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8/10/2019 Numerical Integration Eng
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Numerical Integration
Chapter 4
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Numerical Integration
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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Numerical Integration
Cuadraturenumerical computing of
the simple integrals;
Cubature- numerical computing of the
dubleintegrals.
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Numerical Integration
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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Numerical Integration4.1Newton-Cotes Cuadrature Formulae
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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Numerical Integration4.1Newton-Cotes Cuadrature Formulae
Lagrange interpolation formula is now:
n
1i
iin
n
ij
1n y
)!in()!1i()1(
1jq
xL
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Numerical Integration4.1Newton-Cotes Cuadrature Formulae
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
Numerical Integration4.1Newton-Cotes Cuadrature Formulae
where,
n
1i
ii
b
a
yAdxxf
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Numerical Integration4.1Newton-Cotes Cuadrature Formulae
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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Numerical Integration4.1Newton-Cotes Cuadrature Formulae
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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Numerical Integration4.2Trapezoidal Rule
NC Formula
n
1i ii
b
a
yH)ab(dxxf
)yy(2hdxxf 21
b
a
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Numerical Integration4.2Trapezoidal Rule
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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Numerical Integration4.2Trapezoidal Rule
THE GENERALIZED
TRAPEZOIDAL RULE
By generalizing,
Cuadrature formula ofpractical interest.
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Numerical Integration4.2Trapezoidal Rule
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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Numerical Integration4.2Trapezoidal Rule
where,
f(xi)=yi, i=1, 2, ..., n.
2
y
y2
y
hdx)x(f
n1n
2ii
1
b
a
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Numerical Integration4.2Trapezoidal Rule
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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Numerical Integration4.3Simpsons Rule
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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Numerical Integration4.3Simpsons Rule
The approximation accuracy can be improved by
using the GENERALIZED SIMPSONS RULE.
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Numerical Integration4.3Simpsons Rule
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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Numerical Integration4.3Simpsons Rule
)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