16-Point Gaussian Quadrature

[Albert Chan]

(04-09-2021 05:17 AM)Gamo Wrote:  Here is the problem: ∫ 0 to 2 √4 - X² dx = Pi = 3.141592654

Both Gaussian and Romberg quadrature are based on fitting the curve to polynomial.
When X ≈ 2, the curve fall off a cliff, which is hard to curve fit by polynomial.

HP-15C does a u-substitution, which in this case, removed the cliff entirely.

int(√(4-x^2), x=0..2.)
= int(4*√(1-x^2), x=0..1.)
= int(24*u*(1-u) * √(1 - (u*u*(3-2*u))^2), u = 0..1.)

Try this u-transformed function, 16-points should converge to pi pretty good.

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Another way is to avoid the cliff.

y = √(4-x²), x≥0       → x = √(4-y²)

Note the x/y symmetry, instead of integrating from 0 to 2, we can do 0 to √2
With limit = [0,√2], doubled the result over-counted by a square, area = (√2)² = 2

int(√(4-x^2), x=0..2.) = int(√(4-x^2), x=0..√(2.)) * 2 - 2