This study examined the characteristics of a variable three-point Gauss quadrature using a variable set of weighting factors and corresponding optimal sampling points. The major findings were as follows. The one-point, two-point, and three-point Gauss quadratures that adopt the Legendre sampling points and the well-known Simpson’s 1/3 rule were found to be special cases of the variable three-point Gauss quadrature. In addition, the three-point Gauss quadrature may have out-of-domain 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 higher-order 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 Gauss-Legendre quadrature [
The Gauss-Legendre 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 3-point 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 one-point, two-point, and three-point Gauss quadratures adopting the Legendre sampling points and Simpson’s 1/3 rule were actually special cases of the variable three-point Gauss quadrature. The order of the polynomial that can be integrated precisely by conventional three-point Gauss quadrature is 5. On the other hand, the use of varied sampling points with a variable three-point 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 shear-locking 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 out-of-domain sampling points. A new method, adopting quadratically extrapolated integrals and a nonlinearity index, using integrals of the variable three-point 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 near-zero center-weight 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 three-point Gauss quadrature of variable sampling point.
Integral
Without loss of generality, we assumed the range of integration
Equation (
Equation (
In the variable three-point Gauss integration formula, the term of “variable” was adopted because the weighting factor
Table
Different types of 3-point Gauss integration of variable sampling points and the integration characteristics.
Type | Weight ratio |
Weighting factor |
Integration weights | Sampling points | Integration characteristics |
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(a) | 0 | 0.0 | 1.00000000 |
±0.57735027 |
Conventional two-point rule |
(b) |
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0.99995000 |
±0.57736470 |
Quasi two-point rule |
(c) | 1 | 2/3 | 0.66666667 |
±0.70710678 |
Three-point rule |
(d) | 8/5 | 8/9 | 0.55555556 |
±0.77459667 |
Conventional three-point rule |
(e) | 2 | 1.0 | 0.50000000 |
±0.81649658 |
Three-point rule |
(f) | 4 | 4/3 | 0.33333333 |
±1.00000000 |
Three-point rule |
(g) | 22 | 22/12 | 0.08333333 |
±2.00000000 |
Three-point rule |
(h) | 52 | 52/27 | 0.03703704 |
±3.00000000 |
Three-point rule |
(i) |
|
2.0 | 0.00000000 |
|
Conventional one-point rule |
The variable three-point rule of a Gauss quadrature includes the conventional Gauss-Legendre quadrature of the one-point rule
Various sets of integration weights and optimal sampling points with respect to the weight ratio
We present three corollaries regarding three-point Gauss quadrature of variable sampling point.
The three-point Gauss quadrature of Lemma
The three-point Gauss quadrature of Lemma
The three-point Gauss quadrature of Lemma
To test the characteristics of variable three-point 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 3-point Gauss integration of variable sampling pints for 5 monomial integrands.
Weight ratio ( |
Relative errors for each monomial integrand | ||||
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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
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( |
( |
( |
( |
( |
−0.018783611 | 0.40846778 | 1.2206317 | 0.40846778 | −0.018783611 |
As in the proof of Lemma
Equation (
Integral
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−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
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( |
( |
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( |
( |
( |
0.0051149999 | −0.041801110 | 0.45450278 | 1.1643667 | 0.45450278 | −0.041801110 | 0.0051149999 |
As in the proof of Theorem
Equation (
Equation (
Integral
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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 (end-point Gauss quadrature) with Simpson’s 1/3 rule.
Integrand | Error | Exact | Simpson | Formula I | Formula II | Formula III | Formula IV |
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Value |
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% | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | |
Ratio | — | — | — | — | — | — | |
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% | 0 |
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Ratio | — | 1.0 | 0.97 | 0.17 | 0.48 | 0.0080 | |
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Exp( |
Value |
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Ratio | — | 1.0 | 0.90 | 0.13 | 0.44 | 0.013 |
Table
Comparison of the integrals using four formulae of extrapolation methods (end-point Gauss quadrature) with Boole’s rule.
Integrand | Error | Exact | Boole | Formula I | Formula II | Formula III | Formula IV |
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Value |
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% | 0.0 | 0.0 |
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0.0 | 0.0 | 0.0 | |
Ratio | — | — | ∞ | — | — | ||
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% | 0 |
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Ratio | — | 1.0 | 4.2 | 0.31 | 0.90 | 0.015 | |
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Exp( |
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Ratio | — | 1.0 | 3.3 | 0.46 | 1.6 | 0.050 |
(1) A lemma for the variable three-point 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 (NRF-2012R1A1A2008903).