Error analysis in the series evaluation of the exponential type integral ezE1(z)

The method of convolution algebra is used to compute values of the exponential type integral ezE1(z), by expansion of the integrand in a string of Taylor seriesʹ along the real s-axis for any complex parameter z, accurate within ±1 of the last digit of seven-digit computation. Accuracy is verified by comparison with existing tables of E1 and related integrals. This method is used to assess the accuracy of the error estimates of all subsequent computations. Three errors of a Taylor series are identified. These consist of a Taylor series truncation error, a digital truncation error, and a stability error. Methods are developed to estimate the error. By iteration a numerical radius of convergence for a given accuracy is determined. The z-plane is divided into three regions in which three different types of series are used to expand the function f(z) directly in z. Around the center the well-known Frobenius series is used. In the outer region the well-known asymptotic approximation is used. Their accuracy boundaries are determined. In the near-annular region in between, a set of Taylor series is introduced. As the result, the function f(z) can be computed fast with the appropriate series for any complex argument z, to an accuracy within less than relative error of 5 × 10−7.