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 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 | SMALL | MEDIUM | LARGE Below is the OCR-scanned text from this page: INTRODUCTION XI by the last rounding, that is, .4403X10-6, plus 3 X and so certainly cannot exceed .8X lod6. In this example, the relevant formula is the 5-point one, given by ( 2 ) Lagrange's formula. f=A-2 (PV-a +A-,(p)f-1 +Ao(p)fo+A, (PYI + 442 (PI f a Tables of the coefficients Ak(p) are given in chapter 25 for the range p=O(.O1)1. We evaluate the formula for p=.52, .53 and .54 in turn. Again, in each evaluabion we accumulate the An@) in the multiplier register since their sum is unlty. We now have the following subtable. 2 X P E i (2) 7.952 .89772 9757 7.953 .89774 0379 -2 7.954 .89775 0999 10622 10620 n zn yn=xezEl(x) -6 8.0 . 89823 7113 1 7. 9 . 89717 4302 . 89773 44034 2 8. 1 . 89927 7888 . 89774 48264 .89' 3 7. 8 . 89608 8737 2 90220 The numbers in the third and fourth columns are the first and second differences of the values of xezE,(x) (see below); the smallness of the second difference provides a check on the three interpola- tions. The required value is now obtained by linear interpolation : fp=.3(.89772 9757)+.7(.89774 0379) =.89773 7192. In cases where the correct order of the Lagrange polynomial is not known, one of the preliminary interpolations may have to be performed with polynomials of two or more different orders as a check on their adequacy. ( 3 ) Aitken's method of iterative linear interpola- tion. The scheme for carrying out this process in the present example is as follows: YO, n YO, I, n YO. 1.2. n YO.l,2,3.n Z n - X . 0473 -. n m 4 8. 2 . 90029 7306 4 98773 5 7. 7 . 89497 9666 2 35221 Here If the quantities xn-x and xm-x are used as multipliers when forming the cross-product on a desk machine, their accumulation (xn---x) - (2,-x) in the multiplier register is the divisor to be used a t that stage. An extra decimal place is usually carried in the intermediate interpolates to safe- guard against accumulation of rounding errors. The order in which the tabular values are used is immaterial to some extent, but to achieve the maximum rate of convergence and at the same time minimize accumulation of rounding errors, we begin, as in this example, with the tabular argument nearest to the given argument, then take the nearest of the remaining tabular argu- ments, and so on. The number of tabular values required to achieve a given precision emerges naturally in the course of the iterations. Thus in the present example six values were used, even though it was known in advance that five would suffice. The extra row confirms the convergence and provides a valuable check. (4) Difference formulas. We use the central difference notation (chapter 25), . 773 71499 . 1473 1216 16 89773 71930 . 2473 2394 .89773 71938 -. 1527 2706 43 30 -. 2527 f o fl fa fa Sfill Sfaiz Sf512 S4f2 Sfvia z 4 f4 Here ~ f l a = f l - f o , ~f3iz=fa-f1, . . . 1 , SZf 1 = Sf312 - Sfl/Z =fa - 2fl +fo Saf3/z= Safz - Safi =f3 - 3fa + 3fi -fo S4f2= 6'f5/z - sy3/a=fr- 4f3 + 6fa- 4fi+f0 and so on. In the present example the relevant part of the difference table is as follows, the differences being written in units of the last decimal place of the function, as is customary. The smallness of the high differences provides a check on the function values X x e z E l (x) SZf S4f 7. 9 . 89717 4302 -2 2754 -34 8 . 0 .89823 7113 -2 2036 -39 Applying, for example, Everett's interpolation formula f p = (1 - p)fo+ Ez(P) aafo+ E4(p)G4fo+ . . . +pfl+Fz(p)Safl+F4(p)S%+ . * and taking the numerical values of the interpola- tion coefficients Ea(p), Ed ), F2(p) and F,(p) from Table 25.1, we find t t a t 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.