Question

or the following two data sets, construct a divided difference table.

What conclusions can you make about the data?

Would you use a low-order polynomial as an empirical model? If so, what order?

DATA SET 1.    

x	0	1	2	3	4	5	6	7
y	2	8	24	56	110	192	308	464

DATA SET 2.

x	0	1	2	3	4	5	6	7
y	1	4.5	20	90	403	1808	8103	36316

Answer 1

Here is your answer...
for data set 1 x o 1 2 8 2 3 4 5 6 7 24 56 0) 192 308 464 0 We can take Newton's divided interpollation formula f Ixorx17 = froid-fino) w-no second dluded difference given by '? f(x01 x 11 x) = f(x1, x2) - f 180 1x1) 22 - Yo table is as follows :- X fixil el xixit flxi.- , xjf2 f(x -Xi+3) faxi - Vi+4) f(xi-... vits) xli... Xi+6) If fri- xit7) 0 2 6 5 1 0 o 1 2 3 4 5 6 7 824 56 10 192 308 464 16 32 64 82 116 156 8 11 14 17 20 1 0 0 0 o o Oo oo PIX) = 2+61X-0) + 5(x-00%-1) + X-JC-utt 2

for data set 2. flxi, xit) foi xi+2) flxi. xits) fixit. aith) flxi. x1 +5) fwi... xitol flxi. - ditt) 0 I 3.5 6 7.083 6.083 4.288 2.464 1.234 1 2 3 4 5 6 7 4.5 20 90 403 1808 8103 36316 15.5 70 313 1.4056 3 6.29 E3 2.82 Ey 27.25 1213 546 2.49563 1.0984 31.417 141.3 633 2.83863 27.521 122.875 351.25 19.071 81.675 11.101 PIX) = ++ 3.5lx -0) +604-0) L-1+7.08333333(4-0) ( )( x- t) + 6 6333333? (to) (4-1)(x-2)(x-3) +. 4.2875 N·D)(x^) (X~2) (x-7)(x-4) 4 2.463808889(x-olga (x-2)(x-3)(x-4) (x+5) + 1.2338968 (XO) (XN) (x-2) 6 (7-4) (2.5) (4-0)

By Analysing the divided difference table for both the data we see that for data eet- the forward ualues are much larger and can't be calculated manually So we can chooce to use lower order polynomial by woking at divided difference when the divided difference become small, we can truncate our polynomial there.