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A li d N i lA li d N i lApplied Numerical Applied Numerical AnalysisAnalysisAnalysisAnalysis

Differentiation and IntegrationDifferentiation and IntegrationLecturer: Emad FatemizadehLecturer: Emad FatemizadehLecturer: Emad FatemizadehLecturer: Emad Fatemizadeh

Applied Numerical MethodsApplied Numerical MethodsE. FatemizadehE. Fatemizadeh

Numerical DifferentiationNumerical DifferentiationNumerical DifferentiationNumerical Differentiation

Need for numerical differentiation:Need for numerical differentiation:Need for numerical differentiation:Need for numerical differentiation:•• No explicit function (x(t) No explicit function (x(t) v(t)=?)v(t)=?)•• Too complex functionToo complex function•• Too complex functionToo complex function



Initial Ideas:Initial Ideas:Initial Ideas:Initial Ideas:( ) ( ) ( ) ( ) ( )2 31 1

2! 3!f x h f x hf x h f x h f x′ ′′ ′′′+ = + + + +

( ) ( ) ( ) ( ) ( )21 12! 3!

f x h f xf x hf x h f x

h+ −

′ ′′ ′′′⇒ = + + +

( ) ( ) ( ) ( ) ( )

( ) ( )

2 31 12 3!

1 1

f x h f x hf x h f x h f x

f x f x h

′ ′′ ′′′− = − + − +

− −( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

2

2

1 12 3!1

f x f x hf x hf x h f x

hf x h f x h

f h f

′ ′′ ′′′⇒ = − + −

+ − −′ ′′′


( ) ( ) ( ) ( )2

2 3!f f

f x h f xh

′ ′′′= + +


Forward and Backward EstimationForward and Backward EstimationForward and Backward EstimationForward and Backward Estimation

( ) ( )Forward Estimation:

f x h f x+( ) ( ) ( )

Backward Estimation:

f x h f xf x

h+ −

′ ≈

( ) ( ) ( )f x f x hf x

h− −

′ ≈

( ) ( ) ( )Central:

f x h f x hf x

+ − −′ ≈


( )2

f xh

≈


Richardson Method:Richardson Method:Richardson Method:Richardson Method:

( ) ( ) ( ) ( )2 8 8 2f h f h f h f h+ + + +( ) ( ) ( ) ( ) ( )

( ) ( )54

2 8 8 212

f x h f x h f x h f x hf x

hh f c

− + + + − − + −′ ≈

( )30

h f cE =



Example:Example:Example:Example:( ) ( )cos , 0.8, 0.01f hθ θ θ= = =

MethodMethod ForwardForward BackwardBackward AverageAverage RichardsonRichardson

ErrorError --0.00350.0035 0.00350.0035 1.1956e1.1956e--005005 2.3911e2.3911e--010010



22ndnd DerivativeDerivative22 DerivativeDerivative

( ) ( ) ( ) ( )22f x h f x h f x h f x′′+ + − = + +( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )422

2

2 112

f x h f x h f x h f x

f x h f x f x hf x h f c

h

+ + = + +

+ − + −′′⇒ = +

( ) ( ) ( ) ( )2

122

hf x h f x f x h

f xh

+ − + −′′ ≈



Gauss Method:Gauss Method:Gauss Method:Gauss Method:•• We have a set of We have a set of xxii and and ffii for for ii==00,,11,…,,…,NN

N( ) ( )

( )0

1

Nk

i i ii

n

f x A f

f x x x=

= ∑

•• Now solve for (n+1) unknown parameters.Now solve for (n+1) unknown parameters.( ) 1, , ,f x x x= …



Gauss MethodGauss Method--Example (1Example (1stst):):Gauss MethodGauss Method Example (1Example (1 ):):•• We have (xWe have (xii,f,fii), (x), (xi+1i+1,f,fi+1i+1))

( ) ( ) ( )( ) ( ) ( )

1

11 0i i i

i i

f x Af x Bf x

f x Af x Bf x A B+′ = +

= ⇒ = + = +( ) ( ) ( )( ) ( ) ( )

( ) ( )

1

1 1

1 0

1

1

i i

i i i i

f x Af x Bf x A B

f x x Af x Bf x Ax Bx

f x f x

+

+ +

⇒ + +

= ⇒ = + = +

−( ) ( ) ( )1

1 1

1 i ii

i i i i

f x f xA B f x

x x x x+

+ +

−′= − = ⇒ =

− −



Gauss MethodGauss Method--Example(1Example(1stst):):Gauss MethodGauss Method Example(1Example(1 ):):•• We have (xWe have (xii--11,f,fii--11), (x), (xii,f,fii), (x), (xi+1i+1,f,fi+1i+1))•• xx == h xh x =0 x=0 x =h=h•• xxii--11==--h, xh, xii=0, x=0, xi+1i+1=h=h

( ) ( ) ( ) ( )1 1i i i if x Af x Bf x Cf x+ −′ = + +

( )( ) ( ) ( ) ( )

1 0

1 0

f x A B C

f x x A h B C h

= ⇒ = + +

= ⇒ = − + + +

( ) ( ) ( ) ( )

( ) ( ) ( )

