Unit 4: Interpolation and Extrapolation
Binomial Expansion
The steps are as follows:
1. Find the number of known values (n) of dependent variable Y.
2. For the two missing values, first take
∆𝑛0 = 0 𝑡ℎ𝑎𝑡 𝑖𝑠 (𝑦 − 1) 𝑛 = 0
And ∆1𝑛 = 0
is written by increasing the suffix value by one in
∆𝑛0 = 0
3. The left hand side of the equation is expanded using the binomial expansion. Suppose 4 values
of Y are known, 4th order (leading) difference will be zero.
∆40 = 0
(𝑦 − 1) 4 = 𝑦4 − 4𝑦3 + 6𝑦2 − 4𝑦2 + 𝑦0 = 0
We get one missing value within the range of the data.
4. To get the other missing value, the same binomial expansion as in case of
∆40 = 0 Is written with suffixes of each ‘raised by 1.
𝑦5 − 4𝑦4 + 6𝑦3 − 4𝑦2 + 𝑦1 = 0
The successive numerical co-efficient of ‘y’ can be obtained using binomial coefficients table or y the
formula
𝑇ℎ𝑒 𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑒𝑛𝑡 𝑜𝑓 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 𝑦 × 𝑠𝑢𝑓𝑓𝑖𝑥 𝑜𝑟 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 𝑦
𝑆𝑒𝑞𝑢𝑒𝑛𝑡𝑖𝑎𝑙 𝑜𝑟𝑑𝑒𝑟 𝑜𝑓 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 𝑡𝑒𝑟𝑚
The numerical coefficient of the first term will be always 1.
Newton’s Advancing Difference Method
𝑦𝑥 = 𝑦0 + 𝑥∆10 +
𝑥=
𝑥(𝑥 − 1)
2!
2
0
+
𝑥(𝑥 − 1)(𝑥 − 2)
3!
3
0+
𝑥(𝑥 − 1)(𝑥 − 2)(𝑥 − 3)
4!
𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑋 𝑡𝑜 𝑏𝑒 𝑖𝑛𝑡𝑒𝑟𝑝𝑜𝑙𝑎𝑡𝑒𝑑 − 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑋 𝑎𝑡 𝑜𝑟𝑖𝑔𝑖𝑛
𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑡𝑤𝑜 𝑎𝑑𝑗𝑜𝑖𝑛𝑖𝑛𝑔 𝑣𝑎𝑙𝑢𝑒𝑠 𝑜𝑓 𝑋
𝑥=
𝑡ℎ𝑒 𝑦𝑒𝑎𝑟 𝑜𝑓 𝑖𝑛𝑡𝑒𝑟𝑝𝑜𝑙𝑎𝑡𝑖𝑜𝑛 − 𝑡ℎ𝑒 𝑦𝑒𝑎𝑟 𝑜𝑓 𝑜𝑟𝑖𝑔𝑖𝑛
𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑡𝑤𝑜 𝑎𝑑𝑗𝑜𝑖𝑛𝑖𝑛𝑔 𝑦𝑒𝑎𝑟𝑠