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 POLYNOMIALMathematical 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 -7y 1 -23

Example:

xy

HOW TO?Method 1: Solving a linear system

Determine coefficients

Method 2: Lagrange Form of Interpolation

Determine basis

Method 3: Newton Form of InterpolationUse 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 3f(x) 1 2 4