Generalized quadrature formula for convex functions

Yong-Ming Chen, Ping Lin, Yong He

College of Biosystems Engineering and Food Science

Zhejiang University

268 Kaixuan Road, 310029, Hangzhou, China

yhe@zju.edu.cn

Abstract - A generalized quadrature formula was

introduced to estimate the numerical solutions of the

integral of convex functions. The formula was composed of

the difference between two parts of the areas of the

trapezoid and the weighted convex patch. The range of

weight coefficient of the convex patch was from 0 to 1

evaluated by the Hadamard’s inequality. Tuning the

weighted value of the convex patch could be considered as

using different order curve to approximate the integrand.

The classical trapezoid and Simpson quadrature formulas

could also be reformulated and composed of the difference

between the areas of the trapezoid and the patch weighted

by 0 and 2

3, respectively. Thus those classical quadrature

formulas were generalized. The numerical experiments

were performed to compare the computing performance of

our proposed equation using different weight values. In

addition, the generalized formula looks much simple and

understandable compared with the classical.

Keywords  ‐  Quadrature  formula;  numerical  solution; 

convex function; weight; hadamard’s inequality 

I. INTRODUCTION

We studied quadrature formulas for evaluating the

numerical solution of a definite integral of a function

1( )f x on the interval 1,a b

:

( )b

aS f x dx (1)

The numerical solution of above integral could be

obtained by calculating the area beneath a graph of the

integrand ( )f x . The classical trapezoid and Simpson

rules are simple and efficient for evaluating the integral

function. These algorithms are still widely used and

continuously studied [1-5].

The well-known Hadamard’s inequality [6-7] can be

used to estimate the range of the solution of a definite

integral of any convex function 1( )f x on the

interval 1,a b , which is defined as:

1 ( ) ( )( ) ( )

2 2

b

a

a b f a f bf f x dx

b a

(2)

In this paper, we worked out a novel and generalized

quadrature formula for estimating the numerical solutions

of the integral of convex functions. The formula was

composed of the difference between two parts of the areas

of the trapezoid and the weighted convex patch. Tuning

the weighted value of the convex patch could be

considered as using different order curve to approximate

the integrand. The numerical experiments were performed

to compare the computing performance of the proposed

equation using distinct weight values.

II. REFORMULATE QUADRATURE FORMULAS

We studied the numerical solutions of the integral of

convex functions. Generally the integrand takes on

convexity or concavity on a subinterval whose width is

small enough, when the integrand is smooth and

continuous. The Trapezoid and Simpson quadrature rules

use 1-order straight line function

( 1( ) , ,f x ax b a b ) and 2-order parabola

function ( 2 12 1 2 1( ) , , ,f x a x a x b a a b ) to

approximate the integrand, respectively. Line function

and parabola function could be proved as convex

2010 International Conference on Artificial Intelligence and Computational Intelligence

978-0-7695-4225-6/10 $26.00 © 2010 IEEE

DOI 10.1109/AICI.2010.267

136

2010 International Conference on Artificial Intelligence and Computational Intelligence

978-0-7695-4225-6/10 $26.00 © 2010 IEEE

DOI 10.1109/AICI.2010.267

136

 

 

 

 

function, so the estimated areas under the integrands have

convexity or concavity [8]. These methods are equal to

transform the non-convex integrands into convex or

approximately convex functions for study and estimation

to some extent. Based on the upper analysis, we

considered that the methods for the convex function

quadrature could be applied to ordinary functions.

We first presented the Trapezoid and Simpson

quadrature formulas for evaluating the numerical

solutions of the integral of convex functions, and then

reformulated them as the difference expressions between

the areas of the trapezoid (DBEG ) and the weighted

convex patch ( ABCD ) as follows:

Trapezoid quadrature:

1 32 0 22

0

T

T Patch

f fS h eh

S S

(3)

Simpson quadrature:

1 2 3

1 31 2 3

1 3

43

22 ( 2 )

2 3

22 (2 )

2 32

3

S

T Patch

hS f f f

f fh h f f f

f fh eh

S S

(4)

where 1 3 2

1( )

2e f f f is the Euclidean

distance between the points A and B ,

2 1 3 2h x x x x and 2PatchS eh is the area

of the unweighted patch (see Fig. 1).

