# AlidN i lApplied Numerical Analysis - دانشگاه صنعتی شریف - خانه fatemizadeh/Courses/ANA/07-Differentiation...AlidN i lApplied Numerical Analysis Differentiation and Integration Lecturer: Emad Fatemizadeh Applied Numerical Methods E. Fatemizadeh Numerical DifferentiationNumerical Differentiation Need for numerical differentiation:

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

• 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

• Numerical DifferentiationNumerical DifferentiationNumerical DifferentiationNumerical Differentiation

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

= +

+

( ) ( ) ( ) ( )22 3!

f ff x h f x

h = + +

• Numerical DifferentiationNumerical DifferentiationNumerical DifferentiationNumerical Differentiation

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 xh

( ) ( ) ( )Central:

f x h f x hf x

+

( )2

f xh

• Numerical DifferentiationNumerical DifferentiationNumerical DifferentiationNumerical Differentiation

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 =

• Numerical DifferentiationNumerical DifferentiationNumerical DifferentiationNumerical Differentiation

Example:Example:Example:Example:( ) ( )cos , 0.8, 0.01f h = = =

MethodMethod ForwardForward BackwardBackward AverageAverage RichardsonRichardson

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

• Numerical DifferentiationNumerical DifferentiationNumerical DifferentiationNumerical Differentiation

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

+ +

• Numerical DifferentiationNumerical DifferentiationNumerical DifferentiationNumerical Differentiation

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=

• Numerical DifferentiationNumerical DifferentiationNumerical DifferentiationNumerical Differentiation

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

+

+ +

+ +

= = + = +

( ) ( ) ( )11 1

1 i ii

i i i i

f x f xA B f x

x x x x+

+ +

= = =

• Numerical DifferentiationNumerical DifferentiationNumerical DifferentiationNumerical Differentiation

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 12

i iif x h

+ =

• Numerical DifferentiationNumerical DifferentiationNumerical DifferentiationNumerical Differentiation

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 122i i i

i

f x f x f xf x

h+ + =

• Numerical DifferentiationNumerical DifferentiationNumerical DifferentiationNumerical Differentiation

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 =

• Numerical DifferentiationNumerical DifferentiationNumerical DifferentiationNumerical Differentiation

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

• Numerical DifferentiationNumerical DifferentiationNumerical DifferentiationNumerical Differentiation

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

+ +

• Numerical DifferentiationNumerical DifferentiationNumerical DifferentiationNumerical Differentiation

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

• Numerical DifferentiationNumerical DifferentiationNumerical DifferentiationNumerical Differentiation

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.

• Numerical DifferentiationNumerical DifferentiationNumerical DifferentiationNumerical Differentiation

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+ +

+ +