How to Calculate Weights for Gauss-Kronrod Quadrature Using Nodes and Degree?

• I
• Vick
In summary, the Gauss-Kronrod quadrature uses the zeros of the Legendre Polynomials of degree n and the zeros of the Stieltjes polynomials of degree n+1 as nodes for the quadrature. This means that for a Gauss polynomial of degree 7, you will need the Stieltjes polynomial of degree 8, resulting in a total of 15 nodes according to the quadrature rule (2n+1). However, the process of calculating the weights for the Gauss-Kronrod quadrature given the nodes and degree is not as readily available as it is for the Gauss-Legendre quadrature. Further research, such as using Google, may be necessary to find the necessary information.
Vick
TL;DR Summary
How to compute the weights once you have the nodes?
The Gauss-Kronrod quadrature uses the zeros of the Legendre Polynomials of degree n and the zeros of the Stieltjes polynomials of degree n+1. These zeros are the nodes for the quadrature. For example using the Gauss polynomial of degree 7, you will need the Stieltjes of degree 8 and both makes up to 15 nodes in accordance to the quadrature rule (2n+1).

But how would you calculate the weights given the nodes and the degree?

mathman said:
I tried but nothing comes in handy.

For example the Gauss-Legendre quadrature weights is easily understood in this link: Legendre weights
So I'm looking for something like this for the Gauss-Kronrod!

Related to How to Calculate Weights for Gauss-Kronrod Quadrature Using Nodes and Degree?

Gauss-Kronrod quadrature is a numerical integration method used to approximate the definite integral of a function over a given interval. It is a combination of the Gauss-Legendre quadrature and the Kronrod extension, which uses additional points to improve the accuracy of the approximation.

The method works by dividing the interval into subintervals and approximating the integral over each subinterval using a combination of Gauss-Legendre and Kronrod points. The final approximation is then calculated by combining the results from each subinterval.

Gauss-Kronrod quadrature is a highly accurate method for numerical integration, especially for functions with oscillatory or rapidly changing behavior. It also allows for the calculation of the error in the approximation, which can be useful for evaluating the accuracy of the result.

What are the limitations of Gauss-Kronrod quadrature?

One limitation of this method is that it can be computationally expensive, especially for higher order approximations. Additionally, it may not be suitable for functions with singularities or discontinuities within the interval of integration.

How is the accuracy of Gauss-Kronrod quadrature determined?

The accuracy of the method is determined by the number of points used for the approximation, with higher order approximations using more points and therefore providing a more accurate result. The error in the approximation can also be estimated using the difference between the results from the Gauss-Legendre and Kronrod points.

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