(SR-52) In defense of linear quadrature rules
12-22-2019, 02:14 PM
Post: #1
 SlideRule Senior Member Posts: 1,052 Joined: Dec 2013
(SR-52) In defense of linear quadrature rules
An extract from In Defense of Linear Quadrature Rules, William Squire (Aerospace Engineering Dept., West Virginia University), Comp. & Maths with Apple, Vol. 7, pp. 147.-149, Pergamon Press Ltd., 1981

Abstract--lt is shown that appropiate linear quadrature rules can handle integrands with singularities at or near the end points more effectively than the nonlinear methods proposed by Werner and Wuytack. A special 10 point Gauss rule gives good results. A method with exponential convergence gives high accuracy with a moderate number of nodes. Both procedures were implemented on a programmable hand calculator.
INTRODUCTION
The purpose of this note is to demonstrate that:
(1) a special 10 point Gauss rule for integrands with singularities at or near the endpoints proposed by Harris and Evans [2] will give results comparing favourably to any other procedure using a comparable number of nodes.
(2) A quadrature rule, which Stenger [3] has shown to have exponential convergence, gives very accurate results for such integrands with a moderate number of function evaluations.
Both these procedures were implemented on an SR 52 programmable hand calculator.

The SR-52 implementation is given in Appendix A …

The method was implemented on an SR-52 as described in Appendix B …

APPENDIX A
Harris-Evans 10 point rule …

APPENDIX B
SR-52 program for Stenger quadrature (equation 1) …

BEST!
SlideRule
12-22-2019, 06:23 PM (This post was last modified: 01-16-2020 02:13 PM by Albert Chan.)
Post: #2
 Albert Chan Senior Member Posts: 850 Joined: Jul 2018
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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