The use of arithmetic to obtain at least approximate answers to very complicated problems pre-dated the development of the calculus by Newton and Leibniz. The amount of arithmetic done by some of the early astronomers as they attempted to explain the movement of the planets would bring anguish to any person today who might attempt to duplicate their computations, even with the aid of a hand calculator. The introduction of integration and differentiation, however, added a new impetus to the search for methods which would permit more efficient and more accurate approximations for results which even the new mathematics could not provide in a formal sense. We have frequently referred to the fact that formulas fail to exist for the evaluation of much used integrals. Problems such as this are the object of the continuing search for improved numerical methods. In fact, the development of the modern high speed digital computer was hastened by the need for just such computation in time of war. It is our purpose in this chapter to explore some of the more commonly used techniques for differentiation and integration using numerical methods. In many cases the results will be aided by the use of a hand calculator or a computer, although the techniques can be used in hand computation, just as they have been used for decades, and in many cases centuries.
KeywordsError Term Truncation Error Trapezoidal Rule Decimal Place Richardson Extrapolation
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