2 2 2

1 1

2 0 0i

i i

f x x x A h B C h

f x f xf + −

= ⇒ = = + +

−′


( ) ( ) ( )1 1

2i i

if xh

+′ =


Gauss MethodGauss Method--Example(2Example(2ndnd):):Gauss MethodGauss Method Example(2Example(2 ):):•• We have (xWe have (xii--11,f,fii--11), (x), (xii,f,fii), (x), (xi+1i+1,f,fi+1i+1))•• xx == h xh x =0 x=0 x =h=h•• xxii--11==--h, xh, xii=0, x=0, xi+1i+1=h=h

( ) ( ) ( ) ( )1 1i i i if x Af x Bf x Cf x+ −′′ = + +

( )( ) ( ) ( ) ( )

1 0

0 0

f x A B C

f x x A h B C h

= ⇒ = + +

= ⇒ = − + + +

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

2 2 2

1 1

2 0

2i i i

f x x A h B C h

f x f x f x+

= ⇒ = + +

− +′′


( ) ( ) ( ) ( )1 12

2i i ii

f x f x f xf x

h+ −+

′′ =


Introduction the Introduction the zz operator:operator:Introduction the Introduction the zz operator:operator:

( )( ) ( )1i iz f x f x +=

( )( ) ( )( )( ) ( )

22

11

i i

i i

z f x f x

z f x f x+

−−

=

=( )( ) ( )( )( ) ( )2

2i iz f x f x−−=



Summary of Method for 1Summary of Method for 1stst derivative:derivative:Summary of Method for 1Summary of Method for 1 derivative:derivative:1z

h−

•

1

1

1 zh

−

−

−•

1

1

24 3

z zh

z z −

−•

− + −

2 1 2

4 328 8

z zh

z z z z− −

+•

− + − +•


12h•


Summary of Method for 2Summary of Method for 2stst derivative:derivative:Summary of Method for 2Summary of Method for 2 derivative:derivative:1

2

2z zh

−− +•

1 2 2 1

2 2

4 5 2 4 5 2h

z z z z z zh h

− − −− + − + − + − +• ↔

2 1 2

2

3 1 3 22

z z z zh

− −− + − − +•

2 1 2

2

16 30 1612

z z z zh

− −− + − + −•



Summary of Method for 3rd derivative:Summary of Method for 3rd derivative:Summary of Method for 3rd derivative:Summary of Method for 3rd derivative:1 2 2 1

3 3

3 3 3 3z z z z z zh h

− − −− + − − + −• ↔

2 1 2

3

2 22

h hz z z z

h

− −− + −•

Summary of Method for 3rd derivative:Summary of Method for 3rd derivative:

2 1 2

4

4 6 4z z z zh

− −− + − +•


h


Differentiation Using Interpolation:Differentiation Using Interpolation:Differentiation Using Interpolation:Differentiation Using Interpolation:•• Find an interpolator or do curve fitting:Find an interpolator or do curve fitting:•• Take DerivativeTake Derivative•• Take Derivative.Take Derivative.



Example (Lagrange Polynomial):Example (Lagrange Polynomial):Example (Lagrange Polynomial):Example (Lagrange Polynomial):

( ) ( ) ( )1 1 1 1 1 1, , , , , ,k k k k k k k k k kx f x f x f x x x x h− − + + + −− = − =

( ) ( )( )( )( )

( )( )( )( )

1 1 11

1 1 1 1 1

k k k kk k

k k k k k k k k

x x x x x x x xf x f f

x x x x x x x x+ − +

−− − + − +

− − − −= +

− − − −

( )( )( )( )

( )( ) ( )( ) ( )( )

11

1 1 1

k kk

k k k k

x x x xf

x x x x−

++ − +

− −+

− −

( ) ( )( ) ( )( ) ( )( )1 1 1 11 12 2 22 2

k k k k k kk k k

x x x x x x x x x x x xf x f f f

h h h+ − + −

− +

− − − − − −= − +



Example (Lagrange Polynomial):Example (Lagrange Polynomial):Example (Lagrange Polynomial):Example (Lagrange Polynomial):

( ) ( ) ( ) ( ) ( )1 1 1 11 12 2 2 20 0

2 2k k k k k k k k

k k k k k

x x x x x x x xf x f f f f

h h h h+ + − −

− +

− − − −′ = + − − + +

( ) ( ) ( ) ( ) ( )1 12 2 22

2 2

2 2k k k k k

h h h hh h h h

f x f f f fh hh h− +

− −′ = − − +

( ) 1 1

2k k

kf ff x

h+ −−′ =



Two Dimensional Case:Two Dimensional Case:Two Dimensional Case:Two Dimensional Case:•• We deal with Gradient:We deal with Gradient:

( ) ( ) ( ) ( ) ( ) ( ), , , , , , or or

2f x h y f x y f x y f x h y f x h y f x h yf

x h h h+ − − − + − −∂

≈∂

( ) ( ) ( ) ( ) ( ) ( )2

, , , , , , or or

2

x h h hf x y h f x y f x y f x y h f x y h f x y hf

y h h h

∂+ − − − + − −∂

≈∂


Applied Numerical MethodsApplied Numerical MethodsE. E. FatemizadehFatemizadeh

Numerical DifferentiationNumerical Differentiation

MatlabMatlab Simple Command: Simple Command: diffdiff((x,nx,n))•• dydy = = diff(x,ndiff(x,n););

dy(kdy(k)= x(k+1))= x(k+1)--x(k)x(k)h = 0.01;

t = (0:h:1) ;

a = sin(2*pi*1*t); % 1Hz sin, from 0 to 1 sec.

