### Error Estimation in Computer Calculation

is calculated, by some means or other; the nature of the "error in y" which results depends on the conceptual framework in which the calculation is conceived. If the calculation is regarded as the evaluation of a polynomial P(x), where the argument x is exact as represented, the only source of error affecting the result is that arising from rounding or otherwise truncating the results of the arithmetic operations. If x is taken to be only an approximation of some "true" argument, however, this must also be taken into account in attributing an error to the value y = P(x). Finally, one may not wish to think of this as the evaluation of a polynomial at all, but rather as the evaluation of the function log [(1 +x)/(1 -x)] for x near 0; the polynomial is then only an approximation to an infinite series, and the error in y due to neglecting terms of order greater than 2n+1 must also be taken into account. Here one would still have to distinguish between cases according as x is assumed exact or is taken only as an approximation. One sees that care must be taken in making statements about the "error in y," to provide a context for interpretation. It is convenient, and by now more or less traditional, to distinguish three sources of error, designated (here) generated, inherent and analytic. Generated error reflects inaccuracies due to the necessity of rounding or otherwise truncating the numeric results of arithmetic operations, inherent error reflects inaccuracies in initially given arguments and parameters, and analytic error reflects inaccuracies due to the use of a computing procedure which calculates only an approximation to the theoretical result desired. In practice, error from any of these sources is introduced at successive points in a calculation, and affects the subsequent results. The study of the propagation of error in calculations of various types is an important part of error analysis. Error propagation may in general be studied without reference to the classification of error by source; it is usually possible, however, to break down the final error in a calculation and ascribe part of it to each of the several error sources [1, ch. 1]. On the other hand, it is often convenient to adopt a relative point of view concerning sources of error; thus at any stage of a calculation all previously accumulated error, from whatever source, might be regarded as inherent error for subsequent stages. Another expedient which has been successfully used is to