbf{x}\right).$$

(41)

Thus, unlike the Leverrier recurrence, the Faddeev recurrence provides a direct way to construct both the characteristic polynomial and the inverse matrix.

We now derive the D-analogue of the Faddeev method for multiparameter matrices. Let

$$A\left(\mathbf{x}\right)⟷A\left(\mathbf{k}\right), p_q\left(\mathbf{x}\right)⟷P_q\left(\mathbf{k}\right),{ R}_q\left(\mathbf{x}\right)⟷R_q\left(\mathbf{k}\right),q=0, 1,\dots ,n.$$

Since $R_0\left(\mathbf{x}\right)=I_n$, its multidimensional differential spectrum is

$$R_0\left(\mathbf{k}\right)=\{\begin{array}{cc}{I}_n,& \mathbf{k}=\mathbf{0},\\ 0_{n\times n},& \mathbf{k}\ne \mathbf{0}.\end{array}$$

For convenience, define

$$M_q\left(\mathbf{x}\right)=A\left(\mathbf{x}\right)R_{q-1}\left(\mathbf{x}\right), q=\mathrm{1,2},\dots ,n.$$

(42)

Let

$$M_q\left(\mathbf{x}\right)⟷M_q\left(\mathbf{k}\right).$$

Using the convolution rule for the product of matrix-valued functions, we obtain

$$M_q\left(\mathbf{k}\right)=\sum _{\mathbf{s}\le \mathbf{k}}A\left(\mathbf{s}\right)R_{q-1}\left(\mathbf{k}-\mathbf{s}\right),\hspace{1em}\hspace{1em}q=1, 2,\dots ,n.$$

(43)

The spectra of the coefficient functions $p_q\left(\mathbf{x}\right)$ are obtained from (39). Therefore,

$$P_q\left(\mathbf{k}\right)=-{\displaystyle \frac{1}{q}}\mathrm{tr}\left(M_q\left(\mathbf{k}\right)\right),\hspace{1em}q=1, 2,\dots ,n.$$

(44)

Substituting (43) into (44), we get

$$P_q\left(\mathbf{k}\right)=-{\displaystyle \frac{1}{q}}\mathrm{tr}\left(\sum _{\mathbf{s}\le \mathbf{k}}A\left(\mathbf{s}\right)R_{q-1}\left(\mathbf{k}-\mathbf{s}\right)\right),\hspace{1em}\hspace{1em}q=1, 2,\dots ,n.$$

(45)

The differential spectra of the auxiliary matrices ($R_q\left(\mathbf{x}\right)$) are computed from (40) as

$$R_q\left(\mathbf{k}\right)=M_q\left(\mathbf{k}\right)+P_q\left(\mathbf{k}\right)I_n,\hspace{1em}\hspace{1em}q=1, 2,\dots ,n.$$

(46)

Equivalently, using (43), this relation can be written as

$$R_q\left(\mathbf{k}\right)=\sum _{\mathbf{s}\le \mathbf{k}}A\left(\mathbf{s}\right)R_{q-1}\left(\mathbf{k}-\mathbf{s}\right)+P_q\left(\mathbf{k}\right)I_n,\hspace{1em}\hspace{1em}q=1, 2,\dots ,n.$$

(47)

Equations (42)–(47) form the D-analogue of the Faddeev method for multiparameter matrices.

The first steps of the recurrence are as follows:

$$\begin{array}{l}{M}_1\left(\mathbf{k}\right)=A\left(\mathbf{k}\right),\\ P_1\left(\mathbf{k}\right)=-\mathrm{tr}\left(A\left(\mathbf{k}\right)\right),\\ R_1\left(\mathbf{k}\right)=A\left(\mathbf{k}\right)+P_1\left(\mathbf{k}\right)I_n.\end{array}$$

For q = 2,

$$\begin{array}{l}{M}_2\left(\mathbf{k}\right)={\displaystyle \sum _{\mathbf{s}\le \mathbf{k}}}A\left(\mathbf{s}\right)R_1\left(\mathbf{k}-\mathbf{s}\right),\\ P_2\left(\mathbf{k}\right)=-{\displaystyle \frac{1}{2}}\mathrm{tr}\left(M_2\left(\mathbf{k}\right)\right),\\ R_2\left(\mathbf{k}\right)=M_2\left(\mathbf{k}\right)+P_2\left(\mathbf{k}\right)I_n.\end{array}$$

The process is continued until q = n. The condition

$$R_n\left(\mathbf{k}\right)=0_{n\times n},\hspace{1em}\hspace{1em}\mathbf{k}\in {\mathbb{N}}_0^m,$$

serves as a convenient verification condition for the exact recurrence. In truncated computations, the residual values of $R_n\left(\mathbf{k}\right)$ can be used to assess the accuracy of the obtained spectra. According to the inverse multidimensional differential transform, the coefficient functions of the characteristic polynomial are reconstructed as

$$p_q\left(\mathbf{x}\right)={\displaystyle \sum _{K=0}^{\infty }}\sum _{\begin{array}{c}\left|\mathbf{k}\right|=K \mathbf{k}\in {\mathbb{N}}_0^m\end{array}}{P}_q\left(\mathbf{k}\right){\left(\mathbf{x}-{\mathbf{x}}_0\right)}^{\mathbf{k}},\hspace{1em}\hspace{1em}q=1, 2,\dots ,n.$$

(48)

The auxiliary matrix functions are reconstructed as

$$R_q\left(\mathbf{x}\right)={\displaystyle \sum _{K=0}^{\infty }}\sum _{\begin{array}{c}\left|\mathbf{k}\right|=K \mathbf{k}\in {\mathbb{N}}_0^m\end{array}}{R}_q\left(\mathbf{k}\right){\left(\mathbf{x}-{\mathbf{x}}_0\right)}^{\mathbf{k}},\hspace{1em}\hspace{1em}q=0, 1,\dots ,n.$$

(49)

Consequently, the characteristic polynomial is obtained in the form

$$P\left(\lambda ,\mathbf{x}\right)={\lambda }^n+{\displaystyle \sum _{q=1}^n}{p}_q\left(\mathbf{x}\right){\lambda }^{n-q}.$$

(50)

If the matrix $A\left(\mathbf{x}\right)$ is nonsingular in a neighborhood of the expansion center, then the inverse matrix can be obtained from

$$A^{-1}\left(\mathbf{x}\right)=-{\displaystyle \frac{R_{n-1}\left(\mathbf{x}\right)}{p_n\left(\mathbf{x}\right)}}.$$

(51)

For a direct construction of the differential spectrum of the inverse matrix, let

$$g\left(\mathbf{x}\right)={\displaystyle \frac{1}{p_n\left(\mathbf{x}\right)}},\hspace{1em}\hspace{1em}g\left(\mathbf{x}\right)⟷G\left(\mathbf{k}\right).$$

Since

$$p_n\left(\mathbf{x}\right)g\left(\mathbf{x}\right)=1,$$

the spectrum $G\left(\mathbf{k}\right)$ is determined from the convolution equation

$$\sum _{\mathbf{s}\le \mathbf{k}}{P}_n\left(\mathbf{s}\right)G\left(\mathbf{k}-\mathbf{s}\right)=\delta \left(\mathbf{k}\right),$$

(52)

where

$$\delta \left(\mathbf{k}\right)=\{\begin{array}{cc}1,& \mathbf{k}=\mathbf{0},\\ 0,& \mathbf{k}\ne \mathbf{0}.\end{array}$$

If

$$P_n\left(\mathbf{0}\right)\ne 0,$$

then

$$G\left(\mathbf{0}\right)={\displaystyle \frac{1}{P_n\left(\mathbf{0}\right)}},$$

and, for $\mathbf{k}\ne \mathbf{0},$

$$G\left(\mathbf{k}\right)=-{\displaystyle \frac{1}{P_n\left(\mathbf{0}\right)}}\sum _{\begin{array}{c}\mathbf{s}\le \mathbf{k} \mathbf{s}\ne \mathbf{0}\end{array}}{P}_n\left(\mathbf{s}\right)G\left(\mathbf{k}-\mathbf{s}\right).$$

(53)

Finally, if

$$A^{-1}\left(\mathbf{x}\right)⟷B\left(\mathbf{k}\right),$$

then

$$B\left(\mathbf{k}\right)=-\sum _{\mathbf{s}\le \mathbf{k}}G\left(\mathbf{s}\right)R_{n-1}\left(\mathbf{k}-\mathbf{s}\right).$$

(54)

Thus, the D-analogue of the Faddeev method gives the spectra of the characteristic polynomial coefficients and also provides a recurrence-based construction of the inverse matrix spectrum, provided that the matrix is nonsingular at the expansion center.

3.3. Computational Complexity and Truncation Strategy

The computational cost of the proposed method is mainly determined by the number of retained multidimensional spectral coefficients and by the cost of multidimensional convolution operations. Let $m$ be the number of independent variables and let the rectangular truncation vector be $\mathbf{K}=\left(K_1,\dots ,K_m\right).$ The number of retained multi-indices is

$$N_{\mathbf{K}}={\displaystyle \prod _{r=1}^m}\left(K_r+1\right).$$

If the same truncation order $K$ is used for all variables, then $N_{\mathbf{K}}={\left(K+1\right)}^m.$ Thus, the number of retained spectral coefficients grows polynomially with respect to $K$, but exponentially with respect to the number of independent variables $m$. This reflects the standard curse of dimensionality for multidimensional power-series-type representations.

For a scalar multidimensional convolution,

$$\left(U\ast V\right)\left(\mathbf{k}\right)=\sum _{\mathbf{s}\le \mathbf{k}}U\left(\mathbf{s}\right)V\left(\mathbf{k}-\mathbf{s}\right),$$

the number of summands for a fixed multi-index $\mathbf{k}=\left(k_1,\dots ,k_m\right)$ is

$${\displaystyle \prod _{r=1}^m}\left({\mathrm{k}}_r+1\right)$$

Therefore, the total number of scalar convolution summands required for all multi-indices in the rectangular truncation set is

$$C_{\mathbf{K}}={\displaystyle \sum _{k_1=0}^{K_1}}\dots {\displaystyle \sum _{k_m=0}^{K_m}}{\displaystyle \prod _{r=1}^m}\left(k_r+1\right).$$

Since the summations are separable, this gives

$$C_{\mathbf{K}}={\displaystyle \prod _{r=1}^m}{\displaystyle \frac{\left(K_r+1\right)\left(K_r+2\right)}{2}}.$$

For the equal-order case $K_1=\cdots =K_m=K$, this becomes

$$C_{\mathbf{K}}={\left[{\displaystyle \frac{\left(K+1\right)\left(K+2\right)}{2}}\right]}^m.$$

It is important to distinguish the number of convolution sums from the total number of summands in these sums. The number of output coefficients to be computed is $N_{\mathbf{K}}$, whereas $C_{\mathbf{K}}$ counts the total number of elementary scalar products appearing in all convolution sums over the retained rectangular spectral domain.