da = (2*pi*1)*cos(2*pi*1*t);

df = diff(a,1)/h;

subplot(211), plot(t,da,’b’,t(2:end),df,’r’);

subplot(212), plot(t(2:end),abs(da(2:end)-df));

err = norm(df-da(2:end),’fro’)/norm(da(2:end),’fro’); %0.0314

( )2

1( , ' ')

N

inorm a fro a n

=

∼ ∑




Example (Matlab Example (Matlab DiffDiff))Example (Matlab Example (Matlab DiffDiff))•• 22ndnd order derivativeorder derivative

h = 0.01;

t = (0:h:1) ;

a = sin(2*pi*1*t); % 1Hz sin from 0 to 1 seca = sin(2*pi*1*t); % 1Hz sin, from 0 to 1 sec.

d2a = -((2*pi*1)^2)*sin(2*pi*1*t);

d2f = diff(a,2)/(h*h);

subplot(211), plot(t,d2a,’b’,t(3:end),d2f,’r’);

subplot(212), plot(t(3:end),abs(d2a(3:end)-d2f));

f f f


err = norm(d2f-d2a(3:end),’fro’)/norm(d2a(3:end),’fro’); %0.0622




Matlab Commands: GradientMatlab Commands: GradientMatlab Commands: Gradient.Matlab Commands: Gradient.•• 1D case: dy = gradient(f,h);1D case: dy = gradient(f,h);h 0 01h = 0.01;

t = (0:h:1) ;

a = sin(2*pi*1*t); % 1Hz sin, from 0 to 1 sec.a s ( p t); % s , o 0 to sec

da = (2*pi*1)*cos(2*pi*1*t);

df = gradient(a,h);

subplot(211), plot(t,da,’b’,t,df,’r’);

subplot(212), plot(t,abs(da-df));

err = norm(df da ’fro’)/norm(da ’fro’); %err=6 5784e 004


err = norm(df-da, fro )/norm(da, fro ); %err=6.5784e-004




Matlab Commands: GradientMatlab Commands: GradientMatlab Commands: Gradient.Matlab Commands: Gradient.•• 2D case: [fx,fy] = gradient(f,hx,hy);2D case: [fx,fy] = gradient(f,hx,hy);

[x,y] = meshgrid(-2:.2:2, -2:.2:2);z = x .* exp(-x.^2 - y.^2);[px,py] = gradient(z,.2,.2);contour(z) hold on quiver(px py) hold off

•• 3D case: [fx,fy,fy] = gradient(f,hx,hy,hz);3D case: [fx,fy,fy] = gradient(f,hx,hy,hz);

contour(z),hold on, quiver(px,py), hold off





Matlab Programming:Matlab Programming:Matlab Programming:Matlab Programming:

( ) ( ) ( )1n nn

f x f xf x

h+ −

′ ≈

df = [(x(2:end) x(1:end 1))/h 0];

11 22 33 …… NN--11 NN

( ) ( ) ( )1n nn

f x f xf x

h−−

′ ≈

df = [(x(2:end)-x(1:end-1))/h,0];

h11 22 33 …… NN--11 NN


df =[0,(x(2:end)-x(1:end-1))/h];

Numerical DifferentiationNumerical DifferentiationNumerical DifferentiationNumerical Differentiation( ) ( ) ( )1 1

2n n

n

f x f xf x

h+ −−

′ ≈( )2nf

h

11 22 33 …… NN--11 NN

df =[0, (x(3:end)-x(1:end-2))/(2*h),0];

( ) ( ) ( ) ( )8 8f x f x f x f x+ +( ) ( ) ( ) ( )2 1 1 28 812

n n n nf x f x f x f xh

+ + − −− + − +

11 22 33 NN 22 NN 11 NN

df =[0,0, (-x(5:end)+8*x(4:end-1)-8*x(2:end-3)+x(1:end-4))/(12*h) 0 0];

11 22 33 …… NN--22 NN--11 NN


4))/(12*h),0,0];


Comparison:Comparison:Comparison:Comparison:

( ) ( ) ( )1 0.0324n nn err

f x f xf x

h−−

′ ≈ =⇒

( ) ( ) ( )1 0.0324n nn err

hf x f x

f xh

+ −′ ≈ =⇒

( ) ( ) ( )1 1

26.5784e-004n n

n er

hf x f x

f xh

r+ −−′ ≈ =⇒

( ) ( ) ( ) ( ) ( )2 1 1 28 85.1927e-007

12n n n n

n

f x f x f xer

ff x

hr

x+ + − −− + − +′ ≈ =⇒



Matlab Command: Matlab Command: del2del2Matlab Command: Matlab Command: del2del2•• Discrete Laplacian!Discrete Laplacian!

