1. Interpolation Practice-6 Additional chapters of mathematics
Dmitriy Sergeevich Nikitin
Assistant
Tomsk Polytechnic University

2. Vandermonde polynomial
This Vandermonde polynomial is the most straightforward construction of the interpolating polynomial Pn(x). We simply write
Pn(x) = c0xn + c1xn-1 + … + cn.
Then we can immediately form n + 1 linear equations for the n + 1 unknown coefficients c0, c1, …, cn using the n + 1 known points:
y0 = c0x0n + c1x0n-1 + … cn-1x0 + cn.
y1 = c0x1n + c1x1n-1 + … cn-1x1 + cn. …
yn = c0xnn + c1xnn-1 + … cn-1xn + cn.

3. Task 1 Given the following 4 data points, xi 0 1 3 5 yi 1 2 6 7
Find a polynomial in Vandermonde form to interpolate these data. 3

4. Lagrange Interpolation Polynomial
Given (x0, f0) , (x1, f1), …, (xn, fn) with arbitrarily spaced xj. Lagrange had the idea of multiplying each fj by a polynomial that is 1 at xj and 0 at the other n nodes and then taking the sum of these n+1 polynomials. This gives the unique interpolation polynomial of degree n or less.
For general n we obtain
𝐟 𝐱 ≈ 𝐩 𝐧 𝐱 = 𝐤=𝟎 𝐧 𝐋 𝐤 (𝐱) 𝐟 𝐤 = 𝐤=𝟎 𝐧 𝐥 𝐤 (𝐱) 𝐥 𝐤 ( 𝐱 𝐤 ) 𝐟 𝐤
where 𝐋 𝐤 𝐱 𝐤 =𝟏 and 𝐋 𝐤 is 0 at the other nodes, and the 𝐋 𝐤 are independent of the function f to be interpolated.

5. Lagrange Interpolation Polynomial

6. Task 2 Given the following 4 data points, xi 0 1 3 5 yi 1 2 6 7
Find a polynomial in Lagrange form to interpolate these data. 6

7. Newton Interpolation Polynomial
The Newton polynomial is somewhat more clever than the Vandermonde polynomial because it results in a system of linear equations that is lower triangular, and therefore can be solved by forward substitution. The interpolating polynomial is written in the form
which is clearly a polynomial of degree n. The n +1 unknown coefficients given by the c’s can be found by substituting the points (xi, yi) for i = 0, , n:

8. Newton Interpolation Polynomial

9. Task 3 Given the following 4 data points, xi 0 1 3 5 yi 1 2 6 7
Find a polynomial in Newton form to interpolate these data. 9

10. Divided Differences Scheme

11. Divided Differences Scheme

12. Task 4 Given the following 4 data points, xi 0 1 3 5 yi 1 2 6 7
Find a polynomial in Newton form to interpolate these data
using Divided Differences Scheme. 12

13. End of Lecture-4