For matrix-valued functions, each convolution term involves the multiplication of two $n\times n$ matrices. Using the classical matrix multiplication algorithm, the cost of one matrix multiplication is $O\left(n^3\right)$. More generally, if a matrix multiplication algorithm with exponent ${\omega }_{\mathrm{M}\mathrm{M}}$ is used, the cost can be written as $O\left(n^{{\omega }_{\mathrm{M}\mathrm{M}}}\right)$. The best known theoretical bounds give ${\omega }_{\mathrm{M}\mathrm{M}}<2.371339$, although such algorithms are mainly of asymptotic theoretical interest and are not necessarily advantageous in symbolic computations with large intermediate expressions [19]. Therefore, the approximate cost of one full matrix convolution over the retained spectral domain is $O\left(n^{{\omega }_{\mathrm{M}\mathrm{M}}}{C}_{\mathbf{K}}\right).$ In the D-analogue of the Faddeev method, the auxiliary matrices are computed successively for $q=1,\dots ,n$. Therefore, the total leading cost of the recurrence procedure is approximately $O\left(n^{{\omega }_{\mathrm{M}\mathrm{M}}+1}{C}_{\mathbf{K}}\right),$ up to lower-order costs associated with traces, scalar convolutions, and reconstruction. For the classical matrix multiplication algorithm, these estimates reduce to $O\left(n^3{C}_{\mathbf{K}}\right)$ for one full matrix convolution and $O\left(n^4{C}_{\mathbf{K}}\right)$ for the Faddeev recurrence. The Leverrier D-analogue has the same convolutional nature, since it requires the spectra of powers of the matrix and the spectra of trace functions.

The recurrence structure also exposes a significant amount of parallelism. For a fixed recurrence level $q$, the spectra corresponding to different multi-indices $\mathbf{k}$ can be computed independently after the spectra from the previous level have been obtained. Moreover, the summands in each multidimensional convolution and the entries of the resulting matrices can be evaluated by parallel reduction. Therefore, although the number of convolution terms grows rapidly with $m$ and $\mathbf{K}$, the method is naturally suitable for parallel symbolic-numerical implementation.

Compared with direct symbolic computation, the proposed method does not expand the determinant by permutations and does not construct the adjugate matrix explicitly. Direct symbolic determinant computation may lead to a rapid growth of intermediate expressions, especially for complex-valued matrices depending on several parameters. The D-transform approach replaces these symbolic expansions by structured recurrence relations in the spectral domain. This is advantageous when a local representation of bounded order is sufficient, when the matrix spectra are sparse, or when the required characteristics are needed repeatedly for different nearby parameter values.

The choice of the truncation vector $\mathbf{K}$ depends on the structure of the matrix entries and on the required accuracy. In the case of polynomial matrix functions, $\mathbf{K}$ is chosen so that all nonzero spectra of the matrix entries and all required spectra generated by the recurrence relations are included. In this case, the reconstruction is exact. In the case of analytic non-polynomial or rational entries, $\mathbf{K}$ is selected according to the desired approximation accuracy.

A practical adaptive truncation strategy can be formulated as follows. Let $F_{\mathbf{K}}\left(\mathbf{x}\right)$ denote any reconstructed matrix characteristic, for example a coefficient $p_q\left(\mathbf{x}\right)$, the determinant, or an inverse-matrix entry. For nested truncation vectors ${\mathbf{K}}^{\left(j\right)}\subset {\mathbf{K}}^{\left(j+1\right)}$, the truncation order is increased until the change between two successive reconstructions becomes sufficiently small:

$$\parallel F_{\mathbf{K}+\mathbf{1}}\left(\mathbf{x}\right)-F_{\mathbf{K}}\left(\mathbf{x}\right)\parallel \le \epsilon \parallel F_{\mathbf{K}+\mathbf{1}}\left(\mathbf{x}\right)\parallel ,$$

where $\epsilon$ is a prescribed tolerance. For the inverse matrix, an additional residual-based stopping criterion can be used:

$$\parallel A\left(\mathbf{x}\right)B_{\mathbf{K}}\left(\mathbf{x}\right)-I_n{\parallel }_{\mathrm{\infty }}\le \epsilon ,$$

and

$$\parallel B_{\mathbf{K}}\left(\mathbf{x}\right)A\left(\mathbf{x}\right)-I_n{\parallel }_{\mathrm{\infty }}\le \epsilon .$$

These residuals can be evaluated either symbolically, when possible, or numerically at a representative set of points in the parameter domain. The residual-based indicators introduced in Section 2.3 are used below as practical stopping and verification criteria for truncated inverse-matrix reconstruction.

3.4. Algorithmic Implementation

The computational implementation of the proposed D-analogues can be organized as recurrence-based algorithms over the retained multidimensional spectral set

$${\mathcal{T}}_{\mathbf{K}}=\left\{\mathbf{k}\in {\mathbb{N}}_0^m:0\le k_r\le K_r,\hspace{1em}r=1,\dots ,m\right\}.$$

Algorithm 1 summarizes the Leverrier D-analogue for computing the spectra of the characteristic-polynomial coefficients and the determinant. Algorithm 2 presents the Faddeev D-analogue for computing the characteristic-polynomial coefficients, determinant, and inverse matrix, together with residual-based verification criteria. In Algorithm 1 and Algorithm 2, all spectral operations are performed on $\left({\mathcal{T}}_{\mathbf{K}}\right)$, and $\left(\ast \right)$ denotes multidimensional convolution. The algorithms are written in pseudocode form in order to emphasize the order of computations, the recurrence loops, and the stopping criteria.

Algorithm 1. Leverrier D-analogue for multiparameter matrix functions

Input: Matrix function $A\left(\mathbf{x}\right)\in {\mathbb{C}}^{n\times n}$; independent variables $\mathbf{x};$ expansion center ${\mathbf{x}}_0$;    truncation vector $\mathbf{K}$. Output: Spectra $P_q\left(\mathbf{k}\right)$ and reconstructed coefficients $p_q\left(\mathbf{x}\right)$, $q=1,\dots ,n$; characteristic     polynomial ($P\left(\lambda ,\mathbf{x}\right)$); determinant $\det A\left(\mathbf{x}\right).$ Step 1. Compute the multidimensional matrix spectrum $A\left(\mathbf{k}\right)$, $\mathbf{k}\in {\mathcal{T}}_{\mathbf{K}}$. Step 2. Initialize the spectrum of the first matrix power: $B_1\left(\mathbf{k}\right)=A\left(\mathbf{k}\right).$ Compute $S_1\left(\mathbf{k}\right)={\mathrm{tr}B}_1\left(\mathbf{k}\right), P_1\left(\mathbf{k}\right)=-S_1\left(\mathbf{k}\right).$ $\mathrm{S}\mathrm{t}\mathrm{e}\mathrm{p} 3.\mathrm{For} q=2,\dots ,n \mathrm{and} \mathbf{k}\in {\mathcal{T}}_{\mathbf{K}}, \mathrm{compute}$ $B_q\leftarrow A\ast B_{q-1},$ $S_q\left(\mathbf{k}\right)\leftarrow {\mathrm{tr}B}_q\left(\mathbf{k}\right)$,    $P_q\left(\mathbf{k}\right)\leftarrow -{\displaystyle \frac{1}{q}}[S_q\left(\mathbf{k}\right)+{\displaystyle {\sum }_{r=1}^{q-1}}\left(P_{q-r}\ast S_r\right)\left(\mathbf{k}\right)]$. Step 4. Reconstruct the coefficient functions: $p_q\left(\mathbf{x}\right)\leftarrow \sum _{\mathbf{k}\in {\mathcal{T}}_{\mathbf{K}}}{P}_q\left(\mathbf{k}\right){\left(\mathbf{x}-{\mathbf{x}}_0\right)}^{\mathbf{k}},q=1,\dots ,n.$ Step 5. Construct the characteristic polynomial: $P\left(\lambda ,\mathbf{x}\right).$ The determinant is obtained    from $\det A\left(\mathbf{x}\right)={\left(-1\right)}^n{p}_n\left(\mathbf{x}\right).$ Step 6. Stop if a prescribed truncation vector is used. If adaptive truncation is required,    increase $\mathbf{K}$ and repeat Steps 1–5 until the change in the reconstructed quantities    satisfies the prescribed tolerance.

Algorithm 2. Faddeev D-analogue for multiparameter matrix functions

Input: Matrix function $A\left(\mathbf{x}\right)\in {\mathbb{C}}^{n\times n}$; independent variables $\mathbf{x};$ expansion center ${\mathbf{x}}_0$;    initial truncation vector ${\mathbf{K}}^{\left(0\right)}$; tolerance $\epsilon$, if adaptive truncation is used. Output: Spectra $P_q\left(\mathbf{k}\right)$, $R_q\left(\mathbf{k}\right)$, $q=1,\dots ,n$; characteristic polynomial $P\left(\lambda ,\mathbf{x}\right)$;    determinant $\det A\left(\mathbf{x}\right)$; reconstructed inverse matrix $B_{\mathbf{K}}\left(\mathbf{x}\right)$, if it exists; residual    matrices. Step 1. Set $j=0$ and $\mathbf{K}={\mathbf{K}}^{\left(0\right)}$. Step 2. Compute the multidimensional matrix spectrum $A\left(\mathbf{k}\right)$, $\mathbf{k}\in {\mathcal{T}}_{\mathbf{K}}$. Step 3. Initialize $R_0\left(\mathbf{0}\right)=I_n,R_0\left(\mathbf{k}\right)=0_{n\times n},\mathbf{k}\ne \mathbf{0}.$ Step 4. $\mathrm{For} q=1,\dots ,n \mathrm{and} \mathbf{k}\in {\mathcal{T}}_{\mathbf{K}}, \mathrm{compute}$: $M_q\leftarrow A\ast R_{q-1},$ ${\mathrm{P}}_q\left(\mathbf{k}\right)\leftarrow -{\displaystyle \frac{1}{q}}{\mathrm{tr}M}_q\left(\mathbf{k}\right)$,    $R_q\left(\mathbf{k}\right)\leftarrow M_q\left(\mathbf{k}\right)+P_q\left(\mathbf{k}\right)I_n$. Step 5. Reconstruct $p_q\left(\mathbf{x}\right)$, $q=1,\dots ,n$, and form $P\left(\lambda ,\mathbf{x}\right).$ Then $\det A\left(\mathbf{x}\right)={\left(-1\right)}^n{p}_n\left(\mathbf{x}\right).$ Step 6. If the inverse matrix is required, check the local nonsingularity condition    $p_n\left({\mathbf{x}}_0\right)\ne 0.$ If this condition is not satisfied, the inverse-matrix expansion at ${\mathbf{x}}_0$    is not constructed. Otherwise, compute the spectrum $G\left(\mathbf{k}\right)$ of $1/p_n\left(\mathbf{x}\right)$ by the    scalar reciprocal recurrence and then compute $B\left(\mathbf{k}\right)\leftarrow -\left(G\ast R_{n-1}\right)\left(\mathbf{k}\right),\mathbf{k}\in$     ${\mathcal{T}}_{\mathbf{K}}.$ Reconstruct $B_{\mathbf{K}}\left(\mathbf{x}\right).$ Step 7. Compute the residual matrices $E_{1,\mathbf{K}}\left(\mathbf{x}\right), E_{2,\mathbf{K}}\left(\mathbf{x}\right).$ If $\Vert E_{1,\mathbf{K}}\Vert \le \epsilon \hspace{1em}\mathrm{and}\hspace{1em}\Vert E_{2,\mathbf{K}}\Vert \le$     $\epsilon$. stop and accept $B_{\mathbf{K}}\left(\mathbf{x}\right)$ as the reconstructed inverse matrix with the prescribed accuracy. Step 8. If the stopping criterion is not satisfied, increase the truncation vector,    ${\mathbf{K}}^{\left(j+1\right)}>{\mathbf{K}}^{\left(j\right)},$ set $j=j+1$, and repeat Steps 2–7.