2 2

L del2(f hx hy);L del2(f hx hy);

2 22

2 2

f ffx y

∂ ∂∇ = +

∂ ∂

•• L=del2(f,hx,hy);L=del2(f,hx,hy);•• L=del2(f,hx,hy,hz);L=del2(f,hx,hy,hz);



Noise Corrupted case:Noise Corrupted case:Noise Corrupted case:Noise Corrupted case:h = 0.01;

t (0:h:1) ; t = (0:h:1) ;

a = sin(2*pi*1*t); % 1Hz sin, from 0 to 1 sec.

Noisya = a + randn(size(a))*0.1;

da = (2*pi*1)*cos(2*pi*1*t);

df = diff(Noisya,1)/h;

subplot(211), plot(t,da,’b’,t(2:end),df,’r’);

subplot(212), plot(t(2:end),abs(da(2:end)-df));




Numerical IntegrationNumerical IntegrationNumerical IntegrationNumerical Integration

Problem Statement:Problem Statement:Problem Statement:Problem Statement:•• Analytical Function Analytical Function –– Analytical SolutionAnalytical Solution

∞

Analytical Function Analytical Function No SolutionNo Solution

( )0

exp( ) 1f x x dx= − =∫

•• Analytical Function Analytical Function –– No SolutionNo Solution

( )12

2exp( )f x x dx= −∫

•• Discrete Data: ECG dataDiscrete Data: ECG data1∫



NewtonNewton--Cotes Method:Cotes Method: ( ) ( )b b

f x dx P x dx≈∫ ∫NewtonNewton Cotes Method:Cotes Method:

T id l th d f t i tT id l th d f t i t

( ) ( )na a

f x dx P x dx≈∫ ∫

Trapezoidal method for two points:Trapezoidal method for two points:

( ) ( ) ( )b b x a x bf x dx f b f a dx− −⎡ ⎤≈ +⎢ ⎥⎣ ⎦∫ ∫( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )a ab

f f fb a a b

f a f b f a f bf x dx b a h

⎢ ⎥− −⎣ ⎦

+ +≈ − =

∫ ∫

∫ ( ) ( )

( ) ( ) ( )3 3

2 2a h

f

b a hE f c f c−

′′ ′′= =

∫


( ) ( )12 12

E f c f c= − = −


Trapezoidal method for N+1 points:Trapezoidal method for N+1 points:Trapezoidal method for N+1 points:Trapezoidal method for N+1 points:( )0 1, , , Nx x x

x x0

0 1 11 2

N

N N

x xhNf f f ff fI h h h

−=

+ ++0 1 11 2

1

2 2 2

2

N N

N

f f f ff fI h h h

hI f f f

−

−

= + + +

⎡ ⎤= + +⎢ ⎥∑

( ) ( ) ( ) ( ) ( ) ( )

01

3 2 20 0

22 N i

i

N N

I f f f

x x x x h b a hE f c f c f α

=

= + +⎢ ⎥⎣ ⎦

− − −′′ ′′ ′′≈ − ≈ − = −

∑


( ) ( ) ( )212 12 12E f c f c f

Nα≈ − ≈ − = −




Simpson (1/3) method for three points:Simpson (1/3) method for three points:Simpson (1/3) method for three points:Simpson (1/3) method for three points:

( ) 2 00 1 2, , ,

2x xx x x h −

=

⎡ ⎤( ) ( )( )

( )( )( )( )( )( )

( )( )( )( )

2

0

1 2 0 2 0 10 1 2

0 1 0 2 1 0 1 2 2 0 2 1

x b

x a

b

x x x x x x x x x x x xf x dx f f f dx

x x x x x x x x x x x x⎡ ⎤− − − − − −

≈ + +⎢ ⎥− − − − − −⎣ ⎦∫ ∫

( ) ( )

( )

0 1 2

54

43

b

a

hf x dx f f f

h

≈ + +∫( ) ( )4

90hE f c= −


Numerical IntegrationNumerical IntegrationNumerical IntegrationNumerical IntegrationSimpson (1/3) method for (N+1) points (N, even):Simpson (1/3) method for (N+1) points (N, even):p ( / ) ( ) p ( , )p ( / ) ( ) p ( , )

( )0 1

0

, , , N

N

x x xx xh −

( ) ( ) ( )

0

0 1 2 2 3 4 2 14 4 4

N

N N N

hN

h h hI f f f f f f f f f

=

= + + + + + + + + +( ) ( ) ( )0 1 2 2 3 4 2 1

1 2

0

4 4 43 3 3

4 23

N N N

N N

N i i

I f f f f f f f f f

hI f f f f

− −

− −

+ + + + + + + + +

⎡ ⎤= + + +⎢ ⎥

⎣ ⎦∑ ∑

( ) ( ) ( )

01,3, 2,4,

44

80

3

1

N i ii i

b a hE f α

= =⎢ ⎥⎣ ⎦

−≈ −

∑ ∑… …


( )801

Numerical IntegrationNumerical IntegrationNumerical IntegrationNumerical IntegrationSimpson (3/8) method for 4 points:Simpson (3/8) method for 4 points:p ( / ) pp ( / ) p

( )

[ ]0 1 2 3

3, , ,

3 3

x x x xhI f f f f[ ]

( ) ( )

0 1 2 3

54

3 3

38

I f f f f

hE f c

= + + +

≈ −

Another methods:Another methods:

