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Lecture Slides

on

Modeling and Simulation

Lecture: Numerical Derivatives

2

Numerical Differentiation and Integration

• We known from Calculus the mathematical concepts of

differentiation and integration :



b

a

i i x

i i

I f x dx

x

f x x f x

dx

dy

x

f x x f x

x

y

lim

0

 y

 x

f(xi)

f(xi+  x)

x

Numerical Differentiation

 y

 x

f(xi)

f(xi+  x)

x

Numerical Differentiation

 y

 x

f(xi)

f(xi+  x)

x

Numerical Differentiation

xi

f(xi)

x

Finally, we can

see that the

derivative forms

a slope at a point. x

f x x f x

dx

dy

x

f x x f x

x

y

i i x

i i



lim

0

Numerical Differentiation

Derivatives

• We generally use Taylor series to evaluate a function in the

neighborhood of xo when the derivatives of all order are known.

• It can also be used to evaluate approximately the derivatives of f(x)

if the values of function are known at a number of discrete points, { xi } where, i = 1, 2,... N.

• Let us assume that we know the values of f(xo) and f(xo + h), where

h is a small value.

• Then the Taylor series expansion of f(xo + h) can be written as

following:

     ( ) 2

( ) ( ) ( )

2 o o o f x o

h f x h f x hf x

Forward Difference as First Derivative

• When we truncate Taylor’s expansion after the first derivative and

solve for first derivative we obtain an approximation:

• Thus the first derivative at x = xo is approximately estimated by f(xo+

h) and f(xo). By defining the forward differences on the right side is expressed by

• It is termed as forward difference formula for the

derivative.

 ( ) ( ) 1 ( (^) o ) f xo h f xo h

f  x   

 f ( xo ) f ( xo  h ) f ( xo )

  o

o o o x

f x f x h f x h 

   

( ) ( ) ( ) 1

Errors

• We see that approximating the first derivatives by

forward or backward difference processes error and the

error is approximately equal to the (h/2)f’’.

• The error is known as truncation error.
• Another type of error that can be introduced in numerical

calculations is round-off error.

• It is error appearing in computing the difference terms

and may be denoted by .

• The truncation error decreases with h , however the round-off error increases.
• It has been seen that truncation error is dominant when h is large and absolute error has dominating role of round-off error when h is very small.
• Both errors have been compared in Figure.
• For large step size, h, the error in the first derivative is dominated by the truncation error and as h is allowed to reduce, major contribution to the total error becomes due to round-off error.

Errors

 ( 0. 3 ) ( 0. 1 ) 2

1

1. 2

    

f x f x dx h

df x

1. 2

dx x 

df

= 18.5 gm/cm

 ( 0. 4 ) ( 0. 2 ) 2

1

1. 3

    

f x f x dx h

df x

2 ( 0. 1 )

( 7. 2 4. 3 )

1. 3

  dx x 

df = 14.5gm/cm

Let us first calculate the derivative at x = 0.2,

• Similarly at x = 0.3,

Example 1: derivative

Example 1: Derivative

 ( 0. 1 ) 2 ( 0. 2 ) ( 0. 3 )

2

1. 2

2

2       

f x f x f x dx h

d f

x

 2. 1 2. 0 4. 3 5. 8  (^70). ( 0. 1 )

2

1. 2

2

2       dx x 

d f

 ( 0. 2 ) 2 ( 0. 3 ) ( 0. 4 ) 10.

1 2

1. 3

2

2         

f x f x f x dx h

d f

x

and the second derivative at x = 0.2 is given as:

Then the second derivative at x= 0.3 is approximately equal to:

These estimates can also be found using forward and backward difference forms.

Find derivative of polynomial using MATLAB for the

function:

The derivative function is given as following:

Use MATLAB to find the derivative function.

f(x)  2 x^5  2 x^4  5 x^3  10 x  1

( ) 10 8 15 10 4 3 2 f ^ x  x  x  x 

Example 3: derivative Using MATLAB

• Solution:
• The script file for the MATLAB code is give below where

we first find the solution by polyder function and the

employ diff to find the derivative.

• The coefficients of polynomial are identified and then

these are used to find the coefficients of derivative

function.

• Then using polyval command this polynomial is

generated.

• The diff command approximates the derivative as a ratio

of two differences as

( yi  1  yi)/(xi  1  xi)  i  1 , 2 , 

.

Example 3: derivative Using MATLAB