Approximating Functions

Question 1

When does the function stop?

Answer 1

The function stops after the number of terms determined by requirements
The number of data points you are dealing with

Question 2

How do you calculate the difference?

Answer 2

Bigger Term - Smaller Term (Not based on size of value)

Question 3

What is the relationship between the degree of the polynomial and set of data points?

Answer 3

Polynomial of ‘(n-1)’ degrees has ‘n’ datapoints

Question 4

What is ‘h’ in the formula?

Answer 4

‘H’ is the difference between each ‘x’ value of the data points
The method requires constant difference between ‘x’ values

Question 5

What is the formula for Newton’s Forward Difference Interpolation Formula?

Answer 5

F(x) = f(x₀) + ((x - x₀) x Df(x₀)) /// h) + ((x - x₀)(x - x₁) x D²f(x₀)) /// h) + …

Question 6

What is the pattern for Lagrange’s Interpolating Formula?

Answer 6

The ‘y’ value of the current coordinate multiplies the fraction
The denominator has current ‘x’ coordinate subtracted by all other ‘x’ values and multiplied together
The numerator has the constant ‘x’ subtracted by every ‘x’ value other than the current coordinate

Question 7

What is the relationship between the Lagrange polynomial and the number of points required?

Answer 7

P_n requires (n+1) points

Question 8

How does truncating Newton’s Interpolating Polynomial affect the formula?

Answer 8

All subscripts of ‘x’ are added to by the level of truncation