How to derive the quadrature weights for the trapezoid and the Simpsons rule? Trapeziod rule: <m

dumnealorjavgj

dumnealorjavgj

Answered question

2022-05-13

How to derive the quadrature weights for the trapezoid and the Simpsons rule?
Trapeziod rule: a b f ( x ) b a 2 [ f ( a ) + f ( b ) ]
Simpson's rule: a b f ( x ) b a 6 [ f ( a ) + 4 f ( ( a + b ) / 2 ) + f ( b ) ]

Answer & Explanation

Cortez Hughes

Cortez Hughes

Beginner2022-05-14Added 23 answers

The Trapezoid and Simpson's rules are Newton-Cotes formulae. Derivation of the weights come from integrating Lagrange basis polynomials. Let L ( x ) be a lagrange polynomial which interpolates f ( x ) at n + 1 points, x 0 , x 1 , . . . x n . Then
a b f ( x ) d x a b L ( x ) d x = a b i = 0 n f ( x i ) l i ( x ) d x = i = 0 n f ( x i ) a b l i ( x ) d x ,
and here a b l i ( x ) d x is the weight for the i-th interpolation point.

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