# Computing the Matrix Logarithm with the Romberg Integration Method
## 1 Introduction  
Let \(A\in\mathbb{C}^{n\times n}\) be a non-singular matrix whose eigenvalues are found in \(\mathbb{C}-(-\infty,0\,]\). A logarithm of \(A\) is defined as any matrix \(X\in\mathbb{C}^{n\times n}\) such that  
\[A=e^{X}, \tag{1}\]  
where \(e^{X}=\sum\limits_{n\geq 0}\frac{X^{n}}{n! }\) is the matrix exponential of \(X\). Although any non-singular matrix \(A\) has infinite logarithms, we will only consider the principal logarithm represented by \(\log\left(A\right)\), which is the only one whose eigenvalues all belong to the set \(\{z\in\mathbb{C};-\pi<Im(z)<\pi\}\)[1]. The principal matrix logarithm is widely employed in numerous disciplines in science and engineering, such as sociology [2], optics [3], biomolecular dynamics [4], quantum chemistry [5], quantum mechanics [6], mechanics [7], buckling simulation [8], the study of viscoelastic fluids [9, 10], control theory [11], computer graphics [12], computer-aided design (CAD) [13], neural networks [14], machine learning [15, 16, 17, 18, 19], brain-machine interfaces [20], the study of Markov chains [21], graph theory [22], optimization [23], topological distances between networks [24], statistics and data processing [25], and so on. The following is a more detailed description of the aforementioned fields in which the matrix logarithm has direct applicability. Embeddability and identification issues, which are especially pertinent to modeling social phenomena using continuous-time Markov structures, when only fragmentary data are available or when the observations contain errors, are considered in [2]. The logarithm function is required for estimating the intensity matrix. A procedure based on using a linear differential equation expansion, for the extraction of elementary properties of a homogeneous depolarizing medium from the Mueller matrix logarithm is provided in [3]. In a polarimetric measurement, the differential matrix and, consequently, the elementary polarization properties of the medium are obtained using the matrix logarithm of the experimental Mueller matrix. In [4], the method for identifying the most important metastable states of a system with complicated dynamical behavior from time series information was extended to handle arbitrary dimensions and to enlarge the diffusion classes considered. This approach represents the effective dynamics of the full system through a Markov jump process between metastable states and the dynamics within each of these metastable states with diffusions. In this method, the logarithm of the correlation matrix must be computed. In [5], the random phase approximation (RPA) correlation energy was expressed in terms of the exact local Kohn-Sham (KS) exchange potential and the corresponding adiabatic and non-adiabatic exchange kernels for density-functional reference determinants. This approach extends the RPA method and yields correlation energies that are more accurate than the traditional RPA technique. The logarithm matrix function is required in coupling strength integration. A classical upper bound for quantum entropy is identified and illustrated in [6], involving the matrix logarithm of the variance in phase space of the classical limit distribution of a given system. In [7], micro-dilatation theory or void elasticity was extended to both large displacement and large dilatation using thermodynamic principles. The deformation gradient tensor was defined by means of the matrix exponential function. The relationship of the displacement gradient and deformation gradient tensor was implemented using the matrix logarithm function. A finite-element-based computational framework for modeling the buckling distortion of overlap joints due to gas metal arc welding was presented in [8]. The total strain tensor was obtained thanks to logarithm computation of the displacement tensor. In [9], the matrix logarithm conformation representation of a viscoelastic fluid flow was implemented within a finite element method (FEM) context. A different derivation of the log-based evolution equation was also presented.