MHB Gaussian Quadrature: isolated roots

[mathmari]

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)

 

[I like Serena]

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)

 

[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, 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)

 

[I like Serena]

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)