Graph of Function and Lagrange Interpolations
Original Function: \( f(x) = \frac{1}{1 + 25x^2} \)
What is Lagrange Interpolation?
Lagrange interpolation constructs a polynomial that passes through a given set of points. It approximates functions with known values at specific points.
Formula:
\[ P(x) = \sum_{i=1}^n y_i \prod_{\substack{j=1 \\ j \neq i}}^n \frac{x - x_j}{x_i - x_j} \]
\( x_i, y_i \) are the known data points, and \( P(x) \) is the interpolating polynomial.
Types of Point Parameterizations
Uniform Points: Equally spaced points in the given range.
Chebyshev Points: Nodes that minimize interpolation error, defined as:
\[ x_i = \frac{a + b}{2} + \frac{b - a}{2} \cos\left(\frac{(2i + 1) \pi}{2(n + 1)}\right), \, i \in [0, n] \]