From equations (3) and (4), we could see that the

Trapezoid and Simpson quadrature formulas could be

decomposed into two same parts of the trapezoid region

TS and the unweighted patch region PatchS . The

differences between them were the weight coefficients

before the patches PatchS which were 0 and2

3 in

equations (3) and (4), respectively.

x

f(x)

O

f1

f2

f3

x1 x2 x3

f1+ f32 e

h h

A

B

C

D

E F G

 

Fig. 1 Convex function

III. GENERALIZED QUADRATURE FORMULA

We reformulated the Hadamard’s inequality (see

inequality (2) and Fig. 1), thus we got,

2

1 3 1 32

1 3 1 32

1 3

2

2 2 22 2

2 22 2

2 1 22

1T Patch

S h f

f f f fh h h f

f f f fh h f

f fh eh

S S

(5)

We derived the infimum of the area S ，thus we

got,

inf( )

1H

T Patch

S S

S S

(6)

Functions (5) and (6) showed that the areas

estimated by the Hadamard’s inequality could also be

divided into two parts of the areas TS and PatchS

which were the same as the functions (3) and (4). The

differences among them were that the coefficients before

the patches PatchS were 0, 2

3 and 1 in the functions

(3), (4) and (6), respectively. These three equations could

be expressed as one generalized equation when the

coefficients before the patch PatchS were replaced by

the weight parameter w : 

T PatchS S w S (7)

 

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TABLEI. Relative errors of the integral of different functions with distinct weight values

( ) 0,2f x x

2x 6x x

2xe

( )xe 3

1

2 5x x log 7x

ee x

0w

(Trapezoid) 50.00 250.00 25.00 237.93 50.47 160.59 64.01

0.3w 27.50 148.28 15.68 146.46 32.05 104.13 46.32

2

3w

(Simpson)

0.00 23.96 4.29 34.67 9.53 35.12 24.70

0.8w 10.00 21.25 0.15 5.98 1.34 10.02 16.84

0.9w 17.50 55.16 2.96 36.47 4.80 8.80 10.95

where w is defined on the interval 0,1 estimated by

the Hadamard’s inequality. This was geometrically

illustrated in Fig. 2. We could see that the equation (7)

looks much simple and understandable.

IV. NUMERICAL EXPERIMENTS

The equation (7) with different weight values was

used to compute the numerical solutions of the integral of

seven different kinds of functions on the interval 0, 2 .

The test results were described by the relative error and

shown in Table 1. The relative error was computed as

follows,

100measure real

real

S SError

S

(8)

where realS and measureS are the real and measure

values of the integral, respectively. realS was estimated

using the adaptive Simpson quadrature routine quad

provided by Matlab (Version 7.9.0, The MathWorks, Inc.,

USA) excepted for the function log 7xee x

estimated using the adaptive Gauss-Kronrod quadrature

routine quadgk. measureS was evaluated using the

equation (7). From Table 1 we could see that using

different weight values w in equation (7) were fitted

for computing the integrals of different kinds of

functions. Furthermore tuning the weight value w

could be considered to use different order convex

functions to approximate the integrands. It was indicated

that our quadrature formula was more generalized and

applicable than the classical Trapezoid and Simpson

rules.

V. CONCLUSIONS

All the integrals of convex functions could be

composed of the difference bewteen two same parts of the

areas of the trapezoid and the weighted convex patch. The

weight coefficient of the patch was defined on the interval

0,1 estimated by the Hadarmard’s inequality. The

classical Trapezoid and Simpson quadrature formulas

could be described using our proposed formula by setting

the weight values of the patches 0 and 2

3, respectively.

In addition, our proposed equation looks more simple and

understandable than the classical. In view of these results

of numerical experiments it would appear that our

formula was more generalized and applicable for

estimating the numerical solutions of the integral of

different kinds of functions.

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S = ST – ω⋅SPatch

–=x

f(x)

Ox

f(x)

Ox

f(x)

O Fig. 2 Geometrical interpretation of quadrature formula for a convex function

ACKNOWLEDGMENTS

This study was supported by National Science and Technology Support Program (2006BAD10A09), the 863 Project of China (2007AA10Z210), Natural Science Foundation of China (Project No: 30671213), and Natural Science Foundation of Zhejiang (Y307119).

REFERENCES

[1] Boyd, S., Vandenberghe, L.: Convex optimization. Cambridge University Press, Cambridge (2004).

[2] Dragomir, S.S.: Inequalities of Hadamard’s type for Lipschitzian mappings and their applications. J. Math. Anal. Appl. 245, 489-501 (2000).

[3] Fink, A.M.: Hadamard's inequality for log-concave functions. Math. Comput. Model. 32, 625-629 (2000).

[4] Gander, W., Gautschi, W.: Adaptive quadrature – revisited. BIT Numer. Math. 40(1), 84-101 (2000).

[5] Jafar, S.N., Mahdi, H.: Solving linear integral equations of the second kind with repeated modified trapezoid quadrature method. Appl. Math. Comput. 189, 980-985 (2007).

[6] Lyness, J.N.: Notes on the adaptive Simpson quadrature routine. J. Assoc. Comput. Mach. 16(3), 483-495 (1996).

[7] Shampine, L.F.: Vectorized adaptive quadrature in MATLAB. J. Comput. Appl. Math. 211, 131-140 (2008).

[8] Ujević, N.: Error inequalities for an optimal quadrature formula. J. Appl. Math. Comput. 24, 65-79 (2007).

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