# Computing the Matrix Logarithm with the Romberg Integration Method
## 1 Introduction
The matrix logarithm was computed via the eigenvalue decomposition by means of the QR algorithm with a implicit Wilkinson shift. In [21], it was shown how to obtain the most appropriate true or approximate generator matrix \(Q\) for an empirically observed Markov transition matrix \(P\), with particular application to credit ratings. Credit rating transition matrices \(P\) are traditionally considered in credit risk modeling and in the financial industry. Given that empirically estimated matrices \(P\) are mostly for a one year period, there is a need to recover a matrix generator \(Q\), such as \(Q=\log(P)\), so that a transition matrix \(P(t)=e^{tQ}\) can be obtained for any arbitrary period of time \(t\), e.g., with the purpose of assessing a possible default. Systems involving transient interactions arise naturally in many areas, including telecommunications, online social networking, and neuroscience. A new mathematical framework where network evolution is handled continuously over time was presented in [22], providing a representation of dynamical systems for the concept of node centrality. The novel differential equations approach, where the logarithm function of a matrix resulting from the subtraction of the identity and the continuous-time adjacency matrix appears, is convenient for modeling and analyzing network evolution. This new setting is suitable for many digital applications, such as ranking nodes, detecting virality, and making time-sensitive strategic decisions. Two algorithms to solve the frequency-limited Riemannian optimization model order reduction problems of linear and bilinear systems were proposed in [23]. For this purpose, a new Riemannian conjugate gradient scheme based on the Riemannian geometry notions on a product manifold was designed, and a new search direction was then generated. The algorithms are also suitable for generating reduced systems over a frequency interval in band-pass form. Both algorithms involved the computation of matrix logarithms and Frechet derivatives. Various distance measures for networks and graphs in persistent homology were surveyed in [24]. The paper was especially focused on brain networks, but the methodscould be adapted to any weighted graph in other fields. Among these metrics, the log-Euclidean distance can be found, which provides the shortest distance between two edge weights matrices and where the logarithm of both matrices must be computed. Population covariance matrix estimation is an important component of many statistical methods and applications, such as the optimization and classification of human tumors from genomic data, among many others. In [25], a method of estimating the covariance matrix by maximizing the penalized matrix logarithm transformed likelihood function was introduced. In this function, the matrix logarithm of the covariance matrix appears. The matrix logarithm can also be used to retrieve the coefficient matrix of a system driven by the linear differential equation \(y^{\prime}=Xy\) from observations of the vector \(y\), and to calculate the time-invariant element of the state transition matrix in ODEs with periodic time-varying coefficients [26]. The applicability of the matrix logarithm in so many distinct areas has encouraged the development of different approaches to its evaluation. The traditionally proposed methods incorporate algorithms based on the inverse scaling and squaring technique [27], the Schur-Frechet procedure [28], the Pade approximants [29; 30; 31; 32; 33; 34; 35; 36], arithmetic-geometric mean iteration [37], numerical spectral and Jordan decomposition [38], contour integrals [39], or different quadrature formulas [40; 41; 42; 43]. MATLAB incorporates logm as a built-in function that uses the algorithms described in [33; 34] to compute the principal matrix logarithm. Recently, an implementation that used matrix polynomial formulas to efficiently evaluate the [PERSON] approximation of the matrix logarithm was described in [44].