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The present paper focuses on the efficient numerical solving of initial-value problems (IVPs) using digital computers and one-step numerical methods. We start from considering that the integration stepsize is the crucial factor in determining the number of calculations required and the amount of work involved to obtain the approximate values of the exact solution of a certain problem for a given set of points, within a prescribed computational accuracy, is proportional to the number of accomplished iterations. We perform an analysis of the local truncation error and we derive an adaptive stepsize algorithm which coupled with a certain one-step numerical method makes the use of this structure more computationally effective and insures that the estimated values of the exact solution are in agreement with an imposed accuracy. We conclude with numerical computations proving the efficiency of the proposed step selection algorithm.

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The present paper studies the numerical computation of the extreme eigenvalues of a Ò ¢ Ò real symmetric matrix by the means of the Newton's approximate method for the characteristic polynomial È ´ µ An iterative algorithm is also presented,involving the computation of a trace of an appropriate matrix, instead of using the evaluation of È ´ µ and its derivative. Numerical examples solved with this algorithm are to be found within as well.