The Leverrier D-analogue is mainly used for constructing the characteristic polynomial and determinant through the spectra of matrix powers and traces. The Faddeev D-analogue provides a more direct recurrence for the characteristic-polynomial coefficients and, in addition, produces the auxiliary matrix spectra required for reconstructing the inverse matrix. In both algorithms, the recurrence index $q$ is sequential, whereas the computations over different multi-indices $\mathbf{k}$, matrix entries, and convolution summands can be parallelized at each fixed recurrence level.

The proposed algorithms were implemented in a Python environment. A Python-based programmatic representation of the multidimensional differential spectrum was developed together with a computational framework for its manipulation and use. The core of the framework is a data structure that stores transform coefficients and supports direct and inverse transforms, algebraic operations, multidimensional convolution, and symbolic parsing within a unified interface. The entries of complex multiparameter matrices are introduced in symbolic form, including real and imaginary components as functions of the independent variables. The developed programs compute the multidimensional differential spectra and output the required matrix characteristics, including the characteristic polynomial coefficients, determinant, and inverse matrix entries, in symbolic form. Arithmetic and differential operations are handled through operator overloading, while immutable spectral objects help preserve consistency and reproducibility when computations are performed repeatedly or concurrently. This design also enables selected computations over multi-indices, matrix entries, and convolution summands to be performed with a degree of parallelism.

4. Examples

4.1. Example 1

Consider the following multiparameter matrix with complex-valued elements:

$$\mathit{A}(\mathbf{x})=\mathit{A}({\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3)=\left[\begin{array}{cccc}{\mathrm{x}}_1+\mathrm{i}{\mathrm{x}}_2^2+1& {\mathrm{x}}_1{\mathrm{x}}_3^2+(-2+5\mathrm{i}){\mathrm{x}}_1& 5{\mathrm{x}}_3-4\mathrm{i}& {\mathrm{x}}_1\\ 2-{\mathrm{x}}_1& 4+3\mathrm{i}{\mathrm{x}}_1^2& 7{\mathrm{x}}_1{\mathrm{x}}_2& {\mathrm{x}}_2\\ {\mathrm{x}}_1& {\mathrm{x}}_2-7\mathrm{i}& {\mathrm{x}}_1{\mathrm{x}}_2& (1-3\mathrm{i}){\mathrm{x}}_2{\mathrm{x}}_3\\ 1+\mathrm{i}{\mathrm{x}}_3& {\mathrm{x}}_3+\mathrm{i}{\mathrm{x}}_1& 1-2\mathrm{i}{\mathrm{x}}_2& {\mathrm{x}}_1{\mathrm{x}}_2+3\mathrm{i}\end{array}\right]$$

(55)

where

$$\mathbf{x}=\left(x_1,x_2,x_3\right),\hspace{1em}{\mathrm{i}}^2=-\mathbf{1}.$$

The matrix $A\left(\mathbf{x}\right)$ is a fourth-order complex-valued matrix depending on three independent variables. We choose the expansion center and the scale factors as

$${\mathbf{x}}_0=\left(0, 0, 0\right),\hspace{1em}\hspace{1em}{H}_1=H_2=H_3=1.$$

For this matrix, the characteristic polynomial has the form

$$P\left(\lambda ,\mathbf{x}\right)=\det \left(\lambda I_4-A\left(\mathbf{x}\right)\right){=\lambda }^4+p_1\left(\mathbf{x}\right){\lambda }^3+p_2\left(\mathbf{x}\right){\lambda }^2+p_3\left(\mathbf{x}\right)\lambda +p_4\left(\mathbf{x}\right).$$

Using the D-analogue of the Faddeev method, the inverse matrix is obtained in the form

$$A^{-1}\left(\mathbf{x}\right)=-{\displaystyle \frac{R_3\left(\mathbf{x}\right)}{p_4\left(\mathbf{x}\right)}},$$

provided that $p_4\left(\mathbf{x}\right)\ne 0$ in the considered neighborhood of the expansion center.

We first determine the multidimensional differential spectra of the matrix $A\left(\mathbf{x}\right)$. Since all elements of $A\left(\mathbf{x}\right)$ are polynomial functions of ($x_1,x_2,x_3$), its differential spectra are equal to the coefficients of the corresponding monomials. Thus,

$$A\left(\mathbf{x}\right)={\displaystyle \sum _{k_1=0}^{\infty }}{\displaystyle \sum _{k_2=0}^{\infty }}{\displaystyle \sum _{k_3=0}^{\infty }}A\left(k_1,k_2,k_3\right)x_1^{k_1}{x}_2^{k_2}{x}_3^{k_3}$$

For the matrix (55), the nonzero matrix spectra are as follows:

$$\begin{array}{l}A\left(0, 0, 0\right)=\left[\begin{array}{cccc}1& 0& -4\mathrm{i}& 0\\ 2& 4& 0& 0\\ 0& -7\mathrm{i}& 0& 0\\ 1& 0& 1& 3\mathrm{i}\end{array}\right], A\left(1, 0, 0\right)=\left[\begin{array}{cccc}1& -2+5\mathrm{i}& 0& 1\\ -1& 0& 0& 0\\ 1& 0& 0& 0\\ 0& \mathrm{i}& 0& 0\end{array}\right],\\ A\left(0, 1, 0\right)=\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 1& 0& 0\\ 0& 0& -2\mathrm{i}& 0\end{array}\right], A\left(0, 0, 1\right)=\left[\begin{array}{cccc}0& 0& 5& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \mathrm{i}& 1& 0& 0\end{array}\right],\\ A\left(2, 0, 0\right)=\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 3\mathrm{i}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right], A\left(0, 2, 0\right)=\left[\begin{array}{cccc}\mathrm{i}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right], A\left(1, 1, 0\right)=\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 7& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right],\\ A\left(0, 1, 1\right)=\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 1-3\mathrm{i}\\ 0& 0& 0& 0\end{array}\right], A\left(1, 0, 2\right)=\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right],\end{array}$$

All remaining spectra are equal to zero:

$$A\left(k_1,k_2,k_3\right)=0_{4\times 4},$$

$$\left(k_1,k_2,k_3\right)\notin \left\{\left(0, 0, 0\right),\left(1, 0, 0\right),\left(0, 1, 0\right),\left(0, 0, 1\right),\left(2, 0, 0\right),\left(0, 2, 0\right),\left(1, 1, 0\right),\left(0, 1, 1\right),\left(1, 0, 2\right)\right\}$$

Using the obtained matrix spectra, we apply the D-analogue of the Faddeev method to compute the differential spectra of the coefficient functions $p_q\left(\mathbf{x}\right)$, q = 1, 2, 3, 4. In the present example, the characteristic polynomial is written as

$$P\left(\lambda ,\mathbf{x}\right)={\lambda }^4+p_1\left(\mathbf{x}\right){\lambda }^3+p_2\left(\mathbf{x}\right){\lambda }^2+p_3\left(\mathbf{x}\right)\lambda +p_4\left(\mathbf{x}\right).$$

The D-Faddeev recurrence produces the multidimensional spectra of the characteristic-polynomial coefficients $p_q\left(\mathbf{x}\right)$, $q=1,\dots ,4$. Since the complete list of nonzero spectra is relatively long,

Table 1. Summary of nonzero differential spectra of the characteristic-polynomial coefficients in Example 1.

Coefficient	Number of Nonzero Spectra	Highest Total Order |k|	Role in the Characteristic Polynomial
$p_1\left(\mathbf{x}\right)$	5	2	Coefficient of ${\lambda }^3$
$p_2\left(\mathbf{x}\right)$	16	4	Coefficient of ${\lambda }^2$
$p_3\left(\mathbf{x}\right)$	41	6	Coefficient of λ
$p_4\left(\mathbf{x}\right)$	59	8	Constant term; $detA\left(x\right)=p_4\left(x\right)$

summarizes the number of nonzero spectral coefficients and their highest total orders.

Table 2. Nonzero differential spectra of the first two characteristic-polynomial coefficients in Example 1.

k = (k1, k2, k3)	P1(k)	P2(k)
(0, 0, 0)	−5−3i	4 + 15i
(1, 0, 0)	−1	7−3i
(0, 1, 1)	0	−2 + 3i
(0, 2, 0)	−i	−3 + 4i
(1, 0, 1)	0	−5-i
(1, 1, 0)	−2	10 + 51i
(2, 0, 0)	−3i	−11 + 8i
(0, 2, 1)	0	6 + 2i
(1, 0, 2)	0	−2
(1, 2, 0)	0	−7
(2, 1, 0)	0	2
(3, 0, 0)	0	3i
(1, 3, 0)	0	2i
(2, 0, 2)	0	1
(2, 2, 0)	0	−2
(3, 1, 0)	0	6i

lists the nonzero spectra of the first two characteristic-polynomial coefficients. The spectra of $p_3\left(\mathbf{x}\right)$ and $p_4\left(\mathbf{x}\right)$ are longer because of the accumulation of convolution terms in the recurrence procedure. Therefore, the complete coefficient list is moved to Supplementary File S1, Table S1, in order to keep the main text readable while preserving reproducibility.

