5.4 Gaussian quadrature

Source: Ideas from Computational Physics (Landau, Páez, Bordeianu), §5.4. Rewritten in original wording. Continues 5.3 Integration. Labs: 5.4b Gaussian quadrature labs.

Weighting function and exactness for high-degree polynomials

Rewrite the basic weighted sum by splitting off a weight $W(x)$ from a smoother piece $g(x)$:

$$\int_a^b f(x)\,dx\equiv\int_a^b W(x)\,g(x)\,dx\simeq\sum_{i=1}^{N} w_i\,g(x_i).$$

Gaussian quadrature chooses both the nodes $x_i$ and the weights $w_i$ so the rule is exact whenever $g$ is a polynomial of degree at most $2N-1$. That is an extraordinary bargain: $N$ evaluations match what a naive equal-spacing rule would need far more points to approach. The price is that the $x_i$ are not equally spaced — and, for the classical families, never sit exactly at the endpoints $a$ or $b$. Different $N$ give entirely different node sets.

If $g$ is smooth (or can be made smooth by peeling $W$ off a singular or peaked factor), Gauss usually beats trapezoid / Simpson at the same $N$. If the integrand changes character across the domain, split the integral and add the pieces — “smart” library integrators do that automatically.

Integral	Name
$\int_{-1}^{1} f(y)\,dy$	Gauss–Legendre
$\int_{-1}^{1} F(y)/\sqrt{1-y^{2}}\,dy$	Gauss–Chebyshev
$\int_{-\infty}^{\infty} e^{-y^{2}} F(y)\,dy$	Gauss–Hermite
$\int_0^{\infty} e^{-y} F(y)\,dy$	Gauss–Laguerre
$\int_0^{\infty} e^{-y} y^{-1/2} F(y)\,dy$	Associated Gauss–Laguerre

For ordinary Gauss–Legendre on $[-1,1]$, the nodes are the zeros of the Legendre polynomial $P_N$, and the weights involve $P_N'$:

$$P_N(y_i)=0, \qquad w_i=\frac{2}{(1-y_i^{2})\bigl[P_N'(y_i)\bigr]^{2}}.$$

Libraries, classical tables, or a short root finder will produce $(y_i,w_i)$. A four-point check set (positive half; mirror with $\pm$):

$\pm y_i$	$w_i$
$0.339981043584856$	$0.652145154862546$
$0.861136311594053$	$0.347854845137454$

Sum of all four weights on $[-1,1]$ is $2$ (integral of $1$).

Mapping Gaussian points

Elementary Gauss–Legendre data live on $-1<y_i<1$. Write $(y_i,w_i')$ for the reference nodes/weights on $[-1,1]$. Walk each target interval — pick the matching map equation (lookalikes included).

Derivation sketch (Gauss–Legendre)

Walk the argument one sentence at a time — at each step pick the matching equation (two lookalikes included).

Coda. Nodes = zeros of $P_N$ kill the $q$ term in the discrete sum. The classical weights

$$w_i=\frac{2}{(1-y_i^{2})\bigl[P_N'(y_i)\bigr]^{2}}$$

then make the rule exact for every polynomial of degree $\le 2N-1$.

Quiz

Gaussian quadrature

1 / 3

Order the Gauss–Legendre exactness argument (top = first).

OrderDrag the handle to rearrange — top should be first.

1. ⋮⋮1Write f = q P_N + r; orthogonality ⇒ ∫f = ∫r
2. ⋮⋮2Want ∫f = Σ w_i f(x_i) exact for deg ≤ 2N−1
3. ⋮⋮3Expand Σ w_i f(x_i) = Σ w_i q P_N + Σ w_i r
4. ⋮⋮4Pick nodes as zeros of P_N so the q-sum vanishes

Drag items into the correct order (or use ↑↓), then check.

Next: 5.4b Gaussian quadrature labs.