Newton-Cotes Formulas

Definition

Newton-Cotes formulas are numerical integration techniques used to approximate the definite integral of a function. These formulas rely on evaluating the function at equally spaced points and using polynomial interpolation to estimate the area under the curve. They form a fundamental part of quadrature rules, including methods that enhance accuracy, like Gaussian quadrature.

5 Must Know Facts For Your Next Test

1. Newton-Cotes formulas can be classified into open and closed types, with closed formulas using endpoints of intervals while open formulas do not.
2. The accuracy of Newton-Cotes formulas improves with an increasing number of evaluation points, but can also lead to increased error due to polynomial oscillation.
3. Closed Newton-Cotes formulas include the Trapezoidal Rule and Simpson's Rule, each offering different levels of precision based on the polynomial degree used.
4. The error in Newton-Cotes methods depends on the smoothness of the integrand and is typically expressed in terms of the derivatives of the function.
5. Newton-Cotes formulas are often compared to Gaussian quadrature methods, where the latter usually provides better accuracy for fewer function evaluations.

Review Questions

• How do Newton-Cotes formulas utilize polynomial interpolation in numerical integration?
  • Newton-Cotes formulas leverage polynomial interpolation by approximating the function values at equally spaced points to create a polynomial that fits these points. The definite integral is then estimated as the area under this interpolated polynomial. This method allows for varying degrees of accuracy depending on how many points are used in interpolation, effectively transforming discrete data into a continuous representation for integration.
• Compare and contrast closed and open Newton-Cotes formulas in terms of their application and accuracy.
  • Closed Newton-Cotes formulas use the endpoints of an interval as evaluation points, like the Trapezoidal Rule and Simpson's Rule, while open Newton-Cotes formulas use only interior points. Closed methods generally provide more accurate results as they consider boundary values, but open methods can be advantageous in situations where endpoint values are not known or reliable. The choice between them often depends on the specific properties of the function being integrated and desired precision.
• Evaluate how the choice between Newton-Cotes and Gaussian quadrature methods impacts numerical integration results.
  • Choosing between Newton-Cotes and Gaussian quadrature methods significantly affects integration results due to their differing approaches to accuracy and efficiency. While Newton-Cotes relies on equally spaced points and polynomial fitting, Gaussian quadrature selects specific points (Gauss points) based on optimizing precision, often achieving better results with fewer evaluations. This makes Gaussian quadrature particularly effective for functions with rapid changes or high degrees of smoothness, while Newton-Cotes may be easier to implement but can struggle with oscillatory functions, resulting in higher errors.

Related terms

Trapezoidal Rule:

A numerical method for estimating the integral of a function by dividing the area under the curve into trapezoids, providing a linear approximation.

Simpson's Rule:

An integration technique that approximates the area under a curve by using quadratic polynomials, applying it over intervals divided into an even number of subintervals.

Polynomial Interpolation: The process of approximating a function using a polynomial that passes through a given set of points, forming the basis for Newton-Cotes formulas.