Using the values listed in Table 2 and the complete spectra provided in Table S1 of Supplementary File S1, each coefficient function is reconstructed by the inverse multidimensional differential transform. In particular, $p_q\left(\mathbf{x}\right)$ is obtained as the sum of the products of the corresponding spectral values $P_q\left(\mathbf{k}\right)$ and the monomials ${\left(\mathbf{x}-{\mathbf{x}}_0\right)}^{\mathbf{k}}$. Thus, the tabulated spectra provide a direct constructive scheme for obtaining the characteristic polynomial of the matrix $A\left(\mathbf{x}\right)$.

$$p_q\left(\mathbf{x}\right)=\sum _{\mathbf{k}\in {\mathcal{T}}_{\mathbf{K}}}{P}_q\left(\mathbf{k}\right){\left(\mathbf{x}-{\mathbf{x}}_0\right)}^{\mathbf{k}},\hspace{1em}\hspace{1em}q=1,\dots ,4,$$

Since the expansion center is ${\mathbf{x}}_0=\left(0, 0, 0\right)$, this reduces to

$$p_q\left(\mathbf{x}\right)=\sum _{\left(k_1,k_2,k_3\right)\in {\mathsf{\Omega }}_q}{P}_q\left(k_1,k_2,k_3\right)x_1^{k_1}{x}_2^{k_2}{x}_3^{k_3},\hspace{1em}\hspace{1em}q=1, 2, 3, 4,$$

where

$${\mathsf{\Omega }}_q=\left\{\left(k_1,k_2,k_3\right)\in {\mathbb{N}}_0^3\hspace{0.33em}:\hspace{0.33em}{P}_q\left(k_1,k_2,k_3\right)\ne 0\right\}.$$

For the first coefficient, the nonzero spectra listed in Table 2 give

$$p_1\left(\mathbf{x}\right)=-5-3\mathrm{i}-x_1-3\mathrm{i}{x}_1^2-2x_1{x}_2-\mathrm{i}{x}_2^{2.}$$

The coefficient $p_2\left(\mathbf{x}\right)$ is reconstructed in the same way from the values $P_2\left(\mathbf{k}\right)$ listed in Table 2.

$$\begin{array}{l}{p}_2\left(\mathbf{x}\right)=4+15\mathrm{i}+\left(7-3\mathrm{i}\right)x_1+\left(-11+8\mathrm{i}\right)x_1^2+3\mathrm{i}{x}_1^3+6\mathrm{i}{x}_1^3{x}_2\\ \hspace{1em}\hspace{1em} +2x_1^2{x}_2-2x_1^2{x}_2^2+x_1^2{x}_3^2+\left(10+51\mathrm{i}\right)x_1{x}_2+2\mathrm{i}{x}_1{x}_2^3\\ \hspace{1em}\hspace{1em} -7x_1{x}_2^2-2x_1{x}_3^2-\left(5+\mathrm{i}\right)x_1{x}_3+\left(-3+4\mathrm{i}\right)x_2^2\\ \hspace{1em}\hspace{1em} +\left(6+2\mathrm{i}\right)x_2^2{x}_3+\left(-2+3\mathrm{i}\right)x_2{x}_3.\end{array}$$

The longer coefficients $p_3\left(\mathbf{x}\right)$ and $p_4\left(\mathbf{x}\right)$ are reconstructed from the complete spectra provided in Table S1 of Supplementary File S1. The third coefficient has the form

$$\begin{array}{l}{p}_3\left(\mathbf{x}\right)=-3\mathrm{i}{x}_1^4{x}_2^2-6\mathrm{i}{x}_1^4{x}_2+6x_1^3{x}_2^3-x_1^3{x}_2^2-9x_1^3{x}_2{x}_3^2\\ \hspace{1em}\hspace{1em} +\left(27-51\mathrm{i}\right)x_1^3{x}_2+\left(-3+15\mathrm{i}\right)x_1^3{x}_3+\left(21+4\mathrm{i}\right)x_1^3-\mathrm{i}{x}_1^2{x}_2^4\\ \hspace{1em}\hspace{1em} +7x_1^2{x}_2^3+\left(-15-25\mathrm{i}\right)x_1^2{x}_2^2{x}_3+\left(2-39\mathrm{i}\right)x_1^2{x}_2^2\\ \hspace{1em}\hspace{1em} +4x_1^2{x}_2{x}_3^2+\left(14+4\mathrm{i}\right)x_1^2{x}_2{x}_3+\left(-15-33\mathrm{i}\right)x_1^2{x}_2\\ \hspace{1em}\hspace{1em} -3\mathrm{i}{x}_1^2{x}_3^2+x_1^2{x}_3+\left(23+4\mathrm{i}\right)x_1^2+7\mathrm{i}{x}_1{x}_2^4\\ \hspace{1em}\hspace{1em} +\left(51-8\mathrm{i}\right)x_1{x}_2^3+\left(-7+21\mathrm{i}\right)x_1{x}_2^2{x}_3^2+\left(-5-2\mathrm{i}\right)x_1{x}_2^2{x}_3\\ \hspace{1em}\hspace{1em} +\left(7+21\mathrm{i}\right)x_1{x}_2^2-\mathrm{i}{x}_1{x}_2{x}_3^3-x_1{x}_2{x}_3^2+\left(12-\mathrm{i}\right)x_1{x}_2{x}_3\\ \hspace{1em}\hspace{1em} +\left(141-72\mathrm{i}\right)x_1{x}_2+6\mathrm{i}{x}_1{x}_3^2+\left(18-16\mathrm{i}\right)x_1{x}_3+\left(-42-40\mathrm{i}\right)x_1\\ \hspace{1em}\hspace{1em} +\left(2-6\mathrm{i}\right)x_2^4{x}_3+\left(3+2\mathrm{i}\right)x_2^3{x}_3+2\mathrm{i}{x}_2^3\\ \hspace{1em}\hspace{1em} +\left(-30-10\mathrm{i}\right)x_2^2{x}_3+25x_2^2+\left(-15-5\mathrm{i}\right)x_2{x}_3^3\\ \hspace{1em}\hspace{1em} +\left(-9+27\mathrm{i}\right)x_2{x}_3^2+\left(8-11\mathrm{i}\right)x_2{x}_3+15\mathrm{i}{x}_2+70\mathrm{i}{x}_3+56-12\mathrm{i}.\end{array}$$

Thus, the first three coefficient functions of the characteristic polynomial are obtained by reconstructing their multidimensional differential spectra. The same recurrence procedure is then applied to compute and reconstruct $p_4\left(\mathbf{x}\right)$, which is the determinant coefficient and is also used in the construction of the inverse matrix.

$$\begin{array}{l}{p}_4\left(\mathrm{x}\right)=3\mathrm{i}{x}_1^5{x}_2^2-3x_1^4{x}_2^4+8x_1^4{x}_2^2{x}_3^2+\left(-16+43\mathrm{i}\right)x_1^4{x}_2^2\\ \hspace{1em}\hspace{1em} +\left(3-15\mathrm{i}\right)x_1^4{x}_2{x}_3+\left(-15-11\mathrm{i}\right)x_1^4{x}_2+3\mathrm{i}{x}_1^4\\ \hspace{1em}\hspace{1em} +\left(-7-9\mathrm{i}\right)x_1^3{x}_2^3-2x_1^3{x}_2^2{x}_3^2+\left(15+25\mathrm{i}\right)x_1^3{x}_2^2{x}_3\\ \hspace{1em}\hspace{1em} +\left(8+38\mathrm{i}\right)x_1^3{x}_2^2+24\mathrm{i}{x}_1^3{x}_2{x}_3^2+\left(-17-3\mathrm{i}\right)x_1^3{x}_2{x}_3\\ \hspace{1em}\hspace{1em} +\left(-129-46\mathrm{i}\right)x_1^3{x}_2+45x_1^3{x}_3-36\mathrm{i}{x}_1^3-7\mathrm{i}{x}_1^2{x}_2^5\\ \hspace{1em}\hspace{1em} +\left(-25+15\mathrm{i}\right)x_1^2{x}_2^4{x}_3+\left(-48+4\mathrm{i}\right)x_1^2{x}_2^4+\left(3-9\mathrm{i}\right)x_1^2{x}_2^3{x}_3\\ \hspace{1em}\hspace{1em} -7x_1^2{x}_2^3+\left(-21-7\mathrm{i}\right)x_1^2{x}_2^2{x}_3^4+\left(-1+24\mathrm{i}\right)x_1^2{x}_2^2{x}_3^3\\ \hspace{1em}\hspace{1em} +\left(85-110\mathrm{i}\right)x_1^2{x}_2^2{x}_3^2+\left(-109-21\mathrm{i}\right)x_1^2{x}_2^2{x}_3\\ \hspace{1em}\hspace{1em} +\left(-1+30\mathrm{i}\right)x_1^2{x}_2^2+\left(-16+48\mathrm{i}\right)x_1^2{x}_2{x}_3^3\\ \hspace{1em}\hspace{1em} +\left(95+26\mathrm{i}\right)x_1^2{x}_2{x}_3^2+\left(25-26\mathrm{i}\right)x_1^2{x}_2{x}_3\\ \hspace{1em}\hspace{1em} +\left(-100-22\mathrm{i}\right)x_1^2{x}_2+\left(4-7\mathrm{i}\right)x_1^2+\left(21+7\mathrm{i}\right)x_1{x}_2^4{x}_3^2\\ \hspace{1em}\hspace{1em} -\mathrm{i}{x}_1{x}_2^4{x}_3+21x_1{x}_2^4+\left(-12-149\mathrm{i}\right)x_1{x}_2^3\\ \hspace{1em}\hspace{1em} +\left(-12-4\mathrm{i}\right)x_1{x}_2^2{x}_3^3+\left(7-21\mathrm{i}\right)x_1{x}_2^2{x}_3^2\\ \hspace{1em}\hspace{1em} +\left(77-44\mathrm{i}\right)x_1{x}_2^2{x}_3+\left(-13-25\mathrm{i}\right)x_1{x}_2^2\\ \hspace{1em}\hspace{1em} +\left(7-21\mathrm{i}\right)x_1{x}_2{x}_3^3+\left(-37-14\mathrm{i}\right)x_1{x}_2{x}_3^2\\ \hspace{1em}\hspace{1em} +\left(14-31\mathrm{i}\right)x_1{x}_2{x}_3+\left(-189+5\mathrm{i}\right)x_1{x}_2\\ \hspace{1em}\hspace{1em} +\left(-105-60\mathrm{i}\right)x_1{x}_3+\left(-48+98\mathrm{i}\right)x_1+2x_2^5\\ \hspace{1em}\hspace{1em} +\left(-8+24\mathrm{i}\right)x_2^4{x}_3-13\mathrm{i}{x}_2^4+\left(-12-4\mathrm{i}\right)x_2^3{x}_3+\left(7-2\mathrm{i}\right)x_2^3\\ \hspace{1em}\hspace{1em} -5\mathrm{i}{x}_2^2{x}_3^2+\left(15+8\mathrm{i}\right)x_2^2{x}_3+\left(-13+4\mathrm{i}\right)x_2^2\\ \hspace{1em}\hspace{1em} +\left(50+50\mathrm{i}\right)x_2{x}_3^3+\left(25-100\mathrm{i}\right)x_2{x}_3^2\\ \hspace{1em}\hspace{1em} +\left(-52+89\mathrm{i}\right)x_2{x}_3+\left(52-7\mathrm{i}\right)x_2+210x_3-168\mathrm{i}.\end{array}$$

