Validated high precision solution of second order initial value problem with Taylor model

The problem of reliability of computer computations is one of great concern to specialists in many areas of science and engineering. The notion of computing estimates of numerical error in computer simulations is not new. In recent years, considerable progress has been made in determining theoretical and computational techniques that aid to improve the reliability of results of simulations. An important advance in this area has been the recent discovery of methods to determine upper and lower bounds of local approximation error in any given simulation. Different from floating-point computations, interval arithmetic offers a simple mechanism to evaluate an enclosure of a function. Interval arithmetic is the arithmetic defined on sets of intervals, rather than sets of real numbers. The power of the interval arithmetic lay in implementation of interval arithmetic on computers. The fundamental problem in interval methods is computing the ranges of values of real function. The overestimation of the range of a given function by the interval arithmetic expression is strongly dependent on the arithmetic expression of the given function. The reason for this is based on the fact that interval arithmetic does not follow the same rules as the arithmetic for real numbers.

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This work was supported by the State Committee for Scientific Research (KBN) under grant No 4 T11F 001 25 which is gratefully acknowledged.