Numerical Integration

ENGINEERING MATHEMATICS 4ENGINEERING MATHEMATICS 4(BWM 30603 / BDA34003)

Lecture Module 6: Numerical Integration

Waluyo Adi SiswantoUniversiti Tun Hussein Onn Malaysia

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Lecture Module 6 Engineering Mathematics IV 2

Topics

Rectangular rule Trapezoidal rule Simpson's rule

Simpson 1/3 Simpson 3/8

Gauss Quadrature 2-point 3-point


x

yy (x)

x1 xn

You want to calculatethe yellow area,under the curve y(x)

Then you can calculatethe area

Area=x1

xn

y (x)dx


x

y

y (x)

x1 xn

If the function is difficultYou will find it difficult to integrate.

Then you can predictby approximation

Area=Area1+Area 2+Area 3

Area=h y1+h y2+h yn1

Or you can write

Area=h ( y1+ y2+ yn1)

Of course not accurate.If you use smaller division ? smaller division ?

y1

y2

yn1


Rectangular rule

x

y

y1

x1

y2

y (x)

h

xn

If you want to divide by N division

h=xnx1N

x1

xn

y (x)dx = h( y1+ y2++ yn1)

h

yn1

h

Number of data n= N+1


Trapezoidal rule

x

y

y1

x1

y2

y (x)

h

xn

If you want to divide by N division

h=x nx1N

h

yn1

h

x1

xn

y(x)dx = h2( y1+ y2)+

h2( y2+ y3)++

h2( y n1+ yn)

Now the area is not rectangularbut trapezoidaltrapezoidal

x1

xn

y (x)dx = h2( y1+2 y2++2 yn1+ yn)

y3

yn


Trapezoidal m function (trapezoidal.m):

function I = trapezoidal(f_str, a, b, n)%TRAPEZOIDAL Trapezoidal Rule integration.% I = TRAPEZOIDAL(F_STR, A, B, N) returns the Trapezoidal Rule approximation% for the integral of f(x) from x=A to x=B, using N divisions (subintervals), where% F_STR is the string representation of f.I=0;g = inline(f_str);h = (b-a)/n;I = I + g(a);for ii = (a+h):h:(b-h) I = I + 2*g(ii);endI = I + g(b);I = I*h/2;


Simpson's rule

Simpson 1/3

x1

xn

y( x)dx = h3( y1+4 y2+2 y3+4 y4+2 y5++4 yn1+ yn)

Simpson 3/8

x1

xn

y( x)dx = 3h8( y1+3( y2+ y3+ y5+ y6+)+2( y4+ y7+)+ yn)

The number of division must be multiplication of 2

The number of division must be multiplication of 3


Simpson 1/3 m function (simpson13.m):function I = simpson13(f_str, a, b, n)%SIMPRULE Simpson's rule integration.% I = SIMPRULE(F_STR, A, B, N) returns the Simpson's rule approximation% for the integral of f(x) from x=A to x=B, using N subintervals, where% F_STR is the string representation of f.% An error is generated if N is not a positive, even integer.I=0;g = inline(f_str);h = (b-a)/n;

if((n > 0) && (rem(n,2) == 0)) I = I + g(a); for ii = (a+h):2*h:(b-h) I = I + 4*g(ii); end for kk = (a+2*h):2*h:(b-2*h) I = I + 2*g(kk); end I = I + g(b);

I = I*h/3;else disp('Incorrect Value for N')end


Simpson 3/8 m function (simpson38.m):

function int=simpson38(f_str,x1,x2,n)h=(x2-x1)/n;x(1)=x1;f=inline(f_str);

if((n > 0) && (rem(n,3) == 0)) sum=f(x1); for i=2:n x(i)=x(i-1)+h; end for j=2:3:n sum=sum+3*f(x(j)); end for k=3:3:n sum=sum+3*f(x(k)); end for l=4:3:n sum=sum+2*f(x(l)); end sum=sum+f(x2); int=sum*3*h/8;else disp('Incorrect Value for N')end


Example 6-1

Find the approximate value of

By using :

a) Exact integration, use SMath

b) Rectangular rule, 12 division

c) Trapezoidal rule, 12 division

d) Simpson 1/3, 12 division

e) Simpson 3/8, 12 division

1

4 x x+1

dx


Exact integration in SMath

Rectangular rule

N = 12h = 0.25

No data x x/sqrt(x+1)1 1 0.707112 1.25 0.833333 1.5 0.948684 1.75 1.055295 2 1.154706 2.25 1.248087 2.5 1.336318 2.75 1.420099 3 1.5000010 3.25 1.5764811 3.5 1.6499212 3.75 1.7206213 4

15.15060 Result=0.25(15.15060)=3.78765


Trapezoidal rule

N = 12h = 0.25

No data x x/sqrt(x+1)1 1 0.707112 1.25 0.833333 1.5 0.948684 1.75 1.055295 2 1.154706 2.25 1.248087 2.5 1.336318 2.75 1.420099 3 1.5000010 3.25 1.5764811 3.5 1.6499212 3.75 1.7206213 4 1.78885

2.49596 14.44350

Result=0.252

(2.49596+2(14.44350))=3.9229

In FreeMat:


Simpson 1/3 ruleN = 12h = 0.25

No data x x/sqrt(x+1)1 1 0.707112 1.25 0.833333 1.5 0.948684 1.75 1.055295 2 1.154706 2.25 1.248087 2.5 1.336318 2.75 1.420099 3 1.5000010 3.25 1.5764811 3.5 1.6499212 3.75 1.7206213 4 1.78885

2.49596 7.85389 6.58961

Result=0.253

(2.49596+4(7.85389)+2(6.48961))=3.9242

In FreeMat :


Simpson 3/8 rule

N = 12h = 0.25

No data x x/sqrt(x+1)1 1 0.707112 1.25 0.833333 1.5 0.948684 1.75 1.055295 2 1.154706 2.25 1.248087 2.5 1.336318 2.75 1.420099 3 1.5000010 3.25 1.5764811 3.5 1.6499212 3.75 1.7206213 4 1.78885

2.49596 10.47542 3.96808

Result=(3)0.258

(2.49596+3(10.47542)+2(3.96808))=3.9242

In FreeMat :


Example 6-2A racing car velocity record from start to 12 seconds is shown below.You have to calculate the distance of the car from its start position in 12 seconds.The data is taken every 1 second.Use Simpson's rule 1/3.

0 2 4 6 8 10 120

20

40

60

80

100

120

140

160

180

0

5060

7075

8090

110

150160

170 170 170

time (seconds)

spee

d (k

m/h

)


N = 12h= 1/3600

no t v(t)1 0 02 1 503 2 604 3 705 4 756 5 807 6 908 7 1109 8 15010 9 16011 10 17012 11 17013 12 170

170 640 545

Result= 133600

(170+4(640)+2(545))=0.3537 km


Gauss QuadratureThe main idea in Gauss QuadratureGauss Quadrature is to change the integration limits tonatural (dimensionless) coordinate limits from -1 to 1

x

y y (x)

x1 xn

()

1 1

Change tonaturalcoordinates


Gauss QuadratureThe main idea in Gauss QuadratureGauss Quadrature is to change the integration limits tonatural (dimensionless) coordinate limits from -1 to 1

x

y y (x)

x1 xn

()

1 1

I=xnx1

2 11

()d

x=12 [(1) x1+(1+) xn ]

()= y (x)

I=xnx1

2I


I =1

1

()d

I = R1(1) + R2(2) ++ Rn(n)

j is the location of the integration point j relative to the center

R j is the weighting factor for point j relative to the center,and n is the number of points at which is to be calculated()


Gauss QuadratureCoefficients for Gaussian Quadrature

jn R j

1 0.0 2.0

2 0.5773502692 1.0

3 0.7745966692 0.5555555560.0 0.888888889

4 0.8611363116 0.34785484510.3399810436 0.6521451549

5 0.9061798459 0.23692688510.5384693101 0.4786286705

0.0 0.5688888889


Program to generate abscissa and weight for any number of integration pointsfunction [x,A] = GaussNodes(n,tol)% USAGE: [x,A] = GaussNodes(n,tol)% n = order of integration points% tol = error tolerance (default is 1.0e4*eps).format long;if nargin < 2; tol = 1.0e4*eps; endA = zeros(n,1); x = zeros(n,1); nRoots = fix(n + 1)/2; for i = 1:nRoots t = cos(pi*(i - 0.25)/(n + 0.5)); for j = i: 30 p0 = 1.0; p1 = t; for k = 1:n-1 p = ((2*k + 1)*t*p1 - k*p0)/(k + 1); p0 = p1;p1 = p; end dp = n *(p0 - t*p1)/(1 - t^2); dt = -p/dp; t = t + dt; if abs(dt) < tol x(i) = t; x(n-i+1) = -t; A(i) = 2/(1-(t^2))/(dp^2); A(n-i+1) = A(i); break end end if nRoots == 1 x(i) =0; endend


Program to use Gauss Quadrature, any number of integration points

function I = GaussQuadrature(func,a,b,n)% USAGE: I = gaussQuad(func,a,b,n)% func = handle of function to be integrated.% for example --> func= @(x) ((sin(x)/x)^2)% a,b = integration limits.% n = order of integration points% I = integral resultformat long;c1 = (b + a)/2; c2 = (b - a)/2;[x,A] = GaussNodes(n); sum = 0;for i = 1:length(x) y = feval(func,c1 + c2*x(i)); sum = sum + A(i)*y;endI = c2*sum;


()

1 1

Gauss Quadrature1 point

weighting2.0

I = R1(1)

0.0

I = 2.0 (0)


()

1 1

Gauss Quadrature2 points

weighting1.0

I = R1(1) + R2(2)

I = 1.0 (0.5773502692) + 1.0 (0.5773502692)

0.57

7350

2692

1.0

-0.5

7735

0269

2

0.0


()

1 1

Gauss Quadrature3 points

weighting

I = R1(1) + R2(2) + R3(3)

I = 0.555555556 (0.7745966692) +0.888888889 (0.0) +0.555555556 (0.7745966692)

0.77

4596

6692

-0.7

7459

6669

2

0.0

0.8888888890.5555555560.555555556


Example 6-3


By using :

a) Gauss Quadrature 2 points

b) Gauss Quadrature 3 points

1

4 x x+1

dx


Example 6-4


By using :

a) Gauss Quadrature 4 points

b) Check your result in Smath (Exact integration)

c) Check your result, using Freemat (GaussQuadrature)

1

41+x2

x3+1dx


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Numerical Integration