The last coefficient $p_4\left(\mathbf{x}\right)$ of the characteristic polynomial is equal to the determinant of the matrix $A\left(\mathbf{x}\right)$, that is

$$p_4\left(\mathbf{x}\right)=\det (A\left(\mathbf{x}\right)).$$

Therefore, the inverse matrix exists for all parameter values satisfying

$$p_4\left(\mathbf{x}\right)\ne 0.$$

In the D-analogue of the Faddeev method, the inverse matrix is constructed using the auxiliary matrix $R_3\left(\mathbf{x}\right)$. For the considered fourth-order matrix, we have

$${B\left(\mathbf{x}\right)=A}^{-1}\left(\mathbf{x}\right)=-{\displaystyle \frac{R_3\left(\mathbf{x}\right)}{\det (A\left(\mathbf{x}\right))}},$$

Then the entries of the inverse matrix $B\left(\mathbf{x}\right)$ are obtained as

$$b_{ij}\left(\mathbf{x}\right)=-{\displaystyle \frac{r_{ij}\left(\mathbf{x}\right)}{\det (A\left(\mathbf{x}\right))}}, i,j=1, 2, 3, 4.$$

For example, one representative element of the inverse matrix can be written in the form

$$b_{11}\left(\mathbf{x}\right)=-{\displaystyle \frac{r_{11}\left(\mathbf{x}\right)}{\det (A\left(\mathbf{x}\right))}}={\displaystyle \frac{N_{11}\left(\mathbf{x}\right)}{\det (A\left(\mathbf{x}\right))}},$$

where

$$\begin{array}{l}{N}_{11}\left(\mathbf{x}\right)=-3x_1^4{x}_2^2-9\mathrm{i}{x}_1^3{x}_2-7\mathrm{i}{x}_1^2{x}_2^3-5x_1^2{x}_2^2{x}_3\left(5-3\mathrm{i}\right)+4x_1^2{x}_2^2\left(-12+\mathrm{i}\right)+3x_1^2{x}_2{x}_3\left(1-3\mathrm{i}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+7x_1{x}_2^2{x}_3^2\left(3+\mathrm{i}\right)-\mathrm{i}{x}_1{x}_2^2{x}_3+21x_1{x}_2^2-3x_1{x}_2\left(4+49\mathrm{i}\right)+2x_2^3-8x_2^2{x}_3\left(1-3\mathrm{i}\right)-13\mathrm{i}{x}_2^2\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-4x_2{x}_3\left(3+\mathrm{i}\right)+7x_2\end{array}$$

The explicit expressions for all sixteen elements of $A^{-1}\left(\mathbf{x}\right)$ are rather lengthy. Therefore, only one representative element is presented in the main text, while the complete inverse matrix is given in Supplementary File S1. This allows us to demonstrate the construction of the inverse matrix without overloading the example with long rational expressions.

To verify the correctness of the obtained inverse matrix, the products of the original matrix and the reconstructed inverse matrix were computed symbolically. The following identities were confirmed:

$$A\left(\mathbf{x}\right)A^{-1}\left(\mathbf{x}\right)=I_4,$$

and

$$A^{-1}\left(\mathbf{x}\right)A\left(\mathbf{x}\right)=I_{4.}$$

This confirms that the inverse matrix obtained by the D-analogue of the Faddeev method is identical to the inverse matrix computed by direct symbolic inversion.

The considered example demonstrates that the proposed multidimensional differential-transform-based approach provides a constructive procedure for computing the characteristic polynomial and the inverse matrix of a multiparameter complex-valued matrix. The coefficient functions of the characteristic polynomial were reconstructed from their multidimensional differential spectra, while the inverse matrix was obtained using the D-analogue of the Faddeev method. The equality of the reconstructed inverse matrix to the direct symbolic inverse was verified by confirming that both products $A\left(\mathbf{x}\right)A^{-1}\left(\mathbf{x}\right)$ and $A^{-1}\left(\mathbf{x}\right)A\left(\mathbf{x}\right)$ are equal to the identity matrix. This confirms the correctness of the proposed computational scheme for the considered three-parameter complex matrix.

4.2. Example 2: A Three-Degree-of-Freedom Damped Vibration System

To illustrate the applicability of the proposed method to an engineering-motivated matrix function, we consider a three-degree-of-freedom damped vibration system in the frequency domain. Dynamic stiffness matrices of the form $K+\mathrm{i}\omega C-{\omega }^{2}M$ arise naturally in damped vibration and frequency-response analysis of mechanical and structural systems. Such formulations are closely related to quadratic eigenvalue problems involving mass, damping, and stiffness matrices and to industrial frequency-response problems [11,20].

Let $q\left(t\right)\in {\mathbb{R}}^3$ be the vector of generalized displacements. The equations of motion of a linear damped system have the form

$$M\ddot{q}\left(t\right)+C\dot{q}\left(t\right)+Kq\left(t\right)=f\left(t\right),$$

where $M$, $C$, and $K$ are the mass, damping, and stiffness matrices, respectively. Assuming a harmonic response

$$q\left(t\right)=\widehat{q}{e}^{\mathrm{i}\omega t},\hspace{1em}\hspace{1em}f\left(t\right)=\widehat{f}{e}^{\mathrm{i}\omega t},$$

we obtain the frequency-domain algebraic system

$$A\left(\omega \right)\widehat{q}=\widehat{f},$$

where

$$A\left(\omega \right)=K+\mathrm{i}\omega C-{\omega }^{2}M$$

is the dynamic stiffness matrix. The matrix $A\left(\omega \right)$ is complex-valued because of the damping term $\mathrm{i}\omega C$.

To introduce several independent parameters, we assume that the stiffness and damping matrices depend on structural and damping parameters. In particular, let

$$K\left(\alpha ,\gamma \right)=K_0+\alpha K_{\alpha }+\gamma K_{\gamma },$$

and

$$C\left(\beta ,\eta \right)=C_0+\beta C_{\beta }+\eta C_{\eta }.$$

Thus, the dynamic stiffness matrix becomes

$$A\left(\omega ,\alpha ,\gamma ,\beta ,\eta \right)=K\left(\alpha ,\gamma \right)+\mathrm{i}\omega C\left(\beta ,\eta \right)-{\omega }^{2}M.$$

Here,

$$\mathbf{x}=\left(\omega ,\alpha ,\gamma ,\beta ,\eta \right)$$

is the vector of independent parameters. The parameter $\omega$ is the excitation frequency, $\alpha$ and $\gamma$ describe stiffness variations, while $\beta$ and $\eta$ describe damping variations. Hence, the matrix is a five-parameter complex-valued matrix function.

For the considered example, we choose

$$\begin{gathered} M=\left[\begin{array}{ccc}2& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],K_0=\left[\begin{array}{ccc}5& -2& 0\\ -2& 6& -3\\ 0& -3& 7\end{array}\right],K_{\alpha }=\left[\begin{array}{ccc}1& -1& 0\\ -1& 1& 0\\ 0& 0& 0\end{array}\right],\hspace{1em}\hspace{1em}{K}_{\gamma }=\left[\begin{array}{ccc}0& 0& 0\\ 0& 1& -1\\ 0& -1& 1\end{array}\right], \\ {C}_0=\left[\begin{array}{ccc}2& -1& 0\\ -1& 3& -1\\ 0& -1& 2\end{array}\right],\hspace{1em}\hspace{1em}{C}_{\beta }=\left[\begin{array}{ccc}1& -1& 0\\ -1& 1& 0\\ 0& 0& 0\end{array}\right],\hspace{1em}\hspace{1em}{C}_{\eta }=\left[\begin{array}{ccc}0& 0& 0\\ 0& 1& -1\\ 0& -1& 1\end{array}\right]. \end{gathered}$$

Then

$$A\left(\omega ,\alpha ,\gamma ,\beta ,\eta \right)=\left[\begin{array}{ccc}5+\alpha -2{\omega }^2+\mathrm{i}\omega \left(2+\beta \right)& -2-\alpha -\mathrm{i}\omega \left(1+\beta \right)& 0\\ -2-\alpha -\mathrm{i}\omega \left(1+\beta \right)& 6+\alpha +\gamma -{\omega }^2+\mathrm{i}\omega \left(3+\beta +\eta \right)& -3-\gamma -\mathrm{i}\omega \left(1+\eta \right)\\ 0& -3-\gamma -\mathrm{i}\omega \left(1+\eta \right)& 7+\gamma -{\omega }^2+\mathrm{i}\omega \left(2+\eta \right)\end{array}\right],$$

The determinant equation

$$\mathrm{d}\mathrm{e}\mathrm{t}A\left(\omega ,\alpha ,\gamma ,\beta ,\eta \right)=0$$

defines the frequency-dependent singularity condition of the system. In vibration analysis, such conditions are related to resonance-type behavior and loss of invertibility of the dynamic stiffness matrix. If

$$\mathrm{d}\mathrm{e}\mathrm{t}A\left(\omega ,\alpha ,\gamma ,\beta ,\eta \right)\ne 0,$$

then the frequency response is given by

$$\widehat{q}=A^{-1}\left(\omega ,\alpha ,\gamma ,\beta ,\eta \right)\widehat{f}.$$

Thus, the inverse matrix has a direct physical interpretation as a frequency-response operator.

