Gaussian quadrature Definition and 17 Threads

Hey! 😊 If we want to calculate the nodes $x_1, x_2$ and the weight functions $w_1, w_2$ for the Gaussian quadrature of the integral $$\int_{-1}^1f(x)\, dx\approx \sum_{j=1}^2w_jf(x_j)$$ is there a criteria that we have to consider at chosing the weight functions? I mean if we use e.g...

Hello everyone. I am studying this article since I am interested in optimization. The article makes use of Clenshaw–Curtis quadrature scheme to discretize the integral part of the cost function to a finite sum using Chebyshev polynomials. The article differentiates between the case of odd...

## \int_{-1}^{1} \int_{-1}^{1} e^{-(x^2 + y^2)} cos(2π (x^2 + y^2)\,dx\,dy ## ## I = \int_{-1}^{1} \int_{-1}^{1}f(x,y) \,dx\,dy \approx \sum_{i=0}^{n}\sum_{j=0}^{n} w_i w_j f(x_i, y_j) ## ## = w_0 w_0 f(x_0, y_0) + w_0 w_1 f(x_0, y_1) + w_1 w_0 f(x_1, y_0) + w_1 w_1 f(x_1, y_1) ## ## w_0 =...

Hey! :o I want to calculate the integral $$\int_0^1\frac{1}{x+3}\, dx$$ with the Gaussian quadrature formula that integrates exactly all polynomials of degree $6$. The gaussian quadrature integrates exactly polynomials $\Phi (x)$ with maximum degree $2n-1$. In this case we consider $n=4$...

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...

Hi there, I am having some difficulty evaluating a repeated integral, which is the first of two shown in the image. I had hoped to be able to use Gaussian Quadrature to provide a numerical result, however am unsure on if this is possible for a repeated integral? I have attempted to use Cauchy'...

How do you calculate the necessary points in a function to numerically integrate it using the Gaussian Quadrature? If I were to evaluate a function using two points, the Gaussian Quadrature needs the value of the function at ##\displaystyle{\pm \sqrt{\frac{1}{3}}}## with weights of unity. How...

Homework Statement Evaluate the definite integral below numerically (between limits -1 and 1) using a couple of numerical methods, including Gauss-Legendre quadrature - and compare results. Homework Equations $$ \int{(1-x^2)^\frac{1}{2}} dx $$ "Gauss quadrature yields the exact integral if φ...

Hi, I'm studying about Chebyshev Quadrature and i found so little and confused information about this. I don't know if Gauss-Chebyshev Quadrature is the same of Chebyshev Quadrature. The only good information that i found was from Wolfram...

I need to use the four-point Gaussian quadrature rule to do some intense numerical calculations. Could anyone link to this page where it's written out explicitly over an [a,b] interval. I haven't been able to find it, I'm trying to derive it now but it's crucial that I'm 100% correct. I haven't...

My numerical analysis book doesn't explain it. It just tells you to use precomputed tables, and directs you to an out of print book from the 80's that I can't find anywhere. After searching, I found http://en.wikipedia.org/wiki/Gaussian_quadrature#Computation_of_Gaussian_quadrature_rules" in...

Homework Statement approximate this integral: \oint e^(-(x^2)) from 0 to 4 using Gaussian Quadrature with n = 3 Homework Equations can be found at: http://en.wikipedia.org/wiki/Gaussian_quadrature The Attempt at a Solution n = 3 coefficients: c(1) = c(3) = 5/9, c(2) = 8/9...

\int^{1}_{-1}f(x)dx = \sum^{n}_{j=-n}a_{j}f(x_{j}) Why does \sum_{j}a_{j} = 2 ? I know that the aj's are weights, and in the case of [-1,1], they are calculated using the roots of the Legendre polynomial, but I don't understand why they all add up to 2.

Homework Statement 6.3.b highlighted in attachment. Have solved part a (which gives the approximant used in part b) and problem 3.8 (which gives the original function). 3.8 was definitely solved correctly. Part a could be wrong, but the solution seems OK. a = acreage y = yield from 3.8 -...

I'm trying to make a generalized quadrature method and I seem to be running into some bizarre errors. For n=2 my answer is twice what it should be and for n greater the innaccuracy increases (answer/n is close but worse than answer/2 with n=2). My general algorithm is: p = nth legendre...

Anyone care to explain the concept of gaussian quatrature? I've tried some websites but they're a little over my head. An example would be appreciated, thanks!

I am not sure of the spelling, but I heard of the 'gaussian quadature' (or quadrature). It was spoken, and was in a mathematical equation. What the heck is it?