Numerical Integration Eng

8/10/2019 Numerical Integration Eng

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Numerical Integration

Chapter 4


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If

Function f(x)continuous on [a, b];

Its primitive, F(x), is known,

Then, the defined integral,

aFbFdxxfIb

a


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Numerical IntegrationWhen do we need it?

Function f(x)defined by a table;

or

The finding out of f(x) primitive, F(x), by using

analytical methods, involves a computing

significant effort f(x) will be evaluated forseveral arguments, the problem becoming a

first case problem).


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Cuadraturenumerical computing of

the simple integrals;

Cubature- numerical computing of the

dubleintegrals.


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4.1Newton-Cotes Cuadrature Formulae

We are requested to compute the

following defined integral:

b

a

dxxfI


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Numerical Integration4.1Newton-Cotes Cuadrature Formulae

1. The interval [a, b] is divided in n-1 equal subintervals of length

by the points

1n

abh

n,...,2,1i,h)1i(ax i

a=x1b=xnx3x2 xn-1

h h h


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2. Its assumed that the values of the function f(x) are known for the

arguments xi,

The function f(x) will be modeled by Lagrange formula of

interpolation.

n,...,2,1i,xfy

ii

n

1i

in

ij

ji

n

ij j

1n y

xx

xx

xL


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Numerical Integration4.1Newton-Cotes Cuadrature Formulae The xipoints network - EQUIDISTANT

The following notation is introduced:

The produces from the Lagrange interpolation

formula become:

h

xxq 1

n

ij

1nn

ijj 1jqhxx

!)in()!1i(h)1()xx(n

ij

1nin

ji


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Lagrange interpolation formula is now:

n

1i

iin

n

ij

1n y

)!in()!1i()1(

1jq

xL


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3. If its considered that:

The following cuadrature formula results:

b

a

1n

b

a

xd)x(Ldxxf

n

1i

ii

b

a

yAdxxf


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)!in()!1i()1(

qd1jqh

)!in()!1i()1(

xd1jq

Ain

1n

0

n

ijb

a

in

n

ij

i

Change of variable x q


where,

n

1i

ii

b

a

yAdxxf


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The Aicoefficients have the form:

where

Now, the Newton-Cotescuadrature formula can be obtained:

ii H)ab(A

n,...,2,1i,

)1n()!in()!1i()1(

qd1jq

Hin

1n

0

n

ij

i

n

1i

ii

b

a

yH)ab(dxxf


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The HicoefficientsCotes coef f f ic ients

Characterist ics

1.

2.

Remark

The Hicoefficients are independent of:the function to be integrated, f(x);

the integration interval.

1Hn

1i

i

1ini HH


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Numerical Integration4.2Trapezoidal Rule

By considering for the NC coefficients,

n=2, the following values for the Cotes

coefficients are obtained:

1

0

12

1qd)1q(H

n,...,2,1i,)1n()!in()!1i()1(

qd1jq

H in

1n

0

n

ij

i

2

1qdqH

1

0

2


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NC Formula

n

1i ii

b

a

yH)ab(dxxf

)yy(2hdxxf 21

b

a


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NC Formula

n

1i ii

b

a

yH)ab(dxxf

)yy(2

hdxxf 21

b

a

Practically the Trapezoidal Rule is of no interest.


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THE GENERALIZED

TRAPEZOIDAL RULE

By generalizing,

Cuadrature formula ofpractical interest.


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Procedure

The interval [a, b] is divided in n-1 equal subintervals of

length h=(b-a)/(n-1), , by considering n equidistant

points:

The Trapezoidal Rule will be used on each subinterval

[xi, xi+1], i=1, ..., n-1.

1,2,...ni,h)1i(ax i

)yy(2

h...)yy(

2

hdx)x(f n1n21

b

a

b=xna=x1 x3x2 xn-1

hh h


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where,

f(xi)=yi, i=1, 2, ..., n.

2

y

y2

y

hdx)x(f

n1n

2ii

1

b

a


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Geometrically, this formula assumes the replacement of the

functions graph by a polygonal line to link the points (x1,

y1), ..., (xi, yi),..., (xn, yn).


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Numerical Integration4.3Simpsons Rule

Better accuracy than the trapezoidal rule

It comes from the NC formula for n=3

2

0

316

1qd)2q)(1q(

4

1HH

2

0

23

2qd)2q(q

2

1H


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Considering b-a = x3-x1=2h

)yy4y(3

hdx)x(f 321

x

x

3

1

Geometrically, the formula assumes the replacement of the

curve y=f(x) by a parabola y=L2(x)

y=L2(x)

y=f(x)


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The approximation accuracy can be improved by

using the GENERALIZED SIMPSONS RULE.


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Procedure

The interval [a, b] is divided in n-1 equal

subintervals of length h=(b-a)/(n-1), by n

equidistant points, where n is mandatory odd.

On each double subinterval [x1, x3], [x3, x5], ...,

[xn-2, xn], the Simsons Rule is used.

1,2,...ni,h)1i(ax i

b=xna=x1 x3x2 xn-1

hh h


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)yy4y(3

h...)yy4y(

3

h)yy4y(

3

hdx)x(f n1n2n543321

b

a

)y24y(3

hydx

n121

b

a

2/)3n(

1i

1i21 y

2/)1n(

1i

i22 y

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