The expansion center is selected as

$${\mathbf{x}}_0=\left(0,0,0,0,0\right),\hspace{1em}\hspace{1em}{H}_1=H_2=H_3=H_4=H_5=1.$$

Since the entries of $A\left(\mathbf{x}\right)$ are polynomial functions of the parameters, the multidimensional differential spectra coincide with the corresponding polynomial coefficients. Therefore, only finitely many matrix spectra are nonzero.

The nonzero spectra are

$$\begin{gathered} A\left(0,0,0,0,0\right)=\left[\begin{array}{ccc}5& -2& 0\\ -2& 6& -3\\ 0& -3& 7\end{array}\right],A\left(1,0,0,0,0\right)=\left[\begin{array}{ccc}2\mathrm{i}& -\mathrm{i}& 0\\ -\mathrm{i}& 3\mathrm{i}& -\mathrm{i}\\ 0& -\mathrm{i}& 2\mathrm{i}\end{array}\right], \\ A\left(0,1,0,0,0\right)=\left[\begin{array}{ccc}1& -1& 0\\ -1& 1& 0\\ 0& 0& 0\end{array}\right],A\left(0,0,1,0,0\right)=\left[\begin{array}{ccc}0& 0& 0\\ 0& 1& -1\\ 0& -1& 1\end{array}\right], \\ A\left(2,0,0,0,0\right)=\left[\begin{array}{ccc}-2& 0& 0\\ 0& -1& 0\\ 0& 0& -1\end{array}\right],A\left(1,0,0,1,0\right)=\left[\begin{array}{ccc}\mathrm{i}& -\mathrm{i}& 0\\ -\mathrm{i}& \mathrm{i}& 0\\ 0& 0& 0\end{array}\right], \\ A\left(1,0,0,0,1\right)=\left[\begin{array}{ccc}0& 0& 0\\ 0& \mathrm{i}& -\mathrm{i}\\ 0& -\mathrm{i}& \mathrm{i}\end{array}\right]. \end{gathered}$$

All remaining spectra are equal to the zero matrix.

Using the D-analogue of the Faddeev method, the first coefficient of the characteristic polynomial is obtained as

$$p_1\left(\mathbf{x}\right)=4{\omega }^2-2\alpha -2\gamma -18-\mathrm{i}\omega \left(7+2\beta +2\eta \right).$$

The second coefficient can be written as

$$\begin{array}{l}{p}_2\left(\mathbf{x}\right)=5{\omega }^4-63{\omega }^2+94+3\alpha \gamma +21\alpha +17\gamma -5\alpha {\omega }^2-6\gamma {\omega }^2-7\beta {\omega }^2-7\eta {\omega }^2\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} -3\beta \eta {\omega }^2+i(3\alpha \eta \omega +7\alpha \omega +3\beta \gamma \omega +21\beta \omega +17\eta \omega +7\gamma \omega +74\omega \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} -5\beta {\omega }^3-6\eta {\omega }^3-19{\omega }^3),\end{array}$$

The third coefficient is

$$p_3\left(\mathbf{x}\right)=p_{3,R}\left(\mathbf{x}\right)+\mathrm{i}{p}_{3,I}\left(\mathbf{x}\right)$$

where

$$\begin{array}{l}{p}_{3,R}\left(\mathbf{x}\right)=2{\omega }^6-50{\omega }^4+191{\omega }^2-137-40\alpha -31\gamma -8\alpha \gamma \\ \hspace{1em}\hspace{1em}\hspace{1em} +33\alpha {\omega }^2-3\alpha {\omega }^4+29\gamma {\omega }^2-4\gamma {\omega }^4+4\alpha \gamma {\omega }^2\\ \hspace{1em}\hspace{1em}\hspace{1em} +29\beta {\omega }^2-9\beta {\omega }^4+25\eta {\omega }^2-10\eta {\omega }^4\\ \hspace{1em}\hspace{1em}\hspace{1em} +8\beta \eta {\omega }^2-4\beta \eta {\omega }^4+3\alpha \eta {\omega }^2+3\beta \gamma {\omega }^2.\end{array}$$

and

$$\begin{array}{l}{p}_{3,I}\left(\mathbf{x}\right)=-165\omega +109{\omega }^3-12{\omega }^5-29\alpha \omega +9\alpha {\omega }^3\\ \hspace{1em}\hspace{1em}\hspace{1em} -25\gamma \omega +10\gamma {\omega }^3-40\beta \omega +33\beta {\omega }^3-3\beta {\omega }^5\\ \hspace{1em}\hspace{1em}\hspace{1em} -31\eta \omega +29\eta {\omega }^3-4\eta {\omega }^5\\ \hspace{1em}\hspace{1em}\hspace{1em} -8\alpha \eta \omega +4\alpha \eta {\omega }^3-3\alpha \gamma \omega \\ \hspace{1em}\hspace{1em}\hspace{1em} -8\beta \gamma \omega +4\beta \gamma {\omega }^3+3\beta \eta {\omega }^3.\end{array}$$

Since the matrix is of order three and the characteristic polynomial is written in the form $P\left(\lambda ,\mathbf{x}\right)=\det \left(\lambda I_3-A\left(\mathbf{x}\right))\right)$, the determinant is obtained as $\mathrm{d}\mathrm{e}\mathrm{t}A\left(\mathbf{x}\right)=-p_3\left(\mathbf{x}\right).$ The inverse matrix is obtained as

$$A^{-1}\left(\mathbf{x}\right)={\left[b_{ij}\left(\mathbf{x}\right)\right]}_{i,j=1}^3,\mathrm{where} b_{ij}\left(\mathbf{x}\right)={\displaystyle \frac{r_{ij}\left(\mathbf{x}\right)}{\mathrm{d}\mathrm{e}\mathrm{t}\mathrm{A}\left(\mathbf{x}\right)}}$$

$$\begin{array}{l}{r}_{11}\left(\mathbf{x}\right)={\omega }^4-18{\omega }^2+33+7\alpha +7\gamma +\alpha \gamma -\alpha {\omega }^2-2\gamma {\omega }^2\\ \hspace{1em}\hspace{1em}\hspace{1em} -2\beta {\omega }^2-3\eta {\omega }^2-\beta \eta {\omega }^2+\\ \hspace{1em}\hspace{1em}\hspace{1em} +\mathrm{i}(27\omega -5{\omega }^3+2\alpha \omega +7\beta \omega +7\eta \omega +3\gamma \omega \\ \hspace{1em}\hspace{1em}\hspace{1em} \hspace{1em}\hspace{1em}-\beta {\omega }^3-2\eta {\omega }^3+\alpha \eta \omega +\beta \gamma \omega ).\end{array}$$

$$\begin{gathered} \begin{array}{l}{r}_{12}\left(\mathbf{x}\right)=\alpha \gamma -\alpha {\omega }^2+7\alpha -\beta \eta {\omega }^2-2\beta {\omega }^2-\eta {\omega }^2\\ \hspace{1em}\hspace{1em}\hspace{1em} +2\gamma -4{\omega }^2+14+\\ \hspace{1em}\hspace{1em}+\mathrm{i}(\alpha \eta \omega +2\alpha \omega +\beta \gamma \omega -\beta {\omega }^3+7\beta \omega \\ \hspace{1em}\hspace{1em}\hspace{1em} \hspace{1em}\hspace{1em}+2\eta \omega +\gamma \omega -{\omega }^3+11\omega ).\end{array} \\ \begin{array}{l}{r}_{13}\left(\mathbf{x}\right)=\alpha \gamma +3\alpha +2\gamma +6-\beta \eta {\omega }^2-\beta {\omega }^2-\eta {\omega }^2-{\omega }^2\\ \hspace{1em}\hspace{1em}\hspace{1em} +\mathrm{i}\left(\alpha \eta \omega +\alpha \omega +\beta \gamma \omega +3\beta \omega +2\eta \omega +\gamma \omega +5\omega \right).\end{array} \\ \begin{array}{l}{r}_{22}\left(\mathbf{x}\right)=2{\omega }^4-23{\omega }^2+35+7\alpha +5\gamma +\alpha \gamma -\alpha {\omega }^2\\ \hspace{1em}\hspace{1em}\hspace{1em} -2\gamma {\omega }^2-2\beta {\omega }^2-2\eta {\omega }^2-\beta \eta {\omega }^2\\ \hspace{1em}\hspace{1em}\hspace{1em} +\mathrm{i}(24\omega -6{\omega }^3+2\alpha \omega +7\beta \omega +5\eta \omega +2\gamma \omega \\ \hspace{1em}\hspace{1em}\hspace{1em} \hspace{1em}\hspace{1em}-\beta {\omega }^3-2\eta {\omega }^3+\alpha \eta \omega +\beta \gamma \omega ).\end{array} \\ \begin{array}{l}{r}_{23}\left(\mathbf{x}\right)=\alpha \gamma +3\alpha +5\gamma +15-\beta \eta {\omega }^2-\beta {\omega }^2\\ \hspace{1em}\hspace{1em}\hspace{1em} -2\eta {\omega }^2-2\gamma {\omega }^2-8{\omega }^2\\ \hspace{1em}\hspace{1em}\hspace{1em} \mathrm{+}i(11\omega -2{\omega }^3+\alpha \eta \omega +\alpha \omega +\beta \gamma \omega +3\beta \omega \\ \hspace{1em}\hspace{1em}\hspace{1em} \hspace{1em}\hspace{1em}+5\eta \omega +2\gamma \omega -2\eta {\omega }^3).\end{array} \\ \begin{array}{l}{r}_{33}\left(\mathbf{x}\right)=2{\omega }^4-22{\omega }^2+26+7\alpha +5\gamma +\alpha \gamma -3\alpha {\omega }^2\\ \hspace{1em}\hspace{1em}\hspace{1em} -2\gamma {\omega }^2-3\beta {\omega }^2-2\eta {\omega }^2-\beta \eta {\omega }^2\\ \hspace{1em}\hspace{1em}\hspace{1em} +\mathrm{i}(23\omega -8{\omega }^3+3\alpha \omega +7\beta \omega +5\eta \omega +2\gamma \omega \\ \hspace{1em}\hspace{1em}\hspace{1em} \hspace{1em}\hspace{1em}-3\beta {\omega }^3-2\eta {\omega }^3+\alpha \eta \omega +\beta \gamma \omega ).\end{array} \\ {r}_{21}\left(\mathbf{x}\right)=r_{12}\left(\mathbf{x}\right),\hspace{1em}\hspace{1em}{r}_{31}\left(\mathbf{x}\right)=r_{13}\left(\mathbf{x}\right),\hspace{1em}\hspace{1em}{r}_{32}\left(\mathbf{x}\right)=r_{23}\left(\mathbf{x}\right). \end{gathered}$$

The reconstructed characteristic polynomial and determinant were compared with MATLAB direct symbolic computations. The following identities were obtained:

$$P_D\left(\lambda ,\mathbf{x}\right)-P_{\mathrm{M}\mathrm{A}\mathrm{T}\mathrm{L}\mathrm{A}\mathrm{B}}\left(\lambda ,\mathbf{x}\right)=0, \underset{D}{\mathrm{d}\mathrm{e}\mathrm{t}}A\left(\mathbf{x}\right)-\underset{\mathrm{M}\mathrm{A}\mathrm{T}\mathrm{L}\mathrm{A}\mathrm{B}}{\mathrm{d}\mathrm{e}\mathrm{t}}A\left(\mathbf{x}\right)=0.$$

The inverse matrix was verified symbolically by computing

$$A\left(\mathbf{x}\right)A_D^{-1}\left(\mathbf{x}\right)-I_3=0, \mathrm{and} A_D^{-1}\left(\mathbf{x}\right)A\left(\mathbf{x}\right)-I_3=0.$$

Although the inverse matrix was verified symbolically, numerical residuals were also evaluated at representative parameter values in order to assess the numerical behavior of the reconstructed expressions. The residuals in

Table 3. Numerical residuals for the three-degree-of-freedom damped vibration system.

