Gaussian quadrature is a method used for numerical integration that involves approximating the definite integral of a function with a weighted sum of function e...
Gaussian quadrature is a method used for numerical integration that involves approximating the definite integral of a function with a weighted sum of function evaluations at specific points. It is based on the idea of dividing the integration interval into smaller subintervals and then approximating the area of each subinterval by multiplying it by the function value at the endpoint of the subinterval.
In Gaussian quadrature, the function is assumed to be continuous and differentiable on the interval of integration. The function values at the endpoints of each subinterval are weighted according to their distance from the center of the subinterval. This weighting scheme ensures that more weight is placed on subintervals closer to the center of the interval.
The Gaussian quadrature method can be used to approximate the definite integral of a function by summing the weighted function values. The weights can be chosen to be constant, linear, or even more complex functions. The weights that are used in Gaussian quadrature are called absciss weights.
Steps involved in Gaussian quadrature:
Divide the integration interval [a, b] into a finite number of subintervals of equal width (h). The width of the subintervals can be chosen based on the desired accuracy or the function's smoothness.
Choose the absciss weights for each subinterval. The absciss weights are typically chosen to be the same for all subintervals, and they are typically determined based on the subinterval's length.
Evaluate the function at the endpoints of each subinterval. This gives us a set of function values that we can use to approximate the area of the subinterval.
Calculate the weighted sum of the function values from all subintervals. The weights allow us to adjust the contribution of different subintervals according to their relative positions.
The approximate integral is obtained by summing the weighted function values. This gives us a good approximation of the definite integral of the function on the original interval.
Gaussian quadrature is a powerful and versatile numerical integration method that can be used to approximate the definite integral of a function. It is particularly well-suited for problems where the function is continuous and differentiable and where the integration interval is not too wide