Legendre Polynomials, Lagrange Interpolation, and Gaussian Quadrature

This section develops the mathematical foundation behind Gaussian quadrature and orthogonal collocation. The key ingredients are Legendre polynomials, Lagrange interpolating polynomials, and Gaussian integration rules.

Legendre Polynomials¶

Legendre polynomials satisfy

\frac{d}{d\tau}\left((1-\tau^2)\frac{dP_n}{d\tau}\right) +n(n+1)P_n(\tau)=0,

(1)

for $n=0,1,\ldots$ on $\tau\in[-1,1]$ .

The first two are

P_0(\tau)=1,\qquad P_1(\tau)=\tau.

(2)

The recurrence relation is

(n+1)P_{n+1}=(2n+1)\tau P_n-nP_{n-1}.

(3)

Orthogonality¶

Legendre polynomials satisfy

\int_{-1}^{1}P_i(\tau)P_j(\tau)\,d\tau = \frac{2}{2j+1}\delta_{ij}.

(4)

Therefore every polynomial of degree at most $n-1$ is orthogonal to $P_n$ .

Lagrange Interpolation¶

Given distinct support points

\tau_1,\ldots,\tau_N,

(5)

define

L_i(\tau)= \prod_{j\ne i} \frac{\tau-\tau_j}{\tau_i-\tau_j}.

(6)

These satisfy

L_i(\tau_j)=\delta_{ij}.

(7)

Hence an interpolant is

g(\tau)\approx \sum_{i=1}^{N} g(\tau_i)L_i(\tau).

(8)

Gaussian Quadrature¶

Approximate

\int_{-1}^{1}g(\tau)\,d\tau \approx \sum_{i=1}^{N}w_i g(\tau_i),

(9)

where

w_i=\int_{-1}^{1}L_i(\tau)\,d\tau.

(10)

Legendre–Gauss (LG)¶

LG points are the roots of

P_N(\tau).

(11)

Using $N$ points exactly integrates every polynomial of degree

2N-1.

(12)

Legendre–Gauss–Radau (LGR)¶

One endpoint is fixed, typically

\tau_1=-1.

(13)

The remaining points are roots of

P_N(\tau)+P_{N-1}(\tau).

(14)

Exactness:

2N-2.

(15)

Legendre–Gauss–Lobatto (LGL)¶

Both endpoints are fixed,

\tau_1=-1,\qquad\tau_N=1.

(16)

Interior points are roots of

P'_{N-1}(\tau).

(17)

Exactness:

2N-3.

(18)

Comparison¶

Method	Endpoints	Exact degree
LG	None	$2N-1$
LGR	One	$2N-2$
LGL	Both	$2N-3$

Why Gaussian Quadrature Works¶

A polynomial of degree at most $2N-1$ can be decomposed into

g(\tau)=P_N(\tau)f(\tau)+h(\tau),

(19)

where $f$ and $h$ have degree at most $N-1$ . Orthogonality eliminates the first term after integration, while the remainder is represented exactly by the Lagrange interpolant.

Accuracy Growth¶

For LG quadrature:

Points	Exact degree
2	3
3	5
4	7
5	9
6	11
7	13

Each additional collocation point increases the exact polynomial degree by two.

Application to Optimal Control¶

Gaussian quadrature provides highly accurate approximations for:

integral cost functions,
state integration,
orthogonal collocation methods,
pseudospectral optimal control.

Summary¶

Legendre polynomials are orthogonal on $[-1,1]$ .
Lagrange polynomials interpolate exactly at support points.
Gaussian quadrature combines both ideas.
LG, LGR and LGL differ by endpoint constraints.
Gaussian quadrature achieves much higher accuracy than low-order rules using the same number of points.

Connection. Those approximation tools determine both quadrature weights and the endpoint structure of the LGL transcription.