$\mathit{\omega }$	$\mathit{\alpha }$	$\mathit{\gamma }$	$\mathit{\beta }$	$\mathit{\eta }$	${\Vert \mathit{A}{\mathit{A}}_{\mathit{D}}^{-\mathbf{1}}-{\mathit{I}}_{\mathbf{3}}\Vert }_{\mathit{\infty }}$	${\Vert {\mathit{A}}_{\mathit{D}}^{-\mathbf{1}}-{\mathit{A}}_{\mathbf{M}\mathbf{A}\mathbf{T}\mathbf{L}\mathbf{A}\mathbf{B}}^{-\mathbf{1}}\Vert }_{\mathit{\infty }}$
0.200	0.100	0.050	0.030	0.020	$2.764043\times {10}^{-16}$	$9.789594\times {10}^{-17}$
0.600	−0.100	0.080	0.050	−0.020	$2.341486\times {10}^{-16}$	$7.240027\times {10}^{-17}$
1.000	0.200	−0.150	0.100	0.050	$5.264976\times {10}^{-16}$	$1.732681\times {10}^{-16}$
1.500	−0.200	0.100	0.080	0.030	$3.613771\times {10}^{-16}$	$9.728882\times {10}^{-17}$

are of order ${10}^{-16}$, which corresponds to double-precision round-off accuracy. Thus, the numerical evaluation confirms the symbolic identities and demonstrates the stability of the reconstructed inverse matrix at the selected parameter values.

Together with the symbolic identities and the residual results reported in Table 3, this example demonstrates that the proposed method can be applied to a physically motivated complex-valued multiparameter matrix arising in frequency-domain vibration analysis and can provide a verified symbolic representation of the corresponding frequency-response operator.

Representative program outputs for this example are provided in Supplementary File S2. They include the symbolic outputs generated by the developed computational framework for the characteristic polynomial, determinant, and inverse matrix of the three-degree-of-freedom damped vibration system.

4.3. Example 3: Computational Comparison and Truncation Effect for a Dense Seven-Parameter Matrix

To examine the computational behavior of the proposed recurrence-based approach, we considered a dense seven-parameter complex matrix function of order seven. The matrix was chosen in the form of a frequency-domain dynamic stiffness matrix [20]

$$A\left(\mathbf{x}\right)=K_0+{\alpha }_1{K}_1+{\alpha }_2{K}_2+{\alpha }_3{K}_3+\mathrm{i}\omega \left(C_0+{\beta }_1{C}_1+{\beta }_2{C}_2+{\beta }_3{C}_3\right)-{\omega }^{2}M,$$

where

$$\mathbf{x}=\left(\omega ,{\alpha }_1,{\alpha }_2,{\alpha }_3,{\beta }_1,{\beta }_2,{\beta }_3\right).$$

Here, $A\left(\mathbf{x}\right)\in {\mathbb{C}}^{7\times 7}$. The matrices $K_0,K_1,K_2,K_3$ and $C_0,C_1,C_2,C_3$ were chosen as dense stiffness and damping matrices, while $M$ was taken as a diagonal mass matrix․ Their explicit forms are given in Supplementary File S3. Such dense matrices may arise in reduced or condensed frequency-domain models. The matrix is complex valued because of the damping term $\mathrm{i}\omega C$. The entries of $A\left(\mathbf{x}\right)$ are given by

$$\begin{gathered} \begin{array}{l}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} a_{11}\left(\mathbf{x}\right)=46+3{\alpha }_1+2{\alpha }_2+4{\alpha }_3-2{\omega }^2+\mathrm{i}\omega \left(2{\beta }_1+3{\beta }_2+2{\beta }_3+7\right),\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} a_{22}\left(\mathbf{x}\right)=52+4{\alpha }_1+3{\alpha }_2+5{\alpha }_3-{\displaystyle \frac{3}{2}}{\omega }^2+\mathrm{i}\omega \left(3{\beta }_1+2{\beta }_2+3{\beta }_3+8\right),\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} a_{33}\left(\mathbf{x}\right)=49+5{\alpha }_1+4{\alpha }_2+4{\alpha }_3-{\displaystyle \frac{6}{5}}{\omega }^2+\mathrm{i}\omega \left(2{\beta }_1+3{\beta }_2+2{\beta }_3+7\right),\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} a_{44}\left(\mathbf{x}\right)=55+4{\alpha }_1+3{\alpha }_2+5{\alpha }_3-{\displaystyle \frac{11}{10}}{\omega }^2+\mathrm{i}\omega \left(3{\beta }_1+2{\beta }_2+3{\beta }_3+9\right),\\ {\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} a}_{55}\left(\mathbf{x}\right)=51+5{\alpha }_1+4{\alpha }_2+4{\alpha }_3-{\displaystyle \frac{13}{10}}{\omega }^2+\mathrm{i}\omega \left(2{\beta }_1+3{\beta }_2+2{\beta }_3+8\right),\\ {\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} a}_{66}\left(\mathbf{x}\right)=48+4{\alpha }_1+3{\alpha }_2+5{\alpha }_3-{\displaystyle \frac{7}{5}}{\omega }^2+\mathrm{i}\omega \left(3{\beta }_1+2{\beta }_2+3{\beta }_3+7\right),\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} a_{77}\left(\mathbf{x}\right)=44+3{\alpha }_1+2{\alpha }_2+4{\alpha }_3-{\displaystyle \frac{8}{5}}{\omega }^2+\mathrm{i}\omega \left(2{\beta }_1+3{\beta }_2+2{\beta }_3+8\right).\end{array} \\ \begin{array}{l}{a}_{12}\left(\mathbf{x}\right)=-8-{\alpha }_1+{\alpha }_2-\mathrm{i}\omega \left({\beta }_1-{\beta }_2+2\right),\\ a_{13}\left(\mathbf{x}\right)=5+2{\alpha }_1-{\alpha }_2+{\alpha }_3+\mathrm{i}\omega \left({\beta }_3-{\beta }_2+1\right),\\ a_{14}\left(\mathbf{x}\right)=-3+2{\alpha }_2-{\alpha }_3+\mathrm{i}\omega \left({\beta }_1-{\beta }_3+2\right),\\ a_{15}\left(\mathbf{x}\right)=4-2{\alpha }_1+2{\alpha }_3-\mathrm{i}\omega \left({\beta }_1-{\beta }_2+1\right),\\ a_{16}\left(\mathbf{x}\right)=-2+{\alpha }_1-{\alpha }_2-2{\alpha }_3+\mathrm{i}\omega \left({\beta }_3-{\beta }_2+1\right),\\ a_{17}\left(\mathbf{x}\right)=3+{\alpha }_1+2{\alpha }_2+{\alpha }_3-\mathrm{i}\omega \left({\beta }_3-{\beta }_1+2\right),\\ a_{23}\left(\mathbf{x}\right)=-7-{\alpha }_1+{\alpha }_2-\mathrm{i}\omega \left({\beta }_1-{\beta }_2+2\right),\\ a_{24}\left(\mathbf{x}\right)=6+2{\alpha }_1-{\alpha }_2+{\alpha }_3+\mathrm{i}\omega \left({\beta }_3-{\beta }_2+1\right),\\ a_{25}\left(\mathbf{x}\right)=-4+2{\alpha }_2-{\alpha }_3+\mathrm{i}\omega \left({\beta }_1-{\beta }_3+2\right),\\ a_{26}\left(\mathbf{x}\right)=5-2{\alpha }_1+2{\alpha }_3-\mathrm{i}\omega \left({\beta }_1-{\beta }_2+1\right),\\ a_{27}\left(\mathbf{x}\right)=-3+{\alpha }_1-{\alpha }_2-2{\alpha }_3+\mathrm{i}\omega \left({\beta }_3-{\beta }_2+1\right),\\ a_{34}\left(\mathbf{x}\right)=-9-{\alpha }_1+{\alpha }_2-\mathrm{i}\omega \left({\beta }_1-{\beta }_2+2\right),\\ a_{35}\left(\mathbf{x}\right)=6+2{\alpha }_1-{\alpha }_2+{\alpha }_3+\mathrm{i}\omega \left({\beta }_3-{\beta }_2+1\right),\\ a_{36}\left(\mathbf{x}\right)=-5+2{\alpha }_2-{\alpha }_3+\mathrm{i}\omega \left({\beta }_1-{\beta }_3+2\right),\\ a_{37}\left(\mathbf{x}\right)=4-2{\alpha }_1+2{\alpha }_3-\mathrm{i}\omega \left({\beta }_1-{\beta }_2+1\right),\\ a_{45}\left(\mathbf{x}\right)=-8-{\alpha }_1+{\alpha }_2-\mathrm{i}\omega \left({\beta }_1-{\beta }_2+2\right),\\ a_{46}\left(\mathbf{x}\right)=7+2{\alpha }_1-{\alpha }_2+{\alpha }_3+\mathrm{i}\omega \left({\beta }_3-{\beta }_2+1\right),\\ a_{47}\left(\mathbf{x}\right)=-5+2{\alpha }_2-{\alpha }_3+\mathrm{i}\omega \left({\beta }_1-{\beta }_3+2\right),\\ a_{56}\left(\mathbf{x}\right)=-7-{\alpha }_1+{\alpha }_2-\mathrm{i}\omega \left({\beta }_1-{\beta }_2+2\right),\\ a_{57}\left(\mathbf{x}\right)=6+2{\alpha }_1-{\alpha }_2+{\alpha }_3+\mathrm{i}\omega \left({\beta }_3-{\beta }_2+1\right),\\ a_{67}\left(\mathbf{x}\right)=-6-{\alpha }_1+{\alpha }_2-\mathrm{i}\omega \left({\beta }_1-{\beta }_2+2\right).\end{array} \\ {a}_{ji}=a_{ij},\hspace{1em}\hspace{1em}i,j=1,\dots ,7. \end{gathered}$$