( )80

E f c≈

[ ] ( ) ( )674 87 32 12 32 7hI f f f f f E h f[ ] ( ) ( )

[ ] ( ) ( )

670 1 2 3 4

670 1 2 3 4 5

7 32 12 32 7 ,90 9455 27519 75 50 50 75 19 ,

I f f f f f E h f c

hI f f f f f f E h f c

= + + + + = −

= + + + + + = −


[ ] ( )0 1 2 3 4 519 75 50 50 75 19 ,288 12096

I f f f f f f E h f c+ + + + +


Romberg Method:Romberg Method:Romberg Method:Romberg Method:( ) ( ) 2

0 0 0I . . : Trapezoidal/Simpson Method Errorb

a

h h f x dx I h c h= ⇒ = +∫

( ) ( ) ( )20 0 0II. 2 2 2

b

ab

h h f x dx I h c h= ⇒ = +∫

( ) ( ) ( )

( ) ( )

20 0 0III. 4 4 4

2

b

a

h h f x dx I h c h

I h I h

= ⇒ = +

−

∫( ) ( )

( )( ) ( )

0 02 20

22

2I. &II.

1 1 2

2b

I h I hc

h

I h I hh

−⇒ =

−

−∫


( ) ( ) ( ) ( )( ) ( )20 00

0 02 2 20

2II. 2 2 Less Error

2 1 1 2a

I h I hhf x dx I h c hh

′⇒ = + +−∫


Romberg Method:Romberg Method:Romberg Method:Romberg Method:

( ) ( ) ( ) ( ) ( )20 00 02

22 2 Less Error

b I h I hf x dx I h c h

−′⇒ = + +∫ ( ) ( ) ( )

( ) ( )

0 02

2

2 1II. and III.:

2 / 2

a

b

f

I h I h

−∫

( ) ( ) ( ) ( )( )

220 02 2

0 02

2 / 22 2

2 11

b

a

I h I hf x dx I h c h

−′′= + +

−∫

n

1Better Estimate=More Accurate+ (More Accurate- Less Accurate)2 -1



Romberg Method Example:Romberg Method Example:Romberg Method, Example:Romberg Method, Example:2

1.5

0 2

0.652

bx b aI e dx h

=− −

= ⇒ = =∫

( ) ( ) ( ) ( )( )0.2

First Estimate: 2 0.662112

0 65

a

hI h f a f a h f b

=

= + + + =

( ) ( ) ( ) ( ) ( ) ( )( )

0.65Second Estimate: h=2

2 2 2 2 3 0 65948hI h f a f a h f a h f a h f b

⇒

+ + + + + + +( ) ( ) ( ) ( ) ( ) ( )( )2 2 2 2 3 0.659482

0.65947 0.66211Better Estimate: 0.65947 0.658594 1

I h f a f a h f a h f a h f b= + + + + + + + =

−+ =


4 1−


Cubic Spline:Cubic Spline:Cubic Spline:Cubic Spline:

1 :i ix x x +≤ ≤

( ) ( ) ( ) ( )

( )

3 2

4 3 2N

i i i i i i i

x N N N N

f x a x x b x x c x x d

h h hf x dx a b c h d

= − + − + − +

+ + +∑ ∑ ∑ ∑∫ ( )1

1 1 1 14 3 2i i i ii i i ix

f x dx a b c h d= = = =

= + + +∑ ∑ ∑ ∑∫



Gauss Method (1):Gauss Method (1):Gauss Method (1):Gauss Method (1):( ){ }

( )

0,

ni i i

b n

x f

f d A f b

=

∑∫ ( ) 00

1

i i nia

b n

f x dx A f x a b x

f fdx b a A

=

= ≤ < ≤

⎫⇒ ⎪

∑∫

∑∫

{ }

0

2 2

1 iia

b n

ni i

f fdx b a A

b af x fdx A x

=

= ⇒ = − = ⎪⎪⎪− ⎪= ⇒ = = ⎪⎬

∑∫

∑∫ { }0 0

1 1

2 ni ii ia i

b n n n

f x fdx xA

b

= =

+ +

⇒ ⎪ ⇒⎬⎪⎪⎪

∑∫


1 1

01

n n nn n

i iia

b af x fdx A xn

+ +

=

⎪−= ⇒ = = ⎪

+ ⎪⎭∑∫


Gauss Method Example:Gauss Method Example:Gauss Method, Example:Gauss Method, Example:( ) ( ) ( )4

2 3

1, 0.3679 , 2, 0.0183 , 3,1.2341 10−×

22.3

1.25

1 05 0 1930

xI e

A A A A

−=

⎧ ⎧

∫

0 1 2 0

0 1 2 1

1.05 0.19302 3 1.8637 0.90024 9 3 4046 0 0433

A A A AA A A AA A A A

+ + = =⎧ ⎧⎪ ⎪+ + = ⇒ =⎨ ⎨⎪ ⎪+ +⎩ ⎩0 1 2 2

0 0 1 1 2 2

4 9 3.4046 -0.04330.0875

A A A AI A f A f A f

⎪ ⎪+ + = =⎩ ⎩≈ + + =



Gauss For Unknown PointsGauss For Unknown PointsGauss For Unknown PointsGauss For Unknown Points( ) ( ) ( )

