What is Gaussian quadrature: Definition and 17 Discussions
In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration. (See numerical integration for more on quadrature rules.) An n-point Gaussian quadrature rule, named after Carl Friedrich Gauss, is a quadrature rule constructed to yield an exact result for polynomials of degree 2n − 1 or less by a suitable choice of the nodes xi and weights wi for i = 1, ..., n. The modern formulation using orthogonal polynomials was developed by Carl Gustav Jacobi 1826. The most common domain of integration for such a rule is taken as [−1, 1], so the rule is stated as
∫
−
1
1
f
(
x
)
d
x
≈
∑
i
=
1
n
w
i
f
(
x
i
)
,
{\displaystyle \int _{-1}^{1}f(x)\,dx\approx \sum _{i=1}^{n}w_{i}f(x_{i}),}
which is exact for polynomials of degree 2n − 1 or less. This exact rule is known as the Gauss-Legendre quadrature rule. The quadrature rule will only be an accurate approximation to the integral above if f(x) is well-approximated by a polynomial of degree 2n − 1 or less on [−1, 1].
The Gauss-Legendre quadrature rule is not typically used for integrable functions with endpoint singularities. Instead, if the integrand can be written as
f
(
x
)
=
(
1
−
x
)
α
(
1
+
x
)
β
g
(
x
)
,
α
,
β
>
−
1
,
{\displaystyle f(x)=\left(1-x\right)^{\alpha }\left(1+x\right)^{\beta }g(x),\quad \alpha ,\beta >-1,}
where g(x) is well-approximated by a low-degree polynomial, then alternative nodes
x
i
′
{\displaystyle x_{i}'}
and weights
w
i
′
{\displaystyle w_{i}'}
will usually give more accurate quadrature rules. These are known as Gauss-Jacobi quadrature rules, i.e.,
∫
−
1
1
f
(
x
)
d
x
=
∫
−
1
1
(
1
−
x
)
α
(
1
+
x
)
β
g
(
x
)
d
x
≈
∑
i
=
1
n
w
i
′
g
(
x
i
′
)
.
{\displaystyle \int _{-1}^{1}f(x)\,dx=\int _{-1}^{1}\left(1-x\right)^{\alpha }\left(1+x\right)^{\beta }g(x)\,dx\approx \sum _{i=1}^{n}w_{i}'g\left(x_{i}'\right).}
Common weights include
1
1
−
x
2
{\textstyle {\frac {1}{\sqrt {1-x^{2}}}}}
(Chebyshev–Gauss) and
1
−
x
2
{\displaystyle {\sqrt {1-x^{2}}}}
. One may also want to integrate over semi-infinite (Gauss-Laguerre quadrature) and infinite intervals (Gauss–Hermite quadrature).
It can be shown (see Press, et al., or Stoer and Bulirsch) that the quadrature nodes xi are the roots of a polynomial belonging to a class of orthogonal polynomials (the class orthogonal with respect to a weighted inner-product). This is a key observation for computing Gauss quadrature nodes and weights.
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...
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?