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Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (AMS55)
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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
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)
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
-. n m
4 8. 2 . 90029 7306 4 98773
5 7. 7 . 89497 9666 2 35221
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
z 4 f4
~ 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
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
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
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