Lagrange interpolation

Definition

Lagrange interpolation is a polynomial interpolation method used to construct a polynomial that passes through a given set of data points. It enables the estimation of values between known data points and is particularly useful when you need to find a polynomial that fits several data points without explicitly calculating the coefficients of the polynomial. This method is vital in numerical analysis for applications like curve fitting and numerical integration.

5 Must Know Facts For Your Next Test

1. Lagrange interpolation can be expressed using the formula: $$P(x) = \sum_{i=0}^{n} y_i L_i(x)$$, where $$L_i(x)$$ are the Lagrange basis polynomials.
2. The Lagrange basis polynomial for a point $$x_i$$ is given by: $$L_i(x) = \prod_{j \neq i} \frac{x - x_j}{x_i - x_j}$$, ensuring that $$L_i(x_i) = 1$$ and $$L_i(x_j) = 0$$ for all $$j \neq i$$.
3. This interpolation method can lead to Runge's phenomenon, where oscillations occur between the endpoints if too many points are used for interpolation.
4. Lagrange interpolation is computationally more efficient for smaller sets of data but can become inefficient for larger datasets due to the need to recompute all basis polynomials.
5. While Lagrange interpolation provides exact results for the interpolated points, it does not guarantee good behavior outside the range of those points unless the underlying function is well-behaved.

Review Questions

• How does Lagrange interpolation ensure that the constructed polynomial passes through each given data point?
  • Lagrange interpolation constructs a polynomial by combining Lagrange basis polynomials, which are specifically designed to equal one at their respective data points and zero at all others. The formula involves summing these basis polynomials multiplied by their corresponding y-values. This ensures that when evaluated at each data point, the overall polynomial yields the exact y-value, effectively fitting all provided data points.
• Compare and contrast Lagrange interpolation with Newton's divided differences in terms of efficiency and ease of use.
  • Lagrange interpolation is straightforward in that it directly computes the interpolating polynomial using all data points without needing to calculate intermediate coefficients. However, it can become inefficient as more points are added because it requires recalculating all basis polynomials. In contrast, Newton's divided differences offer a more efficient approach for adding new points since it allows for building the polynomial incrementally without starting from scratch. This makes Newton's method preferable for larger datasets or iterative applications.
• Evaluate the implications of Runge's phenomenon in Lagrange interpolation and how it affects practical applications in numerical analysis.
  • Runge's phenomenon illustrates the challenges faced in Lagrange interpolation when using a large number of equidistant points, leading to significant oscillations near the edges of the interval. This behavior can result in poor approximations even though the interpolated values match exactly at specified points. In practical applications like numerical integration or curve fitting, it emphasizes the need for cautious selection of interpolation methods and point distribution to avoid inaccurate results and ensure smooth approximations.

Related terms

Polynomial Interpolation:

A technique used to find a polynomial that passes through a given set of points, providing an exact fit for those values.

Newton's Divided Differences:

A method of constructing an interpolating polynomial using divided differences, offering a systematic way to derive coefficients iteratively.

Basis Polynomial: The individual polynomials in Lagrange interpolation that are combined to form the overall interpolating polynomial, each corresponding to one of the data points.