gauss quadrature rule of integration

04/21/23http://

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Gauss Quadrature Rule of Integration

Major: All Engineering Majors

Authors: Autar Kaw, Charlie Barker

http://numericalmethods.eng.usf.eduTransforming Numerical Methods Education for STEM

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Gauss Quadrature Rule of Integration

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What is Integration?Integration

b

a

dx)x(fI

The process of measuring the area under a curve.

Where:

f(x) is the integrand

a= lower limit of integration

b= upper limit of integration

f(x)

a b

y

x

b

a

dx)x(f


Two-Point Gaussian Quadrature Rule


Basis of the Gaussian Quadrature Rule

Previously, the Trapezoidal Rule was developed by the methodof undetermined coefficients. The result of that development issummarized below.

)(2

)(2

)()()( 21

bfab

afab

bfcafcdxxfb

a



The two-point Gauss Quadrature Rule is an extension of the Trapezoidal Rule approximation where the arguments of the function are not predetermined as a and b but as unknownsx1 and x2. In the two-point Gauss Quadrature Rule, the integral is approximated as

b

a

dx)x(fI )x(fc)x(fc 2211



The four unknowns x1, x2, c1 and c2 are found by assuming that the formula gives exact results for integrating a general third order polynomial, .xaxaxaa)x(f 3

32

210 Hence

b

a

b

a

dxxaxaxaadx)x(f 33

2210

b

a

xa

xa

xaxa

432

4

3

3

2

2

10

432

44

3

33

2

22

10

aba

aba

abaaba



It follows that

3232222102

313

2121101 xaxaxaacxaxaxaacdx)x(f

b

a

Equating Equations the two previous two expressions yield

432

44

3

33

2

22

10

aba

aba

abaaba

3232222102

313

2121101 xaxaxaacxaxaxaac

3223113

222

211222111210 xcxcaxcxcaxcxcacca



Since the constants a0, a1, a2, a3 are arbitrary

21 ccab 2211

22

2xcxc

ab

222

211

33

3xcxc

ab

322

311

44

4xcxc

ab


Basis of Gauss Quadrature

The previous four simultaneous nonlinear Equations have only one acceptable solution,

21

abc

22

abc

23

1

21

ababx

23

1

22

ababx


Basis of Gauss Quadrature

Hence Two-Point Gaussian Quadrature Rule

23

1

2223

1

22

)( 2211

ababf

abababf

ab

xfcxfcdxxfb

a


Higher Point Gaussian Quadrature Formulas


Higher Point Gaussian Quadrature Formulas

)()()()( 332211 xfcxfcxfcdxxfb

a

is called the three-point Gauss Quadrature Rule.

The coefficients c1, c2, and c3, and the functional arguments x1, x2, and x3

are calculated by assuming the formula gives exact expressions for

b

a

dxxaxaxaxaxaa 55

44

33

2210

General n-point rules would approximate the integral

)x(fc.......)x(fc)x(fcdx)x(f nn

b

a

2211

integrating a fifth order polynomial


Arguments and Weighing Factors for n-point Gauss

Quadrature Formulas

In handbooks, coefficients and

Gauss Quadrature Rule are

1

1 1

n

iii )x(gcdx)x(g

as shown in Table 1.

Points

WeightingFactors

FunctionArguments

2 c1 = 1.000000000

c2 = 1.000000000

x1 = -0.577350269x2 = 0.577350269

3 c1 = 0.555555556c2 = 0.888888889c3 = 0.555555556

x1 = -0.774596669x2 = 0.000000000x3 = 0.774596669

4 c1 = 0.347854845c2 = 0.652145155c3 = 0.652145155c4 = 0.347854845

x1 = -0.861136312x2 = -0.339981044x3 = 0.339981044x4 = 0.861136312

arguments given for n-point

given for integrals

Table 1: Weighting factors c and function arguments x used in Gauss Quadrature Formulas.



Quadrature Formulas

Points WeightingFactors

FunctionArguments

5 c1 = 0.236926885c2 = 0.478628670c3 = 0.568888889c4 = 0.478628670c5 = 0.236926885

x1 = -0.906179846x2 = -0.538469310x3 = 0.000000000x4 = 0.538469310x5 = 0.906179846

6 c1 = 0.171324492c2 = 0.360761573c3 = 0.467913935c4 = 0.467913935c5 = 0.360761573c6 = 0.171324492

x1 = -0.932469514x2 = -0.661209386

x3 = -0.2386191860

x4 = 0.2386191860x5 = 0.661209386x6 = 0.932469514

Table 1 (cont.) : Weighting factors c and function arguments x used in Gauss Quadrature Formulas.



Quadrature Formulas

So if the table is given for

1

1

dx)x(g integrals, how does one solve

b

a

dx)x(f ? The answer lies in that any integral with limits of b,a

can be converted into an integral with limits 11, Let

cmtx

If then,ax 1t

,bx If then 1tSuch that:

2

abm



Quadrature Formulas

2

abc

Then Hence

22

abtab

x

dtab

dx2

Substituting our values of x, and dx into the integral gives us

1

1 222)( dt

ababtab

fdxxfb

a


Example 1

For an integral

derive the one-point Gaussian Quadrature

Rule.

,dx)x(fb

a

Solution

The one-point Gaussian Quadrature Rule is

11 xfcdx)x(fb

a


SolutionThe two unknowns x1, and c1 are found by assuming that the formula gives exact results for integrating a general first order polynomial,

.)( 10 xaaxf

b

a

b

a

dxxaadxxf 10)(

b

a

xaxa

2

2

10

2

22

10

abaaba


Solution

It follows that

1101)( xaacdxxfb

a

Equating Equations, the two previous two expressions yield

2

22

10

abaaba 1101 xaac )()( 11110 xcaca



Since the constants a0, and a1 are arbitrary

1cab

11

22

2xc

ab

abc 1

21

abx

giving


Solution

Hence One-Point Gaussian Quadrature Rule

2)()( 11

abfabxfcdxxf

b

a


Example 2

Use two-point Gauss Quadrature Rule to approximate the distance

covered by a rocket from t=8 to t=30 as given by

30

8

892100140000

1400002000 dtt.

tlnx

Find the true error, for part (a).

Also, find the absolute relative true error, for part (a).a

a)

b)

c)

tE


Solution

First, change the limits of integration from [8,30] to [-1,1]

by previous relations as follows

1

1

30

8 2

830

2

830

2

830dxxfdt)t(f

1

1

191111 dxxf


Solution (cont)

Next, get weighting factors and function argument values from Table 1

for the two point rule,

00000000011 .c

57735026901 .x

00000000012 .c

57735026902 .x


Solution (cont.)

Now we can use the Gauss Quadrature formula

191111191111191111 2211

1

1

xfcxfcdxxf

1957735030111119577350301111 ).(f).(f

).(f).(f 350852511649151211

).().( 481170811831729611

m.4411058


Solution (cont)

since

).(.).(

ln).(f 64915128964915122100140000

14000020006491512

8317296.

).(.).(

ln).(f 35085258935085252100140000

14000020003508525

4811708.


Solution (cont)

The absolute relative true error, t, is (Exact value = 11061.34m)

%.

..t 100

3411061

44110583411061

%.02620

c)

The true error, , isb) tEValueeApproximatValueTrueEt

44.1105834.11061

m9000.2

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