Introduction to Numerical Analysis I
MATH/CMPSC 455
Interpolation
CHAPTER 3. INTERPOLATION
A function is said to interpolate a set of data points if it passes through those points
Definition: The function interpolates the data sets if
Note that is required to be a function!
Restriction on the data set:
Main theorem of Polynomial interpolation:If are distinct, there is a unique polynomial of degree such that
How to find this polynomial?
INTERPOLATION POLYNOMIAL
Mathematical Problem: (Interpolate points)Given n+1 points , we seek a polynomial of degree such that Mathematical Problem: (Interpolate a function)A function , assuming its values are known or computable at a set of n+1 points. we seek a polynomial of degree such that ,
LAGRANGE INTERPOLATION
For a data set , the Lagrange form of the interpolation polynomial is
Example:
x 5 -7
y 1 -23
Example:
x
y
HOW TO?Method 1: Solving a linear system
Determine coefficients
Method 2: Lagrange Form of Interpolation
Determine basis
Method 3: Newton Form of Interpolation
Use another basis which is easy to get, and has similar property as the basis for Lagrange form, and determine the coefficient easily.
forms a basis of
Newton form of interpolation polynomial:
Determine the coefficients
NEWTON’S DIVIDED DIFFERENCES
Definition:
Example:
NEWTON FORM OF THE INTERPOLATION POLYNOMIAL
Nested Form:
Definition:
Example:
Example:
x 0 2 3
f(x) 1 2 4
