Extended Matrix Approach for Differential Calculus and Its Application to Reliability Engineering

This paper proposes a new way of executing second order partial differential calculus using 4 by 4 matrices and demonstrates an important application to the reliability engineering field. The existing matrix approach for ordinary or first-order differential calculus prevents an exponential increase in computation time of the post-expression obtained by differential calculus and realizes a linear time increase instead. It was emphasized that this approach is a breakthrough for solving computation problems not only in reliability engineering fields, but also all science and engineering fields, because differential calculus is essential to and commonly used for almost all of them. However, the existing approach can only be applied to ordinary or first-order partial differential calculus. Higher order partial differential derivatives are out of its scope. This paper first extends the matrix approach to second-order partial differential calculus as an important step towards higher order calculus. The proposed method is used to compute the Joint Reliability Importance of System, which is a key index in reliability engineering. We emphasize that our matrix approach is especially useful if a system has a large number of components as in the case of communications systems.