2. Interpolation
Let the function y=f(x) take the values y0, y1,y2,…,yn corresponding
to the values x0,x1,x2,…,xn of x. The process of finding the value of y
corresponding to any value of x=xi between x0 and xn is called
interpolation.
3. Newton’s Forward Interpolation
Let the function y=f(x) take the values y0,y1,y2,…,yn corresponding to the
values x0,x1,x2,…,xn of x. Suppose it is required to evaluate f(x) for x=x0 + rh ,
where r is any real number.
Formula :
Yn (x) = yo + 𝑟∆yo +
𝑟 𝑟−1
2!
∆2yo +
𝑟 𝑟−1 (𝑟−2)
3!
∆3yo + ⋯
where r =
x−x0
ℎ
5. Example: If f(x) is known at the following data points then
find f(0.5) using Newton's forward difference formula.
xi 0 1 2 3 4
fi 1 7 23 55 109
x fi Dfi D2fi D3fi D4fi
0 1
6
1 7 10
16 6
2 23 16 0
32 6
3 55 22
54
4 109
7. Newton’s Backword Interpolation
Let the function y=f(x) take the values y0,y1,y2,…,yn
corresponding to the values x0,x1,x2,…,xn of x. Suppose it is
required to evaluate f(x) for x=x0 + r*h , where r is any real
number.
Formula :
yn(x) = yn +rસyn +
𝑟 𝑟+1
2!
સ2
yn +
𝑟 𝑟+1 (𝑟+2)
3!
સ3
yn + ⋯
where r =
x−xn
ℎ
8.
9. Example:
Consider Following Tabular Values Determine y (300).
X 50 100 150 200 250
y 618 724 805 906 1032
x y સy સ2y સ3y સ4y
50 618
106
100 724 -25
81 45
150 805 20 -40
101 5
200 906 25
126
250 1032