Problem Theoretical background Midpoint Trapezoidal Simpson Romberg Extrapolation

Problem

Theoretical background (Midpoint) (Trapezoidal) (Simpson) (Romberg) (Extrapolation)

Midpoint, Trapezoidal, Simpson h Midpoint Trapezoidal Simpson 0. 8 1. 964800078 0. 172799895 1. 367466684 0. 4 1. 900800015 1. 068799987 1. 623466673 0. 2 1. 716800007 1. 484800001 1. 639466672 0. 1 1. 660300006 1. 600800004 1. 640466672 0. 05 1. 645518755 1. 630550005 1. 640529172 0. 01 1. 640733308 1. 640133358 1. 640533325 0. 001 1. 640291842 1. 64028584 1. 640289841 0. 0001 1. 640533351 1. 640533291 1. 640533331 Answer = 1. 64053334

Midpoint, Trapezoidal, Simpson 2. 5 Integration value 2 1. 5 Mid-point Trapezoidal Simpson TRUE 1 0. 5 0 0 0. 1 0. 2 0. 3 0. 4 0. 5 h value 0. 6 0. 7 0. 8 0. 9

Romberg 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Romberg 0. 172799895 1. 068799987 1. 484800001 1. 600800004 1. 630550005 1. 63803438 1. 639908404 1. 640377093 1. 640494276 1. 640523573 1. 640530897 1. 640532728 1. 640533186 1. 640533329 1. 640533336 1. 640533338 1 2 3 4 5 6 Romberg 0. 172799895 1. 068799987 1. 484800001 1. 600800004 1. 630550005 1. 63803438 Extrapolation 0. 172799895 1. 367466684 1. 640533338 Extrapolation에 의해 수렴 속도가 증가한다.

Romberg 1. 8 1. 6 Integration value 1. 4 1. 2 1 Romberg 0. 8 Extrapolation 0. 6 0. 4 0. 2 0 0 5 10 15 Iteration number 20 큰 차이는 없다 25