Interpolation Practice-6 Additional chapters of mathematics Dmitriy Sergeevich Nikitin Assistant Tomsk Polytechnic University email: NikitinDmSr@yandex.ru

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.

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

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.

Lagrange Interpolation Polynomial

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

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:

Newton Interpolation Polynomial

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

Divided Differences Scheme

Divided Differences Scheme

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

End of Lecture-4