Gaussian quadrature — Gauss-Legendre — Hinglish notes

4.8.17 · Maths › Numerical Methods

WHAT: hum kya karne ki koshish kar rahe hain

$[- 1, 1]$ interval kyun? Kyunki magic hoti hai Legendre polynomials ke saath, jo $[- 1, 1]$ par orthogonal hain. Koi bhi real integral $\int_{a}^{b}$ ko ek linear change of variable se $[- 1, 1]$ mein convert kiya jaata hai (niche dikhaya gaya hai).

HOW: 2-point rule scratch se derive karo

Hum chahte hain $\int_{- 1}^{1} f (x) d x \approx w_{1} f (x_{1}) + w_{2} f (x_{2}),$ degree $\leq 2 n - 1 = 3$ ke liye exact. Chaar unknowns $(w_{1}, w_{2}, x_{1}, x_{2})$ , toh basis $1, x, x^{2}, x^{3}$ par exactness impose karo:

f = 1 : f = x : f = x^{2} : f = x^{3} : w_{1} + w_{2} = \int_{- 1}^{1} 1 d x = 2 w_{1} x_{1} + w_{2} x_{2} = \int_{- 1}^{1} x d x = 0 w_{1} x_{1}^{2} + w_{2} x_{2}^{2} = \int_{- 1}^{1} x^{2} d x = \frac{2}{3} w_{1} x_{1}^{3} + w_{2} x_{2}^{3} = \int_{- 1}^{1} x^{3} d x = 0

Ye equations kyun? Agar rule har monomial ko exactly integrate karta hai, toh linearity se yeh degree $\leq 3$ ke har polynomial ko exactly integrate karega.

Symmetry se guess karo $x_{2} = - x_{1}$ , $w_{1} = w_{2} = w$ .

$f = 1$ se: $2 w = 2 \Rightarrow w = 1$ . (Kyun? dono weights equal hain, sum 2 hai.)
$f = x$ aur $f = x^{3}$ odd symmetry se auto-satisfy ho jaate hain. (Kyun? symmetric nodes par odd function cancel ho jaata hai.)
$f = x^{2}$ se: $x_{1}^{2} + x_{2}^{2} = 2 x_{1}^{2} = \frac{2}{3} \Rightarrow x_{1} = \frac{1}{3}$ .

$\int_{- 1}^{1} f d x \approx f (- \frac{1}{3}) + f (\frac{1}{3})$

Indeed $P_{2} (x) = \frac{1}{2} (3 x^{2} - 1)$ ke roots $\pm 1/ 3$ hain — hamare derivation se match karta hai.

Weight formula (KYUN kaam karta hai)

Weights nodes par Lagrange basis ke integrals hain: $w_{i} = \int_{- 1}^{1} \prod_{j \neq = i} \frac{x - x _{j}}{x _{i} - x _{j}} d x,$ jo degree $\leq n - 1$ ke liye exactness guarantee karta hai; orthogonality ise $2 n - 1$ tak boost karta hai. Ek closed form: $w_{i} = \frac{2}{( 1 - x _{i}^{2} ) [ P _{n}^{'} ( x _{i} ) ] ^{2}} .$

Interval change ( $[a, b] \to [- 1, 1]$ )

Figure — Gaussian quadrature — Gauss-Legendre

Worked examples

Common mistakes

Recall Feynman: 12-saal ke bacche ko samjhao

Socho ek tedhi-medhi saanp ka area nikalna hai. Purane rulers kehte hain fixed, equally-spaced jagahon par measure karo. Gauss kehta hai: "Mujhe khud sabse smart jagahein chunne do aur decide karne do ki har jagah kitni count karti hai." Same number of measurements se woh kaafi zyada tedhe-medhe curves ke liye bilkul sahi ho sakta hai. Uske secret spots special points hain jinhe zeros of Legendre polynomials kehte hain — aisi jagahein jo is tarah arranged hain ki bachi hui error bilkul cancel ho jaaye.

Active recall

n

-point Gauss rule degree

2 n - 1

tak exact kyun ho sakta hai?

Iske paas

2 n

free parameters hain (

n

nodes +

n

weights), jo

2 n

moment conditions

\int x^{k}

for

k = 0..2 n - 1

match karne ke liye kaafi hain.

Gauss–Legendre ke nodes kahan placed hote hain?

[- 1, 1]

par Legendre polynomial

P_{n} (x)

ke roots par.

2-point Gauss–Legendre nodes aur weights kya hain?

x = \pm \frac{1}{3}

w_{1} = w_{2} = 1

P_{n}

ke roots extra accuracy kyun dete hain?

P_{n}

sabhi lower-degree polynomials ke saath orthogonal hai, isliye high-degree remainder

q P_{n}

ka integral 0 ho jaata hai aur nodes par bhi vanish karta hai.

[a, b]

[- 1, 1]

par map karne wala change of variable kya hai?

x = \frac{b - a}{2} t + \frac{a + b}{2}

, extra factor

\frac{b - a}{2}

ke saath.

Closed-form weight formula kya hai?

w_{i} = \frac{2}{( 1 - x _{i}^{2} ) [ P _{n}^{'} ( x _{i} ) ] ^{2}}

Kya Gauss–Legendre endpoints

\pm 1

include karta hai?

Nahi (yeh Gauss–Lobatto hai). Standard Gauss–Legendre nodes strictly interior hote hain.

[a, b]

par correctly scale kiya hai ya nahi, sanity check kya hai?

f = 1

integrate karo toh

b - a

aana chahiye.

Connections

Legendre polynomials — nodes ka source (orthogonality)
Orthogonal polynomials — Gauss-type rules ki general recipe
Newton-Cotes formulas — fixed-node cousins (trapezoid, Simpson)
Gauss-Lobatto quadrature — endpoints include karne wala variant
Polynomial interpolation — weights ke peeche Lagrange basis
Numerical integration error analysis — error $\propto f^{(2 n)} (ξ)$