Stability of the Radau IA and Lobatto IIIC methods for neutral delay differential system

AbstractNumerical stability is considered for several Runge–Kutta methods to systems of neutral delay differential equations. The linear stability analysis is adopted to the system. Adapted with the equistage interpolation process as well as the continuous extension, the Runge–Kutta methods are shown to have the numerical stability similar to the analytical asymptotic stability with arbitrary stepsize, when certain assumptions hold for the logarithmic matrix norm on the coefficient matrices of the NDDE system

An Implementable Version of the Sturm's Algorithm for the Number of Zeros of a Real Polynomial

An algorithm is considered to give the number of real zeros of a real polynomial on an interval rather than their precise locations. The Sturm's algorithm is suitable for such problems because it is not uncommon that the polynomial to be treated is in fact over the rational field Q. While the algorithm is implemented through a symbolic and algebraic manipulation (SAM) software on a computer, computational costs make a significant increase as the degree of polynomial increases. The reason lies in the time-consuming reduction of non-reduced fraction to the irreducible one in SAM. The essential information in the Sturm's algorithm is however not the coefficients of polynomials in the Sturm sequence but their signs at point in the interval. From this viewpoint we have reached an improved version of the algorithm, which drastically reduces the costs in comparison with the original Sturm's algorithm. Some numerical examples arising from a problem in mathematics will be shown