Newton’s Divided Difference Interpolation Formula
Interpolation is an estimation of a value within two known values in a sequence of values. Newton’s divided difference interpolation formula is an interpolation technique used when the interval difference is not same for all sequence of values. Suppose f(x0), f(x1), f(x2)………f(xn) be the (n+1) values of the function y=f(x) corresponding to the arguments x=x0, x1, x2…xn, where interval differences are not same Then the first divided difference is given by
The second divided difference is given by
and so on… Divided differences are symmetric with respect to the arguments i.e independent of the order of arguments. so, f[x0, x1]=f[x1, x0] f[x0, x1, x2]=f[x2, x1, x0]=f[x1, x2, x0] By using first divided difference, second divided difference as so on .A table is formed which is called the divided difference table. Divided difference table:
Advantages of NEWTON’S DIVIDED DIFFERENCE INTERPOLATION FORMULA
- These are useful for interpolation.
- Through difference table, we can find out the differences in higher order.
- Differences at each stage in each of the columns are easily measured by subtracting the previous value from its immediately succeeding value.
- The differences are found out successively between the two adjacent values of the y variable till the ultimate difference vanishes or become a constant.
NEWTON’S DIVIDED DIFFERENCE INTERPOLATION FORMULA
Input: Value at 7
Output: Value at 7 is 13.47
Below is the implementation of Newton’s divided difference interpolation method.
12 1 -0.1667 0.05 13 0.3333 0.1333 14 1 16 Value at 7 is 13.47
Time complexity: O(n2)
Auxiliary space: O(1)