Numerical Integration

Numerical Integration
• Newton-Cotes Numerical Integration Scheme
• See Chapter 10 of E-book – clear and easy to understand
d
derivation of formulae for Trapezoidal & Simpson’s rules
• Replace complicated function (or tabulated data) with some
approximating function that is easy to integrate
n
I
b
= ∫ f
(
x
)
dx
≅ ∫ f
(
x
)
dx
a
b
= ∫ ∫
a
n
n−1
n−
1
f ( x)
= a + a x + L+
a x +
0
1
a
n
x
n
© 2003-2006 Roberto Muscedere 1

Trapezoidal Rule
• Result of integration is
bounded by f(x) and the x
axis between x=a and x=b
• Can be approximated
numerically by dividing the
region into small segments
each of width ∆x
• The area of each segment
can be approximated by a
trapezoid (approximating
each segment of the curve
by a straight line)
© 2003-2006 Roberto Muscedere 2

Trapezoidal Rule
• Trapezoidal Rule is the first of the Newton-
Cotes closed integration formulas (n=1)
I
b
= ∫ f x)
dx ≅
∫ ( )
a
b
( f x dx
1
f ( b)
− f ( a)
f ( x)
= f ( a)
+
( x − a)
1 b − a
• f 1 (x) () is a straight line
a
© 2003-2006 Roberto Muscedere 3

Trapezoidal Rule
• The area under this straight line is an
estimate of the integral of f(x) between the
limits of a and b
I
b
= ∫
a
⎡
⎢
⎣
f
f ( b)
− f ( a)
⎤
(
a
)
+
(
x −
a
)
dx
b − a
⎥
⎦
I
≅
( b
−
a)
f
(
b
)
+
2
f
( a
)
© 2003-2006 Roberto Muscedere 4

Trapezoidal Rule
• Same result if we
calculated l trapezoid
graphically
I
≅
( b
I
⎛
f
(
b
)
−
f
(
a
)
− a)
⎜
⎝ 2
f
(
b
)
+
f
≅ ( b − a)
2
+
f
(
a
)
⎞
( a)
⎟
⎠
I
⎛ f ( b)
− f ( a)
⎞
≅ ( b − a)
⎜ + f ( a)
⎟ ⎝
2
⎠
f ( b)
+ f ( a)
I ≅ ( b − a)
2
© 2003-2006 Roberto Muscedere 5

Trapezoidal Rule: Example
• Numerically integrate:
f
( x
) = 0.2 + 25x
− 200x
+ 675x
2
3
−
900x
4
+
400x
5
a = 0, b = 0.8
(Analytical answer = 1.6405333)
© 2003-2006 Roberto Muscedere 6

Trapezoidal Rule: Example
f
(0)
= 0.2
f
(0.8)
=
0.232
I
≅
0.2
0.232
⎞
0 .8
⎜
⎛ +
⎟ =
⎝ 2 ⎠
0.1728
• Significant difference from analytical solution
of 1.6405333
© 2003-2006 Roberto Muscedere 7

Trapezoidal Rule: Example
• Why so bad?
© 2003-2006 Roberto Muscedere 8

Trapezoidal Rule
• Add a series of intervals
b − a
= x = a x = b
n
n
h
0
x
1
x
2
x n
I
=
∫
f + + +
( x
) dx
∫
f
( x
) dx L ∫
f
( x
) dx
x
0
x
1
x n− 1
© 2003-2006 Roberto Muscedere 9

Trapezoidal Rule
• Substituting the trapezoidal rule for each
integral yields:
f
( x
) +
f
( x
)
f
( x
) +
f
( x
)
I ≅ h
0 1
+ h
1 2
+
2
2
f ( x
L+ + h
n−
)
+
f
( x
1 n
2
)
© 2003-2006 Roberto Muscedere 10

• Grouping the terms:
Trapezoidal Rule
I
≅
h
⎡
⎢
⎣
f
n−1
⎤
( x ) + 2∑
f ( x ) + f ( x )
0 i n ⎥
i=11
⎦
2 i
I
≅
x
− x
n −1
n 0
f ∑
2n
i
⎡
⎢
⎣
⎤
(
x
)
+
2∑
f
(
x
)
+
f
(
x )
0 i n
⎥
=1
⎦
© 2003-2006 Roberto Muscedere 11