1

1 21

f t dt af t bf t+

= +∫

( )( )

1

1 2

2110

a bf ta bat btf t t

−

+ =⎧=⎧ ⎪ = =+ = ⎧⎪ ⎪⎪ ⎪ ⎪( )( )( )

1 2

2 2 22 11 2

3

123 3

f t tt tf t t at bt

f t t

⎪ ⎪=⎪ ⎪ ⎪⇒ ⇒⎨ ⎨ ⎨ = − == + =⎪ ⎪ ⎪⎩⎪ ⎪=⎩ ( )

( )

3 31 2

1

0

1 1

f t t at bt

f t dt f f+

⎪ ⎪=⎩ + =⎪⎩

⎛ ⎞ ⎛ ⎞= − +⎜ ⎟ ⎜ ⎟∫Applied Numerical MethodsApplied Numerical Methods

E. FatemizadehE. Fatemizadeh

( )1 3 3

f t dt f f−

+⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠∫


Gauss For Unknown PointsGauss For Unknown PointsGauss For Unknown PointsGauss For Unknown Points

( )b

f x dx∫( ) ( )

2 2

a

b a t b a b ax dx dt

− + + −= ⇒ =

( ) ( ) ( )1

12 2

b

a

b a b a t b af x dx f dt

+

−

− − + +⎛ ⎞= ⎜ ⎟

⎝ ⎠⎛ ⎞⎛ ⎞ ⎛ ⎞

∫ ∫

( ) ( ) ( ) ( )1 3 1 32 2 2

b

a

b a b a b a b a b af x dx f f dt

⎛ ⎞⎛ ⎞ ⎛ ⎞− − − + + − + +⎜ ⎟≈ +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

∫



Example:Example:Example:Example:

( )2

sinI x dxπ

= ∫

( ) ( )

0

Actual Value: 1sin 0 sin 2π π π+ ⎛ ⎞( ) ( )sin 0 sin 2

Trapezoidal: 0 0.7853982 2 4

G i i 0 998473

π π π

π π π π π

+ ⎛ ⎞− = ≈⎜ ⎟⎝ ⎠

⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟Gauss: sin sin 0.998473

4 4 44 3 4 3π π π π π⎛ ⎞⎛ ⎞ ⎛ ⎞+ + − + ≈⎜ ⎟⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠⎝ ⎠



Gauss in General form:Gauss in General form:Gauss in General form:Gauss in General form:•• Step 1:Step 1:

( ) ( )1b b a b a t b a+ − − + +⎛ ⎞∫ ∫

Step 2:Step 2:

( ) ( ) ( ) ( ) ( )1

,2 2a

b a b a t b aI f x dx I g t dx g t f

−

+ +⎛ ⎞= ⇒ = = ⎜ ⎟

⎝ ⎠∫ ∫

•• Step 2:Step 2:

( ) ( )2 1 : Has n roots in [-1,+1]n n

n n

dP t t= −( ) ( )( ) ( ) ( ) ( )2

0 1 2

[ , ]

1, 2 , 4 3 1

n ndtP t P t t P t t= = = −



•• Step 3Step 3Step 3Step 3

( ) ( ) ( ) ( )1

0 0 1 11

n ng t dt A g t A g t A g t+

−

= + + +∫

( )( )

0 1

0 0 1 1

21 0

2

n

n n

A A Ag t A t A t A tg t t

+ + + =⎧⎪=⎧ + + + =⎪⎪ = ⎪⎪ ( )

( )

( )

2 2 22 0 0 1 1

23n n

g t tA t A t A tg t t

⎪⎪ + + + =⎪⎪ = ⎪⎪ ⎪⇒⎨ ⎨⎪ ⎪( )

( )

( )0 0 1 1

2 1

1 11

nnn n n

n n

n

g t tA t A t A t

ng t t +

⎪ ⎪= − −⎪ ⎪ + + + =⎪ ⎪ +⎪ ⎪=⎩ ⎪


( )2 1 2 1 2 1

0 0 1 1 0n n nn n

g t tA t A t A t+ + +

⎩ ⎪+ + + =⎪⎩


tt are roots of are roots of PP ((xx))ttii are roots of are roots of PPnn((xx))ReplaceReplace ttii in first (n+1) equation and getin first (n+1) equation and getAAAAii



Example n=2Example n=2Example n=2Example n=2

( ) ( )1b

I f x dx I g t dx+

= ⇒ =∫ ∫( ) ( )

( ) ( )1

22 0 1

1 14 3 1 0 ,3 3

a

f g

P t t t t

−

= − = ⇒ = − = +

∫ ∫

( ) ( )

0 1 0

3 32 1

10A A A

AA t A t+ = =⎧ ⎧

⇒⎨ ⎨ =+ = ⎩⎩ 10 0 1 1 10 AA t A t =+ = ⎩⎩




1

⎧⎪ ( )

( )

1

01

Real Value:arctan 4 0.7854

1Trapezoidal: 1+0.5 0.7500

x

dx

π⎪

= ≈⎪⎪⎪⇒ ≈⎨∫ ( )2

0

2 2

Trapezoidal: 1 0.5 0.75001 2

1 1 1Gauss:= + 0.7869

x⇒ ⎨+ ⎪

⎪ ⎛ ⎞=⎪ ⎜ ⎟⎜ ⎟⎪

∫

( ) ( )2 22 1+ 0.211 1+ 0.789⎪ ⎜ ⎟⎜ ⎟⎪ ⎝ ⎠⎩



Subdivide to n section: Subdivide to n section: h=xh=x 11--xxSubdivide to n section: Subdivide to n section: h xh xi+1i+1--xxii

