Integration

Gauss Quadrature

Optimal node placement for maximum precision integration.

1

Recap: Newton-Cotes

Previously, we used equally spaced nodes (Trapezoid, Simpson). This is restrictive and prone to oscillation (Runge's phenomenon).

$\int_a^b f(x)\,dx \approx \sum_{j=0}^n w_j f(x_j)$

Trapezoid (n=1): 2 points, exact for degree 1.
Simpson (n=2): 3 points, exact for degree 3.

2

Gaussian Quadrature Idea

What if we choose the nodes ($x_j$) optimally instead of fixing them?

Degrees of Freedom

With $n+1$ points, we have $n+1$ weights ($w_j$) AND $n+1$ locations ($x_j$).
Total unknowns = $2n+2$.

Result: Exact integration for polynomials of degree $2n+1$.

3

Deriving the 2-Point Rule

Let's find the best 2 points ($x_0, x_1$) and weights ($w_0, w_1$) for $\int_{-1}^1 f(x) dx$.

Equations (Exact for $1, x, x^2, x^3$)

  • $w_0 + w_1 = \int_{-1}^1 1 dx = 2$
  • $w_0x_0 + w_1x_1 = \int_{-1}^1 x dx = 0$
  • $w_0x_0^2 + w_1x_1^2 = \int_{-1}^1 x^2 dx = 2/3$
  • $w_0x_0^3 + w_1x_1^3 = \int_{-1}^1 x^3 dx = 0$

Solution

$w_0 = 1, \quad w_1 = 1$

$x_0 = -\frac{1}{\sqrt{3}}, \quad x_1 = \frac{1}{\sqrt{3}}$

Standard Trapezoid

Trapezoid (End Points)

Gauss 2-Point

Gauss (Interior Points)

4

General Theory

The optimal nodes $x_j$ are the roots of Legendre Polynomials.

Legendre Polynomials

  • $P_0(x) = 1$
  • $P_1(x) = x$
  • $P_2(x) = \frac{3x^2-1}{2}$ (Roots $\pm 1/\sqrt{3}$)
  • $P_3(x) = \frac{5x^3-3x}{2}$ (Roots $0, \pm \sqrt{3/5}$)
Legendre Polynomials

Roots are the nodes.

Gauss Nodes Distribution

Nodes cluster towards the endpoints (like Chebyshev).

5

Using Gauss Quadrature

Gauss rules are defined for $\int_{-1}^1$. To integrate over $[a,b]$, we must change variables.

Transformation Formula

$\int_a^b f(x) dx = \frac{b-a}{2} \int_{-1}^1 f\left( \frac{b-a}{2}t + \frac{b+a}{2} \right) dt$

Example: $\int_0^1 x^2 e^{-x} dx$ ($n=1$, 2 points)

1. Map $[0,1]$ to $[-1,1]$: $x = \frac{1}{2}(t+1)$, $dx = \frac{1}{2} dt$.

2. Integral becomes $\frac{1}{2} \int_{-1}^1 (\frac{t+1}{2})^2 e^{-(t+1)/2} dt$.

3. Evaluate at $t = \pm 1/\sqrt{3}$ with weights 1.

Result $\approx 0.1594$

🧠 Knowledge Check +20 XP

How many points ($n+1$) does Gauss Quadrature need to exactly integrate a polynomial of degree 5?

6

Real World: Finite Elements

In Finite Element Analysis (FEA), we integrate functions over millions of 3D elements (tetrahedrons/bricks). Minimizing function evaluations is critical.

Gauss Quadrature is standard here.

3D Mesh Tetrahedron