Symbolic Construction of Matrix Functions in a Numerical Environment

Within the field of Computational Science, the importance of programs and tools involving systems of differential equations cannot be overemphasized. Many industrial sites, such as nuclear power facilities, are unable to safely operate without these systems. This research explores and studies matrix differential equations and their applications to real computing structures. Through the use of software such as MatLab, I have constructed a toolbox, or collection, of programs that will allow any user to easily calculate a variety of matrix functions. The first tool in this collection is a program that computes the matrix exponential, famously studied and presented by I.E. Leonard and Eduardo Liz. Currently, The Padé approximation is widely adopted to compute matrix exponentials. Unfortunately, this approximation yields significant errors given sufficiently large matrices or ill-conditioned matrices. This program is able to compute these matrix function values with little to no error, providing a distinct advantage over recursive-based methods. Utilizing the results from the matrix exponential, the next programs developed into the study and computation of logarithmic, sinusoidal, and tangential matrix functions. Many of these functions could have numerous uses in the study of dynamical systems that have applications in engineering models