Although the matrix is dense, its multidimensional differential spectrum is sparse. The matrix function is completely defined by nine nonzero spectra:

$$\begin{gathered} A\left(0,0,0,0,0,0,0\right)=K_0, A\left(1,0,0,0,0,0,0\right)=\mathrm{i}{C}_0, A\left(2,0,0,0,0,0,0\right)=-M, \\ A\left(0,1,0,0,0,0,0\right)=K_1,\hspace{1em}\hspace{1em}A\left(0,0,1,0,0,0,0\right)=K_2,\hspace{1em}\hspace{1em}A\left(0,0,0,1,0,0,0\right)=K_3, \end{gathered}$$

and

$$A\left(1,0,0,0,1,0,0\right)=\mathrm{i}{C}_1,\hspace{1em}\hspace{1em}A\left(1,0,0,0,0,1,0\right)=\mathrm{i}{C}_2,\hspace{1em}\hspace{1em}A\left(1,0,0,0,0,0,1\right)=\mathrm{i}{C}_3.$$

The sparse D-Faddeev recurrence was applied for the following truncation vectors:

$$\begin{array}{l}{\mathbf{K}}_1=\left(14,1,1,1,1,1,1\right),{\mathbf{K}}_2=\left(14,2,2,2,2,2,2\right),{\mathbf{K}}_3=\left(14,3,3,3,3,3,3\right), \mathrm{a}\mathrm{n}\mathrm{d}\\ {\mathbf{K}}_4=\left(14,7,7,7,7,7,7\right).\end{array}$$

The numerical results are summarized in

Table 4. Computational characteristics for the dense $7\times 7$ seven-parameter complex matrix.

Truncation	Retained Multi-indices	Nonzero Spectra of ${\mathit{R}}_{\mathbf{6}}$	Terms in ${\mathit{p}}_{\mathbf{7}}$	Recurrence Time, s	Reconstruction Time, s
${\mathbf{K}}_1$	960	448	576	39.108	15.507
${\mathbf{K}}_2$	10,935	1587	2583	145.506	18.659
${\mathbf{K}}_3$	61,440	2244	4212	223.532	27.618
${\mathbf{K}}_4$	3,932,160	2508	5146	352.372	45.158

and

Table 5. Accuracy indicators for the reconstructed inverse matrix.

Truncation	Maximum Residual Indicator $\parallel \mathit{A}{\mathit{B}}_{\mathbf{K}}-{\mathit{I}}_{\mathbf{7}}{\parallel }_{\mathbf{\infty }}$	Maximum Difference from MATLAB Numerical Inverse
${\mathbf{K}}_1$	5.208 × 10−3	1.344 × 10−4
${\mathbf{K}}_2$	1.479 × 10−4	3.672 × 10−6
${\mathbf{K}}_3$	$9.038 \times 1$0−8	2.087 × 10−9
${\mathbf{K}}_4$	4.169 × 10−15	7.529 × 10−17

.

The results show that increasing the truncation orders with respect to the stiffness and damping parameters significantly improves the accuracy of the reconstructed inverse matrix. When passing from ${\mathbf{K}}_1$ to ${\mathbf{K}}_2$, the maximum residual norm decreased from $5.208\times {10}^{-3}$ to $1.479\times {10}^{-4}$, while the maximum difference between the D-Faddeev inverse and MATLAB numerical inversion decreased from $1.344\times {10}^{-4}$ to $3.672\times {10}^{-6}$. For ${\mathbf{K}}_3$, the maximum residual decreased further to $9.038\times {10}^{-8}$, and the corresponding maximum difference from MATLAB numerical inversion was $2.087\times {10}^{-9}$. For the final truncation vector ${\mathbf{K}}_4=\left(14, 7, 7, 7, 7, 7, 7\right)$, the inverse-matrix residuals were reduced to the order of 4.169 × 10⁻¹⁵, while the difference from MATLAB numerical inversion was of the order of 7.529 × 10⁻¹⁷.

The results for ${\mathbf{K}}_4=\left(14, 7, 7, 7, 7, 7, 7\right)$ show that increasing the truncation vector leads to a significant increase in computational cost but also reduces the residual indicators to the order of ${10}^{-15}$. This confirms the role of the truncation vector as an accuracy-control parameter and illustrates the trade-off between computational effort and reconstruction accuracy.

For comparison, MATLAB direct symbolic computation of the determinant of the same dense $7\times 7$ seven-parameter complex matrix required 556.09 s. In contrast, the sparse D-Faddeev recurrence with ${\mathbf{K}}_2$ required 145.506 s for the recurrence computation and 18.659 s for reconstruction. For the larger truncation vector ${\mathbf{K}}_4$, the recurrence required 352.372 s and reconstruction required 45.158 s, while providing significantly higher accuracy. All computations reported in this example were performed on the same computer equipped with an Intel(R) Core (TM) i9-14900K processor (Intel Corporation, Santa Clara, CA, USA) and an NVIDIA GeForce RTX 4070 graphics card (NVIDIA Corporation, Santa Clara, CA, USA). In our computational experiment, direct MATLAB symbolic verification of the inverse matrix by simplifying $A{A}^{-1}-I_7$ and $A^{-1}A-I_7$ proved substantially more time-consuming than determinant computation. Therefore, the reconstructed inverse matrix was assessed by residual norms and by comparison with MATLAB numerical inversion at representative parameter values.

The comparison confirms two important points. First, the proposed recurrence-based method can exploit the sparsity of the multidimensional differential spectrum even when the original matrix is dense. Second, the truncation vector provides a practical mechanism for controlling the accuracy of the reconstructed inverse matrix. Increasing the truncation vector reduces the residual norms and the difference from MATLAB numerical inversion, but it also increases the number of retained multi-indices and the recurrence time. Thus, the example clearly illustrates the trade-off between computational cost and approximation accuracy. This behavior is consistent with the convergence and truncation-error discussion presented in Section 2.3.

The complete input data and the reconstructed determinant for this example are provided in Supplementary File S3. In particular, Supplementary File S3 contains the matrices $M$, $K_0,K_1,K_2,K_3$, $C_0,C_1,C_2,C_3$, and the reconstructed determinant $D_{{\mathbf{K}}_4}\left(\mathbf{x}\right)={\det }_{{\mathbf{K}}_4}A\left(\mathbf{x}\right)=-p_{7,{\mathbf{K}}_4}\left(\mathbf{x}\right),{\mathbf{K}}_4=\left(14, 7, 7, 7, 7, 7, 7\right).$ However, the expanded numerator polynomials $R_{ij,\mathbf{K}}\left(\mathbf{x}\right)$ of the inverse matrix entries are not reproduced in full because of their excessive length. The accuracy of the reconstructed inverse matrix is assessed by the residual indicators reported in Table 5.

5. Conclusions

In this paper, a multidimensional differential-transform-based approach was developed for computing the characteristic polynomial, determinant, and inverse matrix of complex-valued multiparameter matrix functions. The proposed method extends earlier one-parameter D-transform constructions to the multidimensional case and formulates D-analogues of the Leverrier and Faddeev methods in the spectral domain. By using multidimensional convolution, the method replaces direct symbolic expansion of matrix characteristics with structured recurrence relations for the corresponding differential spectra.

The theoretical discussion shows that the inverse multidimensional transform is governed by the analyticity of the considered functions in a neighborhood of the expansion point. For analytic functions, convergence in a polydisc and truncation-error estimates can be obtained from the multidimensional Taylor expansion and Cauchy estimates. For inverse matrices, the additional local nonsingularity condition $\det A\left({\mathbf{x}}_0\right)\ne 0$ is essential. Under this condition, the inverse matrix admits a local analytic representation in a neighborhood of the expansion point where the determinant does not vanish. If $\det A\left(\mathbf{x}\right)=0$ at some parameter values, the inverse matrix is not defined at those values. Thus, the reconstructed determinant also serves as an indicator of parameter values at which invertibility is lost.

The computational examples confirm the applicability of the proposed method. The illustrative example of a $4\times 4$ complex matrix depending on three independent parameters showed the practical implementation of the developed approach. The differential spectra of the matrix were computed, the characteristic-polynomial coefficients were reconstructed, and the inverse matrix was obtained in rational form. The correctness of the inverse matrix was verified by symbolic multiplication, confirming that the products of the original matrix and its reconstructed inverse are equal to the identity matrix. In the three-degree-of-freedom damped vibration example, the reconstructed characteristic polynomial, determinant, and inverse matrix were consistent with MATLAB symbolic and numerical verification. The residual norms at representative parameter values were of the order of double-precision round-off accuracy. In the dense $7\times 7$, seven-parameter example, the sparse D-Faddeev recurrence demonstrated how the truncation vector can be used as an accuracy-control parameter. Increasing the truncation orders reduced the residual norms and the difference from MATLAB numerical inversion, although it also increased the number of retained multi-indices and the recurrence time.

The obtained results indicate that the proposed approach is most useful when the multidimensional differential spectrum of the matrix function is sparse, when a bounded-order local analytical representation is sufficient, or when the same matrix characteristics must be evaluated repeatedly in a neighborhood of a fixed expansion point. At the same time, the method has natural limitations. The number of retained spectral coefficients and convolution terms grows rapidly with the number of independent parameters and truncation orders, reflecting the curse of dimensionality. This limitation is not eliminated in general, but its practical impact can be partly mitigated in favorable cases by exploiting sparse differential spectra, bounded-order local reconstructions, adaptive truncation, and the parallel structure of spectral-domain computations. The recurrence relations are composed of elementary algebraic operations, multidimensional convolutions, trace evaluations, and matrix products that can be organized over independent multi-indices and matrix entries. Thus, although the total number of operations may be large, many of these operations are structurally simple and naturally suitable for parallel symbolic-numerical implementation in the differential-spectrum domain.

Future work will focus on extending the proposed framework to generalized inverse matrices, Padé-type reconstruction of matrix functions, and eigenvalue-related problems for multiparameter complex matrices. Further development of parallel symbolic-numerical implementations is also planned in order to improve the scalability of the method for larger matrices and higher-dimensional parameter spaces.