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• mathmari
In summary: Does your calculation come from a method with a proof?If so, is there any step where it would be a problem if the weight function had one or more isolated roots?
mathmari
Gold Member
MHB
In an exercise I have determined the Gaussian Quadrature formula and I have applied that also for a specific function.

Then there is the following question:

Explain why isolated roots are allowed in the weight function.

What exacly is meant by that? Could you explain that to me? What are isolated roots?

(Wondering)

mathmari said:
In an exercise I have determined the Gaussian Quadrature formula and I have applied that also for a specific function.

Then there is the following question:

Explain why isolated roots are allowed in the weight function.

What exacly is meant by that? Could you explain that to me? What are isolated roots?

Hey mathmari!

Which weight function do you have? (Wondering)

Wiki lists $w_i = \frac{2}{\left( 1-x_i^2 \right) [P'_n(x_i)]^2}$ for the Gauss-Legendre quadrature on [-1,1], where $P_n$ are the normalized Legendre polynomials.

This weight function does not have roots. (Worried)

Anyway, I think an isolated root is a root with multiplicity $1$, which means that it's not a root of the derivative.
For instance $(x-3)^2$ has a root at $3$ with multiplicity $2$, meaning it is not an isolated root. (Nerd)

Klaas van Aarsen said:
Which weight function do you have? (Wondering)

Wiki lists $w_i = \frac{2}{\left( 1-x_i^2 \right) [P'_n(x_i)]^2}$ for the Gauss-Legendre quadrature on [-1,1], where $P_n$ are the normalized Legendre polynomials.

This weight function does not have roots. (Worried)

Anyway, if think an isolated root is a root with multiplicity $1$, which means that it's not a root of the derivative.
For instance $(x-3)^2$ has a root at $3$ with multiplicity $2$, meaning it is not an isolated root. (Nerd)

I had determined the formula $$\int_{-1}^1f(x)\cdot \sqrt{|x|^3}\, dx\approx \sum_{i=1}^nf(x_i)\cdot w_i=f(x_1)\cdot w_1+f(x_2)\cdot w_2$$ So the weight function in this case is $\sqrt{|x|^3}$ which has one root. (Thinking)

mathmari said:
I had determined the formula $$\int_{-1}^1f(x)\cdot \sqrt{|x|^3}\, dx\approx \sum_{i=1}^nf(x_i)\cdot w_i=f(x_1)\cdot w_1+f(x_2)\cdot w_2$$ So the weight function in this case is $\sqrt{|x|^3}$ which has one root.

Does your calculation come from a method with a proof?
If so, is there any step where it would be a problem if the weight function had one or more isolated roots? (Wondering)

Edit: Just realized that a typical problem could be when the weight function would be zero on an interval.
That is, an isolated root is not about multiplicity. Instead it is about whether there's a neighborhood without other roots or not. (Blush)

## 1. What is Gaussian Quadrature?

Gaussian Quadrature is a numerical integration method used to approximate the definite integral of a function by evaluating it at specific points. It is based on the concept of using weighted sums of function values at these points to improve the accuracy of the approximation.

## 2. How does Gaussian Quadrature work?

Gaussian Quadrature works by selecting a set of points, known as the quadrature points, and corresponding weights based on the desired accuracy of the approximation. The function is then evaluated at these points and multiplied by their corresponding weights. The sum of these weighted function values gives an approximation of the definite integral.

## 3. What are isolated roots in Gaussian Quadrature?

In Gaussian Quadrature, isolated roots refer to the points at which the function being integrated changes sign. These points are important in the selection of the quadrature points as they help to improve the accuracy of the approximation.

## 4. How are the quadrature points and weights selected in Gaussian Quadrature?

The selection of quadrature points and weights in Gaussian Quadrature is based on the orthogonal polynomials associated with the function being integrated. These polynomials have the property that their roots are the quadrature points and their coefficients are used as the weights for the approximation.

## 5. What are the advantages of using Gaussian Quadrature over other integration methods?

Gaussian Quadrature offers several advantages over other integration methods, such as the Trapezoidal rule or Simpson's rule. It can achieve higher accuracy with a smaller number of quadrature points, making it more efficient. It also works well for functions with singularities or oscillatory behavior, which can be problematic for other methods. Additionally, Gaussian Quadrature can handle a wider range of integration limits, including infinite limits.

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