Open main menu
This page needs to be proofread.
708
INTERPOLATION


We first find an approximate value for 0: then calculate Px, and find by (6) a more accurate value of 0; then, if necessary, recalculate P1, and thence 0, and so on.

II. CONSTRUCTION or TABLES BY SUBDIVISION OF INTERVALS 6. When the values of u have been tabulated for values of x proceeding by a difference h, it is often desirable to deduce a table in which the differences of x are h/11, where 11 is an integer. If 71. is even it may be advisable to form an intermediate table in which the intervals are ik. For this purpose we have 141=i(U0'l'U1) (7)»

where

U =u°' § 52u'|"i%;;§ 4u* ff§ 5§ ]5611'l" . . . = 14- il5'1¢- f'0l5'“ - 2“4(5°11'“~—)ll (3)-The following is an example; the data are the values of tan x to five places of decimals, the interval in x being 1°. The differences of odd order are omitted for convenience of printing. x 140 -|-HA140. QV1. 1 4;V0. 9V1 -|'1 9V0. M. 73;-6 -22 35 . . .

73°'7 - '39 1 1 —73°°S

-44 71 . . .

73°'9 » '33 54 —74°'0

3'43741 00 3'43741

74°°1l 3'51110 40 '24 53 33 54 -58 12 351052 7402 353479 30 '43 02 44 71 -37 73 3'53392 74°'3 355349 20 '°49 13 39 11 -38 29 3'55701 74 '4 353213 60 *36 89 22 35 "59 24 353159 74°-5 3-60588 oo 3-60588

The following are the values of the coefficients of 141, 52141, 64141, and 66141 in (9) for certain values of 11. For calculating the four terms due to 52141 in the case of n=5 it should be noticed that the third term is twice the first, the fourth is the mean of the first and the third, and the second is the mean of the third and the fourth. In table 3, and in the last column of table 2, the coefficients are Example 5 corrected in the last figure. ii 1-01 W i i J 14;mean of i TABLE 1.-n=5. x. u=tan x. 5-14. 6414. 5514. U. “dues 0fU x. 3 3 } - - ~ co. 14. co. 5214. co. 6414. co. 6614. + + + 7 ~ ~' 7~ 7

1 73° 3-27085 2339 100 5 3-26794 95 10 + - +° 537594 732 ' -2 -032 006556 '00135168=I/740 01>pf0X 74 3-48741 2808 132 23 3-48392 98 -4 -056 'OIO752 °00226304=I/442, , < 3-60588 74%° ~6 ~o64 'OI 1648 -00239616-1/417, , 75° 3~73205 3409 187 18 3-72783 17 866 1° ~8 4 W -048 4 -008064 -00160512=1/623, 3° 71 752 "' ' ' .

76° 4.01078 4197 260 SI 4-00559 22 4.16530 76? TABLE 2   n=IO Y

l,77° 433148 5245, 384 64 402501 07 In' eo. 6214. VA” co. Y6414. V co. 6614. If a new tabledis forma? from tlgese valul es, fthe intervals lbeing %°, + Nl) + " it will be foun that di erences eyond the ourth are neg igible. To subdivide h into smaller intervals than éh, various methods ggi?) gggggégg 88?;;?2gé may be used. One is to calculate the sets of quantities which in .3 ~o455 -00889525 -001887064 the new table will be the successive differences, corresponding to 140, 1 4 .0560 -01075200 -002263040 ul, . . and to find the intermediate terms by successive additions. 5 06, ,, ,), Ou7}§ 75 004441406 A better method is to use a formula due to ]. D. Everett. If we .6 ~o6io '0116480O '0O;396I6O write ¢=1 -0, Everett's formula is, in its most symmetrical form, ,7 O5§ 5 01044225 002115799 1 9 0 0- 0 0 0 6- 6- -8 ~ 8 - 8 6 ' ~ 6

u0=, , u1+< +035 1>, .u, +< .+2>< +055 or 21,41 + () W my V p g;8<§ My gg4;4=;§ ;>p ggtggggljjg . 9 .

+¢uD+(¢+1>;>!<¢ r)6, u0+(¢+2><¢+1>;<¢ 1>(¢-2)5.u0+ n TABLE 3 n=I2 For actual calculations a less symmetrical form may be used. i Mic0 M- CO- 5214- C0-5414 CO- 55%. Denoting, ' >~—~' 7' 7 7

+ - + -

“'*'>§ »“” ”11+'””“”'”+'>§ ~'” '”" ”~~1+~~-<1°> 111; °:1;9z11§ 332323592 '2°°58§ r1 - - -0, 0 2 - 0114 22

by eV, , we have, for interpolation between ug and 141, 3(1; 2g;2g;431§ 301530505 4/1 9 - 02032211

“H = ""+"A“°+@V1+1-HV” (1 'lf 5/ 12 0573381 1 7 010979453 002307557 the successive values 01 0 being 1/n, 2/n, .... (11-I)/n. For l 6/12 062500000 011718750 '002441400 interpolation between 141 and 142 we have, with the same succession 7/ 12 064139660 011736667 »o02419911 of values of 0, Sy; 0gIg§§ ~;95 310973937 '002§§§ 432 = ~ 9 1 0 4 oo 09399414 'OOI 275

ffl* “VNV” V'+'-OV' (I2)' 10/12 'O42438272 -007014103 'O01387048 The values of, ,, Vi in (12) are exactly the same as those of eV, 11/12 024402006 003855178 -000748981 in (11), but in the reverse order. The process is therefore that (i.) ' ' ' 7 we find the successive values of 110-i-0A14¢, , &c., i.e. we construct a table, with the required intervals of x, as if we had only to take first differences into account; (ii.) we construct, in a parallel column, a table giving the values of oV1, &c.; (iii.) we repeat these latter values, placing the set belonging to each interval h in the interval next following it, and writing the values in the reverse order; and (iv.) by adding horizontally we get the final values for the new table. As an example, take the values of tan x by intervals of 4° in x, as found above (Ex. 5). The first diagram below is a portion of this table, with the differences, and the second shows the calculation of the terms of (11) so as to get a table in which the intervals are 0-1 of 1°. The last column but one in the second diagram is introduced for convenience of calculation.

Example 6.

u tan x 21/L 5114. 5414.

III. GENERAL OBSERVATIONS

7. Derivation of Iformulae.-The advancing-difference formula (1) may be written, in the symbolical notation of finite differences, “9=(1'l“A)9%o=E91»¢o (I3);

and it is an extension of the theorem that if 11 is a positive integer un=u0+nAuo+7i7§ °QA2u0+. . (14), 1

the series being continued until the terms vanish. The formula (14) is identically true: the formula (13) or (1) is only formally true, but its applicability to concrete cases is due to the fact that the series in (1), when taken for a definite number of terms, differs from the true value of 14,9 by a “ remainder” which in most cases is very small when this definite number of terms is properly chosen. Everett's formula (9), and the central-difference formula obtained by substituting from (4) in (2), are modifications of a standard 11147

11847

formula

Lg u., +os14;+, a214+(”+'W'9 ' .sw

745 36088 o

x. = . 5u. 5 .

+ + + +

62

74°~0 3'43741 700 8

70

°- - 5 77 9

~ A 3 3 79

4 0(0-1) -)

= -. 2. ° " 3: ' 1+

(6+1)6(0-1)(0-2)

4.

12617 K, —54uo+. . . (151