( ) ( ) ( )2

b n

i ihf x dx f C f D≈ +⎡ ⎤⎣ ⎦∑∫ ( ) ( ) ( )

1

1

2

3 3, ,2 6 6

ia

i ii i i i i

x xA C h A D h A

=

−

⎣ ⎦

+= = − + = + +

∑∫

ExampleExample::

2 6 6

ExampleExample::1

20

0.25, 0.785401

dxhx

= ≈+∫


0


Example n=3Example n=3Example n=3Example n=3

( ) ( )1b

I f x dx I g t dx+

= ⇒ =∫ ∫( ) ( )1

0 1 23 30

a

I f x dx I g t dx

t t t

−

⇒

= − = = +

∫ ∫

0 1 2

0 2 1

, , 0,5 5

5 8,9 9

t t t

A A A

+

= = =9 9




Actual Value: 0 30117⎧1

0

Actual Value: 0.30117sin Simpson=0.30005

Gauss 0.30117:=x xdx

⎧⎪⇒ ⎨⎪⎩

∫G⎩



Subdivide to n section: Subdivide to n section: h=xh=x 11--xxSubdivide to n section: Subdivide to n section: h xh xi+1i+1--xxii

( ) ( ) ( ) ( )5 8 518

b n

i i ihf x dx f C f A f D≈ + +⎡ ⎤⎣ ⎦∑∫

1

1

18

3 3, ,2 5 2 5 2

ia

i ii i i i i

x x h hA C A D A

=

−

⎣ ⎦

+= = − + = + +

∫

2 5 2 5 2


Multiple IntegralsMultiple IntegralsMultiple IntegralsMultiple Integrals

Formulation:Formulation:Formulation:Formulation:

( ) ( )b d b d

f x y dydx f x y dy dx⎧ ⎫

= ⎨ ⎬∫ ∫ ∫ ∫( ) ( ), ,a c a c

f x y dydx f x y dy dx= ⎨ ⎬⎩ ⎭

∫ ∫ ∫ ∫

d

a b

c


a b


Trapezoidal:Trapezoidal:Trapezoidal:Trapezoidal:

( ) ( ) ( ) ( ), , , ,2

b d b d b d cf x y dydx f x y dy dx f x d f x c dx⎧ ⎫ −

= ≈ +⎡ ⎤⎨ ⎬ ⎣ ⎦⎩ ⎭

∫ ∫ ∫ ∫ ∫( ) ( ) ( ) ( )

( )( ) ( ) ( ) ( ) ( )

2

, , , ,4

a c a c a

b a d cf a d f b d f a c f b c

⎣ ⎦⎩ ⎭

− −≈ + + +⎡ ⎤⎣ ⎦

∫ ∫ ∫ ∫ ∫

( ) ( ) ( ) ( )4 ⎣ ⎦



Simpson:Simpson:Simpson:Simpson:( ) ( ), ,

b d b d

a c a c

f x y dydx f x y dy dx⎧ ⎫

= ⎨ ⎬⎩ ⎭

∫ ∫ ∫ ∫

( ) ( )( ) ( ), 4 , 2 ,6

b

a

d c f x d f x d c f x c dx− ⎡ ⎤≈ + + +⎣ ⎦∫( ) ( ) ( ) ( ), 4 , ,

36 2d c b a a bf a d f d f b d

d b d

− − +⎡ ⎛ ⎞≈ + + +⎜ ⎟⎢ ⎝ ⎠⎣+ + +⎛ ⎞ ⎛ ⎞ ( )( )

( ) ( )

4 , 16 , 4 , 22 2 2

4

d c a b d cf a f f b d c

a bf f f b

+ + +⎛ ⎞ ⎛ ⎞+ + + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

+ ⎤⎛ ⎞⎜ ⎟


( ) ( ), 4 , ,2

f a c f c f b c ⎤⎛ ⎞+ +⎜ ⎟ ⎥⎝ ⎠ ⎦


Matrix formMatrix formMatrix formMatrix formd

1 4 1⎡ ⎤1 4 11 4 16 4

361 4 1

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

c

1 4 1⎢ ⎥⎣ ⎦

a b



Gauss Method:Gauss Method:Gauss Method:Gauss Method:

Th b ti i t f ll Th b ti i t f ll ( ) ( )

1 1 1

1 1 11 1 1

, , , ,n n n

i j k i j ki j k

f x y z dxdydz a a a f x y z+ + +

= = =− − −

= ∑∑∑∫ ∫ ∫The above equation is correct for all The above equation is correct for all polynomial polynomial xxααyyββzzγγ of degree s(≥ of degree s(≥ αα++ββ++γγ))

