1. Interpolation Theory Section 1
Content
Introduction - Why we need interpolation
Interpolating methods - Polynomial interpolation
Interpolation with Lagrange basis functions
2019

2. Why do we need interpolation…..?
To approximate some quantity using information adjacent to it
(constructing new details using known data points)
To make relationships between quantities using available information
(fitting curves which satisfy known data points, not the regression)
To make easier approximations to complicated functions
(lower degree polynomial approximations)
eg. Live births registered in Sri Lanka
(Census data:
Year
1992
1996
1998
2001
2005
2007
2010
Births
356842
340649
322672
358583
370731
386573
364565
Interpolation
What would be the number of births in 1993 and 2002 ?
What would be the number of births in 1990 and 2011 ?
Extrapolation

3. Interpolating methods
Nearest-neighbor interpolation
White dots – Weather stations
Weather data required on red line ?
(Ref: wikipedia)

4. Interpolating methods ctd…
Linear Polynomial Interpolation
Here we use linear polynomials to approximate values between two known data points. In interpolation theory, these known data points are called node points or simply the nodes. Sometimes the term node refers only to the coordinate of the independent variable of the point.
Dots – Node points
Line segments (in blue) – Linear interpolating polynomials
Line (in green) - Linear regression line
(Ref: wikipedia)

5. Eg. (linear interpolation) Number of births in Sri Lanka for the years 1992 and 1998 are and respectively. Approximate the number of births in 1996 using linear interpolation.
(1992, )
Let y be the no. of births in 1996.
(1996, ? )
(1998, )

6. Linear interpolation - General formula
Equation of the straight line AB leads
to the formula of linear polynomial interpolation corresponding to the node points A and B.

7. Remarks:
Not only the linear polynomial, higher degree polynomials are also used in interpolation.
Polynomials owe this popularity since their simple structure (continuity, differentiability, easier simplifications, etc.).

8. Quadratic Polynomial Interpolation
Interpolating methods ctd…
Quadratic Polynomial Interpolation

13. Applicability: Reduction in polynomial degree