Week 3 Flashcards
What is double zero case
First 2 terms of Taylor series exp are zero
Because f(x) = 0 and fâ(x) = 0
What is pip
Basic building block of Lagrange interpolation
They are fixed by: each polynomials of deg <=n
pi(xi) = 1
And pi(xj) = 0 for all j!=i
Lagrange interpolation formula
Where p_i(x) is pip
Linear ansatz in polynomial fit
Where f_i(x) is some predetermined basis of m+1 linearly independent functions
Sum of squares of deviations for polynomial fit
Minimising S , polynomial fit
Find a_j for which S is minimal by requiring
Form system of linear equations for polynomial fit
For polynomial fit: if f_j(x) = x^j
What is polynomial approximation
Given a set of n points
Find the polynomial that best describes this data
There is always a unique polynomial of degree n which goes through all these points (assuming x_I are pairwise distinct)
Limitations of polynomial interpolation
Works well in middle of interpolation interval but often produces unwanted oscillations at ends
What are chebyshev points and how to use?
Effectively projecting cos function (from [-1,1]) taken at regular angles, onto x axis (rescaled to [a,b])
Therefore makes more densely populated at ends of interval
Formula for chebyshev points
Typical basis of functions for fitting data
fi(x) = xi
Idea behind fitting data
Taking a linear ansatz in some basis of linearly independent functions
We are trying to minimise the coefficients of this basis in the ansatz
What if m = n for fitting data
This means the basis of polynomials has as many base functions as there are points. Therefore produce exact interpolation
