Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (AMS55)
Purchase the electronic edition of this book in Adobe PDF format!
FIRST | PREVIOUS | NEXT | LAST | CONTENTS | PAGE | ABOUT | MEDIUM | LARGE

Below is the OCR-scanned text from this page:
XI1 INTRODUCTION 109f.5z,-.473(89717 4302) +.061196(2 2754) -.012(34) + .527(89823 7113) + .063439(2 2036) - .012(39) =89773 7193. We may notice in passing that Everett’s formula shows that the error in a linear interpolate is approximately can be used. We first compute as many of the derivatives f‘”’(x0) as are significant, and then evaluate the series for the given value of x. An advisable check on the computed values of the derivatives is to reproduce the adjacent tabular values by evaluating the series for X = X - ~ and xl. Since the maximum value of IEz(p)+Fz(p)l in the range O<p<l is )S, the maximum error in a linear interpolate is approximately (5) Taylor’s series. In cases where the succes- sive derivatives of the tabulated function can be computed fairly easily, Taylor’s expansion With x0=7.9 and x-x0=.0527 our computations are as follows; an extra decimal has been retained in the values of the terms in the series to safeguard against accumulation of rounding errors. k f‘k’(zo)/k! (Z - ~ o ) ‘f“) ( ~ 0 ) / k ! 0 39717 4302 .89717 4302 1 .01074 0669 .00056 6033 3 3 .00012 1987 .OOQOO 0017 9 2 -.00113 7621 -.OOOOO 3159 5 With linear interpolation there is no difference in principle between direct and inverse interpola- tion. In cases where the linear formula provides an insUaciently accurate answer, two methods are available. We may interpolate directly, for example, by Lagrange’s formula to prepare a new table at a fine interval in the neighborhood of the approximate value, and then apply accurate inverse linear interpolation to the subtabulated values. Alternatively, we may use Aitken’s method or even possibly the Taylor’s series method, with the roles of function and argument interchanged. It is important to realize that the accuracy of an inverse interpolate may be very different from that of a direct interpolate. This is particularly true in regions where the function is slowly varying, for example, near a maximum or mini- mum. The maximum precision attainable in an inverse interpolate can be estimated with the aid of the formula df dx Ax Afl - in which Af is the maximum possible error in the function values. Given xezEl(x) = .9, find x from the table on page X. The formula Example. (i) Inverse linear interpolation. for p is p=~?J-fo)lCf1-f0). In the present example, we have 72 2112 .708357 .9- .89927 7888 ’=.go029 7306- 39927 7888=1-= 239773 7194 5. Inverse Interpolation The desired x is therefore Z = Z O + P ( Z ~ - Z O ) =8.1+ .708357(.1) ~ 8 . 1 7 0 8 3 57 To estimate the possible errur in this answer, we recall that the maximum error of direct linear interpolation in this table is Af=3X10-6. An approximate value for dfldx is the ratio of the first difference to the argument interval (chapter 25), in this case .010. Hence the maximum error in x is approximately 3X 10-6/(.010), that is, .0003. (ii) Sub tabulation method. To improve the approximate value of x just obtained, we inter- polate directly for p=.70, .71 and .72 with the aid of Lagrange’s 5-point formula, 2 z e z E l (2) 6 62 8. 170 . 89999 3683 8. 171 . 90000 3834 - 2 1 0151 1 0149 8.172 .go001 3983 Inverse linear interpolation in the new table gives .9- 39999 3683= .6223 ’= .00001 0151 Hence x=8.17062 23. An estimate of the maximum errur in this result is _- df 1 X lo-, dx .010 Afl --z -- (iii) Aitken’s method. This is carried out in the same manner as in direct interpolation.