( )1 1 1 1 1 1

, ,f x y z x y z

I d d d d d d

α β γ

α β γ α β γ+ + + + + +

=

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟∫ ∫ ∫ ∫ ∫ ∫

1 1 1 1 1 1

n n n n n n

I x y z dxdydz x dx y dy z dz

I a x a y a z a a a x y z

α β γ α β γ

α β γ α β γ

− − − − − −

= = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞⎛ ⎞ ⎛ ⎞= =⎜ ⎟⎜ ⎟ ⎜ ⎟

∫ ∫ ∫ ∫ ∫ ∫

∑ ∑ ∑ ∑∑∑Applied Numerical MethodsApplied Numerical Methods

E. FatemizadehE. Fatemizadeh

1 1 1 1 1 1i i j j k k i j k i j k

i j k i j k

I a x a y a z a a a x y z= = = = = =

= =⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠∑ ∑ ∑ ∑∑∑



( )( )1 1 11 1 1

16xI u v e dxdudv

+ + +

= − +∫ ∫ ∫ ( )( )1 1 1

2 2 3

16Use two term for u and v and three terms for x

1

− − −∫ ∫ ∫

( )( )1 2 1

1 2

1I= 1 116

1

kxi j k i j

i j ka a b u v e

a a= = =

− +

= =

∑∑∑

1 2

1 3 1

15 8,9 9

a a

b b b= = =


Matlab CommandMatlab CommandMatlab CommandMatlab Command

Simpson method:Simpson method:Simpson method:Simpson method:•• I = quad(I = quad(funfun,a,b);,a,b);

I=quad(@I=quad(@myfunmyfun 0 1);0 1);

b

a

fdx∫I=quad(@I=quad(@myfunmyfun,0,1);,0,1);I=quad(‘exp(I=quad(‘exp(--x.^2’,1,2);x.^2’,1,2);

•• I = quad(I = quad(funfun,a,b,Tol); Tol = 1e,a,b,Tol); Tol = 1e--6 by 6 by I quad(I quad(funfun,a,b,Tol); Tol 1e,a,b,Tol); Tol 1e 6 by 6 by default.default.

y = myfun(x)

y = 4./(1+x.^2);I = quad(@myfun,0,1);

err = (pi-I)/pi;


err = 1.8 e-8


Double IntegralDouble IntegralDouble IntegralDouble Integral•• dblquad(dblquad(funfun,,XminXmin,,XmaxXmax,,YminYmin,,YmaxYmax))•• I=dblquad('exp(I=dblquad('exp(--x ^2x ^2--y ^2)'y ^2)' --1 +11 +1 --2 +2);2 +2);•• I=dblquad( exp(I=dblquad( exp(--x. 2x. 2--y. 2) ,y. 2) ,--1,+1,1,+1,--2,+2);2,+2);•• dblquad(@myfun,dblquad(@myfun,--1,+1,1,+1,--2,+2)2,+2)

Trilpele IntegralTrilpele IntegralTrilpele IntegralTrilpele Integral•• triplequad(fun,triplequad(fun,XminXmin,,XmaxXmax,,YminYmin,,YmaxYmax,,ZminZmin

,,ZmaxZmax););,,ZmaxZmax););•• triplequad('exp(triplequad('exp(--x.^2x.^2--y.^2y.^2--z.^2)',z.^2)',--1,+1,1,+1,--2,+2,2,+2,--1,1);1,1);•• triplequad(@myfun,triplequad(@myfun,--1,+1,1,+1,--2,+2,2,+2,--1,1);1,1);



Another method:Another method: bAnother method:Another method:•• I = quadl(I = quadl(funfun,a,b);,a,b);

I=quadl(@I=quadl(@myfunmyfun 0 1);0 1);a

fdx∫I=quadl(@I=quadl(@myfunmyfun,0,1);,0,1);I=quadl(‘exp(I=quadl(‘exp(--x.^2’,1,2);x.^2’,1,2);

•• I = quadl(I = quadl(funfun,a,b,Tol); Tol = 1e,a,b,Tol); Tol = 1e--6 by 6 by I quadl(I quadl(funfun,a,b,Tol); Tol 1e,a,b,Tol); Tol 1e 6 by 6 by default.default.

y = myfun(x)

y = 4./(1+x.^2);I = quadl(@myfun,0,1);

err = (pi-I)/pi;


err = 1.7 e-8


A nice example:A nice example:A nice example:A nice example:1

2 2 2

1 1 1 1,I dx dx dx t∞ ∞

= = + =∫ ∫ ∫2 2 20 0 1

1 1

2 2

,1 1 1

1 1

x x x x

dx dt

+ + +

= +

∫ ∫ ∫

∫ ∫2 20 0

1

2

1 1

121

x t

dx

+ +

=

∫ ∫

∫ 20 1 x+∫

Applied Numerical MethodsApplied Numerical MethodsE. E. FatemizadehFatemizadeh


A nice example:A nice example:A nice example:A nice example:

2 2 21 1x x xI e dx e dx e dx t

∞ ∞− − −+∫ ∫ ∫

2

0 0 1

11 1

,

e t

I e dx e dx e dx tx

−

= = + =∫ ∫ ∫

2

20 0

e

0 7468 0 1394 0 8862

txe dx dt

t−= +

= + =

∫ ∫0.7468 0.1394 0.8862= + =

Applied Numerical AnalysisApplied Numerical AnalysisE. E. FatemizadehFatemizadeh