Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (AMS55)
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x INTRODUCTION 3. Auxiliary Functions and Arguments One of the objects of this Handbook is to pro- vide tables or computing methods which enable the user to evaluate the tabulated functions over complele ranges of real values of their parameters. In order to achieve this object, frequent use has been made of auxiliary functions to remove the infinite part of the original functions a t their singularities, and auxiliary arguments to co e with cedure clear. The exponential integral of positive apgument is given by infinite ranges. An example will make t % e pro- x 22 xa =7+lnx+-+-+-+ . . . 1.1! 2:2! 3.3! The logarithmic singularity precludes direct inter- polation near z=O. The functions Ei(x)---In z and cc-’[Ei(z)-ln 2-71, however, are well- behaved and readily interpolable in this region. Either will do as an auxiliary function; the latter was in fact selected as it yields slightly higher accuracy when Ei(z) is recovered. The function P[Ei(z)-ln 2-71 has been tabulated to nine decimals for the range O<x<$. For 3 5 2 5 2 , Ei(2) is sufficiently well-behaved to admit direct tabulation, but for larger values of 2, its expo- nential character predominates. A smoother and more readily interpolab€e function for large 2 is ze-$Ei(z); this has been tabulated for 2 52 510. Finally, the range 10 5x50~ is covered by use of the inverse argument z-l. Twenty-one entries of ze-”Ei(x), corresponding to ~-~=.1(-.005)0, suf- fice to produce an interpolable table. 4. Interpolation The tables in this Handbook are not provided with differences or other aids to interpolation, be- cause it was felt that the space they require could be better employed by the tabulation of additional functions. Admittedly aids could have been given without consuming extra space by increasing the intervals of tabdation, but this would have con- flicted with the requirement that linear interpola- tion is accurate to four or five figures: For a plications in which linear interpolation is insdciently accurate it is ’intended that Lagrange’s formula or Aitken’s method of itera- tive linear interpolation3 be used. To help the user, there is a statement at the foot of most tables of the maximum error in a linear inte olate, Lagrange’s formula or Aitken’s method to inter- polate to full tabular accuracy. As an example, consider the following extract from Table 5.1. and the number of function values nee 3 ed in X xezEl(x) X xe*El (x) 7.5 .89268 7854 8.0 .89823 7113 7.6 .89384 6312 8. 1 .89927 7888 7. 7 .89497 9666 8. 2 .90029 7306 7. 8 . 89608 8737 8. 3 .90129 6073 7. 9 . 89717 4302 8.4 .go227 4695 “-”I The numbers in the square brackets mean that the maximum error in a linear interpolate is 3 X and that to interpolate to the full tabular accuracy five points must be used in Lagrange’s and Aitken’s methods. * A. 0 Aitkea On inte olation b iteration of roportional parts with- out the use of diherences, %c. Edingurgh Math. 8oc. 3,66-76 (l932j. Let us suppose that we wish to compute the value of ze”El(z) for 2=7.9527 from this table. We describe in turn the application of the methods of linear interpolation, Lagrange and Aitken, and of alternative methods based on differences and Taylor’s series. (1) Linear interpolation. The formula for this process is given by j P = (l--P)jO+Pfl where jo, jl are consecutive tabular values of the function, corresponding to arguments xo, zl, re- spectively; p is the given fraction of the argument interval and jp the required interpolate. instance, we have P = (-zo)/(21 -20) In the present fo=.89717 4302 fiz.89823 7113 p=.527 The most convenient way to evaluate the formula on a desk calculating machine is to set jo and ji in turn on the keyboard, and carry out the multi- plications by 1-p and p cumulatively; a partial check is then provided by the multiplier dial reading unity. We obtain f.ag.r= (1 -.527) (39717 4302) +.527(.89823 7113) = 39773 4403. Since it is known that there.is a possible error of 3 x in the linear formula, we round off this result to 39773. The maximum possible error in this answer is composed of the error committed