This study examined the characteristics of a variable threepoint Gauss quadrature using a variable set of weighting factors and corresponding optimal sampling points. The major findings were as follows. The onepoint, twopoint, and threepoint Gauss quadratures that adopt the Legendre sampling points and the wellknown Simpson’s 1/3 rule were found to be special cases of the variable threepoint Gauss quadrature. In addition, the threepoint Gauss quadrature may have outofdomain sampling points beyond the domain end points. By applying the quadratically extrapolated integrals and nonlinearity index, the accuracy of the integration could be increased significantly for evenly acquired data, which is popular with modern sophisticated digital data acquisition systems, without using higherorder extrapolation polynomials.
Numerical integration methods may be grouped in two categories. One is the rule for discrete data and the other is for function of continuous data. The GaussLegendre quadrature [
The GaussLegendre quadrature uses function values at interior sampling points with corresponding best weights to result in a very accurate result in spite of the relatively small number of sampling points. However, this quadrature is inapplicable to discrete data points because it does not use boundary point data. In this study, we provide a lemma with a formula for the new 3point Gauss quadrature of variable sampling points which include the Legendre point as well.
An examination of the effect of these varied sampling points found that the onepoint, twopoint, and threepoint Gauss quadratures adopting the Legendre sampling points and Simpson’s 1/3 rule were actually special cases of the variable threepoint Gauss quadrature. The order of the polynomial that can be integrated precisely by conventional threepoint Gauss quadrature is 5. On the other hand, the use of varied sampling points with a variable threepoint Gauss quadrature does not allow exact integration of a polynomial order 5.
Despite their reduced accuracy, variable Gauss quadratures can be applied effectively to special situations, for example, the shearlocking problems that arise when using the finite element method for a plate/shell when the ratio of the thickness to the width is quite small, as reported previously [
Accordingly, this study examined the characteristics of various groups of weighting factors and sampling points and tested the performance of the extended end points quadrature using the outer outofdomain sampling points. A new method, adopting quadratically extrapolated integrals and a nonlinearity index, using integrals of the variable threepoint Gauss integrations of the 1st and 2nd extended end points and conventional end point integration, was applied to the integration of evenly acquired discrete data to obtain new four kinds of numerical integration formulae.
The modification of the Gauss integration formula with a nearzero centerweight factor was included in a previous study [
The general expressions for the sampling points and weights of the Gauss integration can be driven as follows. In (
From (
We present a lemma regarding threepoint Gauss quadrature of variable sampling point.
Integral
Without loss of generality, we assumed the range of integration
Equation (
Equation (
In the variable threepoint Gauss integration formula, the term of “variable” was adopted because the weighting factor
Table
Different types of 3point Gauss integration of variable sampling points and the integration characteristics.
Type  Weight ratio 
Weighting factor 
Integration weights  Sampling points  Integration characteristics 

(a)  0  0.0  1.00000000 
±0.57735027 
Conventional twopoint rule 
(b) 


0.99995000 
±0.57736470 
Quasi twopoint rule 
(c)  1  2/3  0.66666667 
±0.70710678 
Threepoint rule 
(d)  8/5  8/9  0.55555556 
±0.77459667 
Conventional threepoint rule 
(e)  2  1.0  0.50000000 
±0.81649658 
Threepoint rule 
(f)  4  4/3  0.33333333 
±1.00000000 
Threepoint rule 
(g)  22  22/12  0.08333333 
±2.00000000 
Threepoint rule 
(h)  52  52/27  0.03703704 
±3.00000000 
Threepoint rule 
(i) 

2.0  0.00000000 

Conventional onepoint rule 
The variable threepoint rule of a Gauss quadrature includes the conventional GaussLegendre quadrature of the onepoint rule
Various sets of integration weights and optimal sampling points with respect to the weight ratio
We present three corollaries regarding threepoint Gauss quadrature of variable sampling point.
The threepoint Gauss quadrature of Lemma
The threepoint Gauss quadrature of Lemma
The threepoint Gauss quadrature of Lemma
To test the characteristics of variable threepoint Gauss integration formulae, the error ratios of the integral from 0 to 2 for the 5 types of monomial integrand were compared (Table
Relative errors of the different types of 3point Gauss integration of variable sampling pints for 5 monomial integrands.
Weight ratio ( 
Relative errors for each monomial integrand  





 






















































The italic numbers mean improper results.
According to the formula of Gauss quadrature of variable three sampling points with corresponding best weights, one can see that the sampling points may be located at the boundary points (integral
In other words, when one integrates a function from −1~1 in local coordinate
Then, we extrapolate these integrals to compute the integral
The process of extrapolation is presented here. The integral, which is a function of outer sampling point
On the other hand, the error of the integral choosing the outer sampling point as
After obtaining the coefficients,
We present four theorems regarding extrapolated Gauss quadrature of extended end points.
Integral






( 
( 
( 
( 
( 
−0.018783611  0.40846778  1.2206317  0.40846778  −0.018783611 
As in the proof of Lemma
Equation (
Integral






−1/90  34/90  114/90  34/90  −1/90 
−0.011111111  0.37777778  1.2666667  0.37777778  −0.011111111 
As in the proof of Theorem
Equation (
Equation (
Integral









( 
( 

( 
( 
( 
0.0051149999  −0.041801110  0.45450278  1.1643667  0.45450278  −0.041801110  0.0051149999 
As in the proof of Theorem
Equation (
Equation (
Integral








16.32/12960  −241.92/12960  5140.80/12960  16089.60/12960  5140.80/12960  −241.92/12960  16.32/12960 
0.0012592593  −0.018666667  0.39666667  1.2414815  0.39666667  −0.018666667  0.0012592593 
As in the proof of Theorem
Equation (
Equation (
Through this extrapolation to obtain the integral
We compared the four formulae with Simpson’s 1/3 rule for integration of the discrete data from 1 to 3. The discrete data are given at integer point of
Comparison of the integrals using four formulae of extrapolation methods (endpoint Gauss quadrature) with Simpson’s 1/3 rule.
Integrand  Error  Exact  Simpson  Formula I  Formula II  Formula III  Formula IV 
 

Value 






%  0.0  0.0  0.0  0.0  0.0  0.0  
Ratio  —  —  —  —  —  —  
 

Value 






%  0 






Ratio  —  1.0  0.97  0.17  0.48  0.0080  
 
Exp( 
Value 






% 






Ratio  —  1.0  0.90  0.13  0.44  0.013 
Table
Comparison of the integrals using four formulae of extrapolation methods (endpoint Gauss quadrature) with Boole’s rule.
Integrand  Error  Exact  Boole  Formula I  Formula II  Formula III  Formula IV 


Value 






%  0.0  0.0 

0.0  0.0  0.0  
Ratio  —  —  ∞  —  —  
 

Value 






%  0 






Ratio  —  1.0  4.2  0.31  0.90  0.015  
 
Exp( 
Value 






% 






Ratio  —  1.0  3.3  0.46  1.6  0.050 
(1) A lemma for the variable threepoint Gauss quadrature has been presented. Based on that, comprehensive sets of weighting factors and corresponding optimal sampling points were presented.
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF2012R1A1A2008903).