Lagrange Interpolation Polynomial: Algorithm and Applications

Lagrange interpolation polynomial is a fundamental method in numerical analysis used to approximate a function by a polynomial that passes through a given set of points. This technique is widely employed in various fields such as engineering, physics, and computer science for interpolation and polynomial fitting. In this article, we will explore the algorithm behind Lagrange interpolation, its implementation, and applications.

Steps to Construct Lagrange Interpolation Polynomial

The construction of a Lagrange interpolation polynomial involves several steps:

Select Points

The first step is to select n 1 data points, (x0, y0), (x1, y1), ..., (xn, yn), where yi f(xi).

Lagrange Basis Polynomials

For each data point i (where i 0, 1, ..., n), the Lagrange basis polynomial Li(x) is defined as:

Li(x) ∑00≤j≤nj≠ix * xj, xi * xj- xixi - xj

This polynomial Li(x) is constructed such that:

Li(xi) 1 Li(xj) 0 for all j ≠ i

Construct the Interpolating Polynomial

The Lagrange interpolation polynomial P(x) is given by the weighted sum of the basis polynomials:

P(x) ∑i0n yi Li(x)

This polynomial will pass through all the points (xi, yi).

Evaluate the Polynomial

To find the value of the polynomial at a specific point x, substitute x into the polynomial P(x).

Example

Suppose you have three points: (1, 2), (2, 3), (3, 5).

Define Points

x0 1, y0 2 x1 2, y1 3 x2 3, y2 5

Calculate Lagrange Basis Polynomials

L0(x) (x - 2)/(1 - 2) (x - 2)/(-1) -(x - 2) L1(x) (x - 1)/(2 - 1) x - 1 L2(x) (x - 1)/(3 - 1) (x - 1)/2

Construct P(x)

P(x) 2L0(x) 3L1(x) 5L2(x)

Evaluate P(x)

To find P(1.5), substitute x 1.5 into the polynomial.

Complexity and Practical Considerations

The computational complexity of evaluating the Lagrange polynomial can be high, especially for large n, as it involves n multiplications for each evaluation. In practice, using methods like Newtons divided difference or Chebyshev nodes can lead to more efficient interpolations.

This method is particularly useful for small datasets or when exact polynomial interpolation is required. However, be cautious of Runges phenomenon when using high-degree polynomials over evenly spaced points.

Keywords

Lagrange Interpolation Polynomial Approximation Interpolation Technique