Summary of "Lagrange Interpolation |Wierstrass Approximation Theorem|Linear Interpolation|Prof. (Dr.) SK Paikray"

Main Ideas and Concepts

Introduction to Interpolation:
Interpolation is a numerical method used to estimate the value of a function at non-tabulated points based on known values at tabulated points (nodes). It includes techniques like Lagrange Interpolation, Newton divided difference Interpolation, forward Interpolation, and backward Interpolation.
Definitions:
- Nodes: The tabulated points where the function values are known (e.g., \(x_0, x_1, ..., x_n\)).
- Interpolation: Estimating values within the range of known data points.
- Extrapolation: Estimating values outside the range of known data points.
Polynomial Interpolation:
A polynomial \(P(x)\) is an interpolating polynomial if it matches the function \(f(x)\) at the known nodes and their derivatives to a certain order. The polynomial approximation is preferred due to the Weierstrass Approximation Theorem, which states that every continuous function can be approximated by a polynomial.
Types of Interpolation:
- Equally Spaced Nodes: Use Newton forward and backward Interpolation.
- Unequally Spaced Nodes: Use Lagrange Interpolation and Newton divided difference Interpolation.
Lagrange Interpolation:
The Lagrange interpolating polynomial is constructed using the known function values and the corresponding nodes. The formula for the Lagrange polynomial for \(n\) nodes is:

P(x) = \sum_{i=0}^{n} L_i(x) f(x_i)

Where \(L_i(x)\) is the Lagrange basis polynomial defined as:

L_i(x) = \prod_{\substack{0 \leq j \leq n \\ j \neq i}} \frac{x - x_j}{x_i - x_j}
Linear Interpolation:
A specific case of Lagrange Interpolation using two points. The formula for linear Interpolation between two points \( (x_0, f(x_0)) \) and \( (x_1, f(x_1)) \) is:

P(x) = \frac{(x - x_1)f(x_0) + (x - x_0)f(x_1)}{x_0 - x_1}
Error and Approximation:
The error in Interpolation is within a tolerated limit, and the approximation is valid as long as the function is continuous over the interval.
Examples and Application:
Several examples were provided to illustrate how to compute the Lagrange polynomial and how to evaluate it at specific points.

Methodology for Lagrange Interpolation

Step-by-Step Instructions:
- Identify the known nodes and their corresponding function values.
- Construct the Lagrange basis polynomials \(L_i(x)\) for each node.
- Formulate the Lagrange polynomial using the basis polynomials and the function values.
- Simplify the polynomial to find \(P(x)\).
- Evaluate \(P(x)\) at the desired point(s).