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

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: www.statistics.gov.lk) 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

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

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)

Eg. (linear interpolation) Number of births in Sri Lanka for the years 1992 and 1998 are 356842 and 322672 respectively. Approximate the number of births in 1996 using linear interpolation. (1992, 356842) Let y be the no. of births in 1996. (1996, ? ) (1998, 322672)

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.

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.).

Quadratic Polynomial Interpolation Interpolating methods ctd… Quadratic Polynomial Interpolation

Applicability: Reduction in polynomial degree