New Online Book! Handbook of Mathematical Functions (AMS55) Conversion & Calculation Home >> Reference Information 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: INTRODUCTION XIII n yn=xe~El(x) xn 4 . n zO.1.n zO.1.2.n ~ , 1 , 2 . 3 . n Yn-Y n .1)nn29 7306 8.2 nnnzn 7mfi . - - - - - . - - i .89927 7888 S. i 8.17083 5712 -. 00072 2112 3 . 89823 7113 8. 0 8. 17113 8043 2 5948 8. 17062 2244 -. 00176 2887 5 . 89717. 4302 7. 9 8. 17144 0382 2 8142 23 1 265 -.00282 5658 2 . 90129 6033 8. 3 8. 17023 1505 8. 17061 9521 .00129 6033 4 . 90227 4695 8. 4 8. 16992 9437 1 7335 415 8. 17062 2318 . 00227 4695 The estimate of the maximum error in this result is the same as in the subtabulation method. An indication of the error is also provided by the discrepancy in the highest .interpolates, in this case . z \$ . ~ , and G.I .2 ,3 ,5- 6. Bivariate Interpolation Bivariate interpolation is generally most simply performed as a sequence of univariate interpola- tions. We carry out the interpolation in one direction, by one of the methods already described, for several tabular values of the second argument in the neighborhood of its given value. The interpolates are differenced as a check, and interpolation is then carried out in the second direction. An alternative procedure in the case of functions of a complex variable is to use the Taylor’s series expansion, provided that successive derivatives of the function can be computed without much difficulty. 7. Generation of Functions from Recurrence Relations Many of the special mathematical functions which depend on a parameter, called their index, order or degree, satisfy a linear difference equa- tion (or recurrence relation) with respect to this parameter. Examples are furnished by the Le- gendre function P,(x), the Bessel function J,(x) and the exponential integral E,(x), for which we have the respective recurrence relations (n+1)Pn+1-- (2n+l)xPn+nPn-l=0 n En+] + x En = e-a. Particularly for automatic work, recurrence re- lations provide an important and powerful com- puting tool. If the values of qn(x) or J,(x) are known for two consecutive values of n, or E,(x) is known for one value of n, then the function may be computed for other values of n by successive applications of the relation. Since generation is carried out perforce with rounded values, it is vital to know how errors may be propagated in the recurrence process. If the errors do not grow relative to the size of the wanted function, the process is said to be stable. If, however, the relative errors grow and will eventually over- whelm the wanted function, the process is unstable. It is important to realize that stability may depend on (i) the particular solution of the differ- ence equation being computed; (ii) the values of x or other parameters in the difference equation; (iii) the direction in which the recurrence is being applied. Examples are as follows. Stability-increasing n Pnb), pF(z) Qnb), Q F b ) (\$d Fn(q, p) (Coulomb wave function) Illustrations of the generation of functions from their recurrence relations are given in the pertinent chapters. It is also shown that even in cases where the recurrence process is unstable, it may still be used when the starting values are known to sufficient accuracy. Mention must also be made here of a refinement, due to J. C. P. Miller, which enables a recurrence process which is stable for decreasing n to be applied without any knowledge of starting values for large n. Miller’s algorithm, which is well- suited to automatic work, is described in 19.28, Example 1. The page scan image above, and the text in the text box above, are contributions of the National Institute of Standards and Technology that are not subject to copyright in the United States. ©2000 ConvertIt.com, Inc. All rights reserved. Terms of Use.