MHB Gaussian Quadrature: isolated roots

AI Thread Summary
The discussion focuses on the application of Gaussian Quadrature and the concept of isolated roots in weight functions. Isolated roots are defined as roots with multiplicity one, meaning they do not affect the derivative of the function in their vicinity. The weight function used in the example, $\sqrt{|x|^3}$, has one isolated root, which raises questions about potential issues when the weight function equals zero over an interval. Participants clarify that isolated roots are characterized by their lack of nearby roots, rather than their multiplicity. Understanding these concepts is crucial for correctly applying Gaussian Quadrature methods.
mathmari
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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)
 
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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)
 
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