Lecture 5 - Interpolation

Question 1

Interpolation

Answer 1

Estimation of the values of a quantity between the individual data points

Question 2

NAMES

Answer 2

f(x)- original function
F(x) - interpolating function
(xi,yi) - input pairs, f(xi)=yi
xi-node

Question 3

Polynomial interpolation

Answer 3

Given a set of n+1 data points where no to xi are the same, we can finde UNIQUE n-degree polynomial which goes through all n+1 points

Question 4

Calculating coefficients of polynomial intepolation

Answer 4

We can write a set of equations of n-degree polynomial for each node.
Create a matrices from this Ma=y and calculate a= y*inv(M)

Question 5

Vandermonde matrix

Answer 5

Matrix M which shows up while calculating coefficients. Its a matrix that consists n powers of every yi starting from the n power end ending with ones (zero powers)

Question 6

Polynomial interpolation Drawbacks

Answer 6

M is composed with very large and very small numbers. It makes this method prone to round-off errors.
Also this method has poor conditioning - functions start to looks similar as n increases

Question 7

Lagrange interpolation

Answer 7

The obtained function F(x) is the same as with polynomials but this method is stable and easy to implement. We don’t have to solve a system of linear equations -> less prone to round-off errors

Question 8

Lagrange interpolation equation

Answer 8

PHOTO

Question 9

Runge’s phenomenon

Answer 9

Problem of oscillations on the edges which occurs when using polynomial interpolation with polynomials of high degree over a set of interpolation point with equivalent spaces between them