Trapezoidal Rule: Example
• Numerically integrate (using two segments, n=2):
f
( x
) = 0.2 + 25x
− 200x
+ 675x
2
3
−
900x
4
+
400x
5
a = 0, b = 0.8, n = 2
(Analytical answer = 1.6405333)
© 2003-2006 Roberto Muscedere 12

Trapezoidal Rule: Example
f
(0)
=
0.2
f
(0.4)
=
2.456
f
(0.8)
=
0.232
I
≅
⎛ 0.2 + 2(2.456) + 0.232 ⎞
0 .8
⎜
⎟ =
⎝ 4 ⎠
1.0688
• Significantly better than with one segment of
the trapezoidal rule (0.1728)
© 2003-2006 Roberto Muscedere 13

Trapezoidal Code (p.1)
class Integration {
private:
double x0, xn; // limits of integration
double (*f)(double x); //function to be integrated
public:
Integration(double (*F)(double x), double a,
double b) {
f=F;
x0=a;
xn=b;}
double Trapezoidal(int n);
};
© 2003-2006 Roberto Muscedere 14

Trapezoidal Code (p.2)
double Integration::Trapezoidal(int n)
{
double h=(xn-x0)/n;
double sum=0.0;
double x=x0;
double I;
for(int i=1; i

Trapezoidal Code (p.3)
double FUN(double x) // User supplied function
{
return 0.2 + 25*x - 200*x*x + 675*x*x*x - 900*x*x*x*x
+ 400*x*x*x*x*x;
}
int main()
{
Integration I(FUN,0,0.8);
0 int i;
for(i=1;i

Trapezoidal Code -Output
n=1, I=0.1728
n=2, I=1.0688
n=3, I=1.36957
n=4, I=1.4848
n=5, I=1.53988
n=6, I=1.57027
n=7, I=1.58874
n=8, I=1.6008
n=9, I=1.60909
n=10, I=1.61504
n=11, I=1.61945
n=12, I=1.62281
n=13, I=1.62542
n=14, I=1.6275
n=15, I=1.62918
n=16 16, I=1 1.63055
© 2003-2006 Roberto Muscedere 17

Simpson’s 1/3 Rule
• Simpson’s Rule is the second of the Newton-
Cotes closed integration formulas (n=2) using
Lagrange Interpolating Polynomials
f
x
2 2
∫
f
x
dx
≅
∫
I =
(
)
f
(
x
)
dx
2
n
x
( x)
x
x
0 0
n
n
x −
= ∑ f ( x ) ∏ −
i
i=
0 j=
0
j≠i
x
x x i
© 2003-2006 Roberto Muscedere 18
j
j

Simpson’s 1/3 Rule
( x − x1
)( x − x2)
( x − x0)(
x − x2)
f
2
(
x
) =
f
(
x
0
)
+
f
(
x
1
)
+
( x − x )( x − x ) ( x − x )( x − x )
0
1
0
( x −
( x −
2
2
x
x
)( x
)( x
− x1
)
− x )
0 f
(
x2
0 2 1
• After integration ti and algebraic manipulation:
I
h
3
0
1
+
x −
2
x
h = 0
2
≅
[ f
(
x
)
+
4
f
(
x
) +
f
(
x
)
]
© 2003-2006 Roberto Muscedere 19
1
0
)
2
1
2

Trapezoidal Rule: Example
• What does this represent?
© 2003-2006 Roberto Muscedere 20

Simpson’s 1/3 Rule: Example
• Numerically integrate:
f
( x
) = 0.2 + 25x
− 200x
+ 675x
2
3
−
900x
4
+
400x
5
x 0 = 0, x 2 = 0.8
(Analytical answer = 1.6405333)
© 2003-2006 Roberto Muscedere 21

Simpson’s 1/3 Rule: Example
f
(0)
=
0.2
f
(0.4)
=
2.456
f
(0.8)
=
0.232
I
≅
⎛ 0.2 + 4(2.456) + 0.232 ⎞
0 .8
⎜
⎟ =
⎝ 6 ⎠
1.36746667
• Better than with Trapezoidal Rule with two
segments (1.0688)
© 2003-2006 Roberto Muscedere 22

