(SR-52) In defense of linear quadrature rules
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12-22-2019, 06:23 PM
(This post was last modified: 01-16-2020 02:13 PM by Albert Chan.)
Post: #2
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RE: (SR-52) In defense of linear quadrature rules
Thank you, SlideRule.
I tried this modified point / weight quadrature on \(\int _0^6 e^{x^3}\; dx ≈ 5.963938092 × 10^{91}\) gp/gw from Gaussian Quadrature Weights and Abscissae (n=10, rounded to 10 digits) mp/mw is modified points/weights from In Defense of Linear Quadrature Rules, table 1 lua> gp = {0.1488743390, 0.4333953941, 0.6794095683, 0.8650633667, 0.9739065285} lua> gw = {0.2955242247, 0.2692667193, 0.2190863625, 0.1494513492, 0.06667134431} lua> mp = {0.2295037173, 0.6364758401, 0.9015072053, 0.9928383122, 0.9999843443} lua> mw = {0.4501100825, 0.3483026852, 0.1744679776, 0.02696299772, 0.0001562579734} lua> function integ(f,a,b,p,w) -- 10 points quadrature : local t, m, c = 0, (b-a)/2, (b+a)/2 : for i=1,5 do t = t + (f(-m*p[i]+c) + f(m*p[i]+c)) * w[i] end : return m * t : end lua> function f(x) return math.exp(x^3) end lua> integ(f, 0, 6, gp, gw) → 3.052910317e+089 lua> integ(f, 0, 6, mp, mw) → 5.444304730e+091 Modified point/wieght looks much closer. However, if we integrate only the dominant part, plain guassian points is better. lua> integ(f, 5.50, 6, gp, gw) → 5.942395811e+091 lua> integ(f, 5.50, 6, mp, mw) → 5.577982336e+091 lua> integ(f, 5.75, 6, gp, gw) → 5.963893713e+091 lua> integ(f, 5.75, 6, mp, mw) → 5.907966288e+091 |
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(SR-52) In defense of linear quadrature rules - SlideRule - 12-22-2019, 02:14 PM
RE: (SR-52) In defense of linear quadrature rules - Albert Chan - 12-22-2019 06:23 PM
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