LECTURE 3: Polynomial interpolation and numerical differentiation

Using the recursive formula, we can now construct a divided-differences table for a function
f
x f[] f[, ] f[, , ] f[, , , ]
x 0 f[x 0 ]
f[x 0 ,x 1 ]
x 1 f[x 1 ] f[x 0 ,x 1 ,x 2 ]
f[x 1 ,x 2 ] f[x 0 ,x 1 ,x 2 ,x 3 ]
x 2 f[x 2 ] f[x 1 ,x 2 ,x 3 ]
f[x 2 ,x 3 ]
x 3 f[x 3 ]
Example.
Construct a divided differences table and write out the Newton form of the interpolating
polynomial for the following table of values:
x i 1 3/2 0 3
f(x i ) 3 13/4 3 5/3
Step 1.
The second column of entries is given by f[x i ]= f(x i ). Thus
x f[] f[, ] f[, , ] f[, , , ]
1 3
f[x 0 ,x 1 ]
3/2 13/4 f[x 0 ,x 1 ,x 2 ]
f[x 1 ,x 2 ] f[x 0 ,x 1 ,x 2 ,x 3 ]
0 3 f[x 1 ,x 2 ,x 3 ]
f[x 2 ,x 3 ]
2 5/3
Step 2.
The third column of entries is obtained using the values in the second column:
f[x i ,x i+1 ]= f[x i+1]− f[x i ]
x i+1 − x i
For example,
f[x 0 ,x 1 ]= f[x 1]− f[x 0 ]
x 1 − x 0
= 13/4−3
3/2−1 = 1 2