ES 240: Scientific and Engineering Computation. Chapter 17: Numerical IntegrationIntegration ...

ES 240: Scientific and Engineering Computation. Chapter 17: Numerical Integration

IntegrationIntegration

Definition– Total area within a region

– In mathematical terms, it is the total value, or summation, of f(x) dx over the range from a to b:

I f x a

b dx


Newton-Cotes FormulasNewton-Cotes Formulas

General Idea– replace a complicated function or tabulated data with a polynomial

that is easy to integrate:

– where fn(x) is an nth order interpolating polynomial.

I f x a

b dx fn x a

b dx

dxxaxaxaadxxfdxxfIb

a

nn

b

a n

b

a ... 2

210


Newton-Cotes IllustrationsNewton-Cotes Illustrations

The integrating function can be polynomials for any order - for example, (a) straight lines or (b) parabolas.

The integral can be approximated in one step or in a series of steps to improve accuracy.


The Trapezoidal RuleThe Trapezoidal Rule

Uses straight-line approximation for the function

Uses linear interpolation

2

)( 10

bfafabI

dxaxab

afbfafdxxaadxxfI

b

a

b

a

b

a n


Error of the Trapezoidal RuleError of the Trapezoidal Rule

The error is dependent upon the curvature of the actual function as well as the distance between the points.

Error can thus be reduced by:– breaking the curve into parts or

– using a higher order function


Composite Trapezoidal RuleComposite Trapezoidal Rule

Assuming n+1 data points are evenly spaced, there will be n intervals over which to integrate.

The total integral can be calculated by integrating each subinterval and then adding them together:

n

n

ii

nnnn

x

x n

x

x n

x

x n

x

x n

xfxfxfh

I

xfxfxx

xfxfxx

xfxfxxI

dxxfdxxfdxxfdxxfIn

n

n

1

10

11

2112

1001

22

222

1

2

1

1

00


Trapezoid FunctionsTrapezoid Functions

For inline functions, use the ‘trap functions For tabulated data, use trapz(x,y)

– Matlab built-in function for numerical integration based on trapezoidal rule

– y(x) should be in a tabulated form– can handle unequally spaced data as long as x in ascending order– example:

>> x=[0 .12 .22 .32 .4 .44 .54 .64 .7 .8];

>> y=0.2+25*x-200*x.^2+675*x.^3-900*x.^4+400*x.^5;

>> trapz(x,y)

1.5948


Trapezoid Rule ExamplesTrapezoid Rule Examples

Tabulated

Inline Function

% f(x)=cos(x)+sin(2x) on [0 pi/2]h=(pi/2-0)/10;>> x=0:h:pi/2;>> y=cos(x)+sin(2*x);>> I2=trapz(x,y)1.9897

p=inline('cos(x)+sin(2*x)');>> I3=trap(p,0,pi/2,10)1.9897% plot f(x) and I(x)>> for k=1:11I4(k)=trap(p,0,x(k),20);end;>> plot(x,y,x,I4,'r')


Simpson’s RulesSimpson’s Rules

Increasing the approximation order results in better integration accuracy– Simpson’s 1/3 rule

• based on taking 2nd order polynomial integrations• use two panels (three points) every integral• only for even number of panels

– Simpson’s 3/8 rule is• based on taking 3rd order polynomial integrations• use three panels (four points) every integral• only for three-multiple number of panels


Simpson’s 1/3 RuleSimpson’s 1/3 Rule

Using the Lagrange form for a quadratic fit of three points:– Integration over the three points simplifies to:

Composite

dxxfIx

x n 2

0

fn x x x1 x0 x1

x x2 x0 x2

f x0 x x0 x1 x0

x x2 x1 x2

f x1 x x0 x2 x0

x x1 x2 x1

f x2

2

43

02

210

xxh

xfxfxfh

I

n

n

jj

i

n

ii

i

nnn

xfxfxfxfh

I

xfxfxfh

xfxfxfh

xfxfxfh

I

2

even ,2

1

odd ,1

0

12432210

243

43

43

43

n

abh


Simpson’s 3/8 RuleSimpson’s 3/8 Rule

Basic

Composite

n

abh

3210 338

3

3

0

xfxfxfxfh

dxxfIx

x n

203 xx

h

nnn

b

a n xfxfxfxf

xfxfxfxfxfhdxxfI

125

43210

33...3

3233

8

3


Combined Simpson’s RuleCombined Simpson’s Rule

Combined Simpson’s rule– If n (number of panels) is even, use Simpson’s 1/3 rule– If n is odd, use Simpson’s 3/8 rule once at beginning or

end and use Simpson’s 1/3 rule for the rest of the panels


Error of Simpson’s 1/3 RuleError of Simpson’s 1/3 Rule

If f(x) is a polynomial function of degree 3 or less, Simpson’s rule provides no error.

Use smaller spacing (h decreases) or more panels to reduce the error.

In general, Simpson’s rule is accurate enough for the most of functions f(x) with much less panels compared to that with the trapezoidal rule.


Simpson’s Rule ExampleSimpson’s Rule Example

By Hand

x=[0.1 0.2 0.3 0.4 0.5 0.6 0.7];y=[2.1 1.7 1.6 2.3 2.8 1.7 2.5];0=(0.1/3)*(y(1)+4*y(2)+2*y(3)+ 4*y(4)+2*y(5)+4*y(6)+y(7))>>1.2067

% f(x)=cos(x)+sin(2x) on [0 pi/2] n=10; h=(pi/2-0)/10; % 10panelsx=0:h:pi/2;y=cos(x)+sin(2*x);I2=0;for k=1:2:(n-1)I2=I2+h/3*(y(k)+4*y(k+1)+y(k+2));end2.0001

Fucntions

% define function f(x)p=inline('cos(x)+sin(2.*x)');I3=simps(p,0,pi/2,10)2.0001

• Try different number of panels• Compare with trapezoid rule


LabLab

Ex 17.3

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