Introduction to Numerical Analysis I MATH/CMPSC 455 Interpolation

C HAPTER 3. I NTERPOLATION 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? I NTERPOLATION P OLYNOMIAL 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,

L AGRANGE I NTERPOLATION For a data set, the Lagrange form of the interpolation polynomial is

Example: x5-7 y1-23 Example: x y

H OW T O ? 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

N EWTON ’ S D IVIDED D IFFERENCES Definition: Example:

N EWTON F ORM OF THE I NTERPOLATION P OLYNOMIAL Nested Form: Definition:

Example:

x023 f(x)124