Simpson’s 1/3 Rule
• Add a series of intervals
x − x
h =
2
0
n
x
2
x
4
x n
I
=
∫
f + + +
( x
) dx
∫
f
( x
) dx L ∫
f
( x
) dx
x
0
x
2
x n− 2
© 2003-2006 Roberto Muscedere 23

Simpson’s 1/3 Rule
• Substituting Simpson’s 1/3 rule for each
integral yields:
I
≅
+
f
(
x
)
+
4
f
(
x
)
+
f
(
x
0 1
2 h
2
f ( x ) + 4 f ( x ) + f ( x )
2h
2 3 4
6
f ( x ) + 4 f ( x
+ L
+ 2hh
n−
n−
6
6
)
)
+
( x
2 1 n
f
)
© 2003-2006 Roberto Muscedere 24

Simpson’s 1/3 Rule
• Grouping the terms:
I
≅
⎛
⎜
⎝
x
−
x
⎞
⎟
⎠
⎡
⎢
⎢
f
n−1
∑
( x ) + 4 f ( x )
0 i
n 0 ⎢
i
=
1,3,5
3n
⎢
n − 2
⎢
+ 2
∑
f
( x
)
j= 2,4, 6
⎢
⎣
+
f
( x
j n
⎤
⎥
⎥
⎥
)
⎥
⎦
• n must be even
© 2003-2006 Roberto Muscedere 25

Simpson’s 1/3 Rule: Example
• Numerically integrate:
f
( x
) = 0.2 + 25x
− 200x
+ 675x
2
3
−
900x
4
+
400x
5
x 0 = 0, x 2 = 0.8, n = 4
(Analytical answer = 1.6405333)
© 2003-2006 Roberto Muscedere 26

I
f
Simpson’s 1/3 Rule: Example
(0.4)
≅
f
( 0) = 0.2 f (0.2) = 1.288
= 2.456 f (0.6) = 3.464 f
(0.8)
0.232
⎛ 0.2
+
4(1.288
+
3.464)
+
2(2.456)
+
0.232
⎞
0.8⎜
⎟
⎝
12
⎠
=1.62346667
• Better than with Trapezoidal Rule with eight
segments (1.6008)
=
© 2003-2006 Roberto Muscedere 27

Simpson’s 1/3 Code (p.1)
class Integration {
private:
double x0, xn; // limits of integration
double (*f)(double x); //function to be integrated
public:
Integration(double (*F)(double x), double a,
double b) {
f=F;
x0=a;
xn=b;}
double Trapezoidal(int n);
double Simpsons13(int n);
};
© 2003-2006 Roberto Muscedere 28

Simpson’s 1/3 Code (p.2)
double Integration::Trapezoidal(int n)
{
double h=(xn-x0)/n;
double sum=0.0;
double x=x0;
double I;
for(int i=1; i

Simpson’s 1/3 Code (p.3)
double Integration::Simpsons13(int n)
{
double h=(xn-x0)/n;
double sum=0.0;
double x=x0;
double I;
for(int i=1; i

Simpson’s 1/3 Code (p.4)
double FUN(double x) // User supplied function {
return 0.2 + 25*x - 200*x*x * + 675*x*x*x * * - 900*x*x*x*x* * *
+ 400*x*x*x*x*x; }
int main() {
Integration I(FUN,0,0.8);
int i;
for(i=2;i

Simpson’s 1/3 Code -Output
n=2, I(trap)=1.0688, I(simp)=1.36747
n=4, I(trap)=1.4848, I(simp)=1.62347
n=6, I(trap)=1.57027, I(simp)=1.63716
n=8, I(trap)=1.6008, I(simp)=1.63947
n=10, I(trap)=1.61504, I(simp)=1.6401
n=12, I(trap)=1.62281, I(simp)=1.64032
n=14, I(trap)=1.6275, I(simp)=1.64042
n=16, I(trap)=1.63055, I(simp)=1.64047
© 2003-2006 Roberto Muscedere 32