16-Point Gaussian Quadrature

[robve]

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

On the HP-11C and HP-15 Equation is

[X²] [4] [X<>] [-] [√]

Answer on HP-11C is 3.141722674
Answer on HP-15C is 3.141592653

This looks right and not bad for 16 points. 16 points will only give you a (rough) approximation of \( \int_0^2 \sqrt{4-x^2}\,dx \).

Gausss 10 point: 3.14209916979661
Gauss Kronrod 21 point: 3.14161975053083

By comparison (edited to add more examples):

Romberg 524289 points (1e-9 error threshold) gives 3.14159265256818
Romberg 16385 points (1e-7 error threshold) gives 3.14159265256818
Tanh-Sinh 56 points (1e-9 error threshold) gives 3.14159265358672
Tanh-Sinh 30 points (1e-7 error threshold) gives 3.14159265358975
Adaptive Simpson 417 points (1e-9 error threshold) gives 3.14159111522653
Adaptive Simpson 153 points (1e-7 error threshold) gives 3.14159111489381

- Rob