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Numerical Operational Calculus for Matrices with Applications to Mechanical and Mathematical Problems

The calculation of the matrix exponential e A is important in many problems of mechanics and applied mathematics. In this paper its calculation is based on the Dunford-Taylor integral representation. As a contour line, a polygonal is chosen where the eigenvalues of A lie in its interior, the contour integral is evaluated numerically by a summed Gaussian quadrature formula, and estimates of the discretization error for a mechanical problem are given which are optimal in a certain sense, and which prove the convergence of the described method. It is shown theoretically that the method - called Numerical Operational Calculus - is superior to the methods known so far for sparse matrices of large order, a situation which often occurs in applications. The theoretical considerations are confirmed by numerical tests for the free-vibration problem of a multi-mass vibration chain. We stress that the damping matrix need not be proportional to the mass and/or stiffness matrix. Also, the method is applied to a series of problems from mathematics showing its wide range of applicability.