3.5 Numerical Differentiation (Finite Difference Formulas) Numerical differentiation is a process to find an estimate of a derivative of a given function at a given point. 3.5.1 Finite difference approximation of the derivative

The derivative of a function f(x) is defined by the limit

x

xfxxfxf

dx

xdfx

0lim' .

If x is sufficiently small, equation above can be rewritten as

x

xfxxfxf

'

with x represents a small change in x and it can be either positive or negative. To estimate the value of xf ' , we uses the forward, backward, and central finite difference formulas. Three such formulas are presented as follows with the derivative is calculated from the values of two points, as in Figure 3.7, by setting xxi , xxxi 1 , and xxxi 1 .

Figure 3.7. Finite Difference Approximation of First Derivative Forward difference is the slope of the line that connects points B and C:

x

xfxf

xx

xfxfxf ii

ii

iii

1

1

1'

Backward difference is the slope of the line that connects points A and B:

x

xfxf

xx

xfxfxf ii

ii

iii

1

1

1'

Central difference is the slope of the line that connects points A and C:

xxfxf

xx

xfxfxf ii

ii

iii

2' 11

11

11

xi xi-1 xi+1 x

f (x)

0

A B

C

3.5.2 Finite difference formulas using Taylor series expansion

The forward, backward, and central finite difference formulas are also can be derived by using Taylor series expansion as,

imimm

iiiiii xRxfm

hxf

hxf

hxf

hxhfxfxf !!4

'''!3

''!2

' )4(432

1 (1)

and

ininn

niiiiii xRxfn

hxf

hxf

hxf

hxhfxfxf !

1!4

'''!3

''!2

' 4432

1 (2)

where xh is the step size. mR and nR , called the remainders, are given by

1 !mj

ij

j

im xfj

hxR and

1 !

1nj

ij

jj

in xfj

hxR .

Subtracting Equation (2) from Equation (1) can produce the two-point central difference formula.

211

211

3

11

2

'''!32

'

'''!3

2'2

hOh

xfxf

xfh

h

xfxfxf

xfh

xhfxfxf

ii

iii

i

iiii

Hence it is showing that a more accurate formula as the error is of the order of 2hO has been obtained. 3.5.3 Summary of finite difference formulas

We can use the following difference formulas to compute the various derivatives.

First Derivative Differenc

e Type Formula

Truncation Error

Two-point forward

h

xfxfxf iii

1' hO

Two-point backward

h

xfxfxf iii

1'

hO

Two-point central

h

xfxfxf iii 2

' 11

2hO

Three-point forward

h

xfxfxfxf iiii 2

34' 12

2hO

Three-point backward

h

xfxfxfxf iiii 2

43' 21

2hO

Four-point central

h

xfxfxfxfxf iiiii 12

88' 2112

4hO

Second Derivative

Difference Type

Formula Truncatio

n Error Three-point forward

2

12 2''h

xfxfxfxf iiii

hO

Three-point backward

2

212''h

xfxfxfxf iiii

hO

Three-point central

2

11 2''h

xfxfxfxf iiii

2hO

Four-point forward

2

123 254''h

xfxfxfxfxf iiiii

2hO

Four-point backward

2

321 452''h

xfxfxfxfxf iiiii

2hO

Five-point central

2

2112

12

163016''

h

xfxfxfxfxfxf iiiiii

4hO

Third Derivative Differenc

e Type Formula

Truncation Error

Four-point forward

3

123 33'''h

xfxfxfxfxf iiiii

hO

Four-point backward

3

321 33'''h

xfxfxfxfxf iiiii

hO

Four-point central

3

2112

2

22'''

h

xfxfxfxfxf iiiii

2hO

Five-point forward

3

1234

2

51824143'''

h

xfxfxfxfxfxf iiiiii

2hO

Five-point backward

3

4321

2

31424185'''

h

xfxfxfxfxfxf iiiiii

2hO

Six-point central

3

321123

8

813138'''

h

xfxfxfxfxfxfxf iiiiiii

4hO

Fourth Derivative Differenc

e Type Formula

Truncation Error

Five-point forward

4

12344 464

h

xfxfxfxfxfxf iiiiii

hO

Five-point backward

4

43214 464

h

xfxfxfxfxfxf iiiiii

hO

Five-point central

4

21124 464

h

xfxfxfxfxfxf iiiiii

2hO

Six-point forward

4

123454 3142624112

h

xfxfxfxfxfxfxf iiiiiii

2hO

Six-point backward

4

543214 2112426143

h

xfxfxfxfxfxfxf iiiiiii

2hO

Seven-point central

4

3211234

6

1239563912

h

xfxfxfxfxfxfxfxf iiiiiiii

4hO

Example 3.19

Calculate the first derivative for the function 3 23 2f x x x x at point x = 5 numerically with the forward, backward and central finite difference formulas by using (i) points x = 4, x = 5, and x = 6. (ii) points x = 4.75, x = 5, and x = 5.25. Compare the results with the exact (analytical) derivative. Solution

Analytical differentiation:

2465' ,5when 149' 2

fx

xxxf

Numerical differentiation: (i)

x: 4 5 6 f(x): 228 430 726

Forward finite difference:

2961

430726

56

565'

ff

f %33.20100246

246296

error

Backward finite difference:

2021

228430

45

455'

ff

f %89.17100246

246202

error

Central finite difference:

2492

228726

46

465'

ff

f %22.1100246

246249

error

(ii)

x: 4.75 5 5.25 f(x): 371.390625 430 494.484375

Forward finite difference:

9375.25725.0

430484375.494

525.5

525.55'

ff

f %85.4100246

2469375.257

error

Backward finite difference:

4375.23425.0

390625.371430

75.45

75.455'

ff

f %70.4100246

2464375.234

error

Central finite difference:

1875.2465.0

390625.371484375.494

75.425.5

75.425.55'

ff

f

%08.0100246

2461875.246

error

The results show that the central finite difference formula gives a more accurate approximation. In addition, smaller step size h gives a significantly more accurate approximation. Example 3.20

The following table lists the population of Malaysia from 1980 2010.

Table 3.4. The Population of Malaysia in years 1980 2010

Year 1980 1985 1990 1995 2000 2005 2010 Population (millions) 13.9 15.9 18.1 20.7 23.5 26.5 28.3

Calculate the rate of growth of the population in millions per year for 2010. (i) Use two-point backward difference formula. (ii) Use three-point backward difference formula. (iii) Based on part (ii) above, apply the two-point central difference formula to predict the population in the year 2015. Solution

(i) yearper millions 36.05

5.263.28

20052010

200520102010'

ff

f

(ii)

10

5.235.2643.283

52

200020054201032010'

ffff

yearper millions 24.0

(iii) 5220052015

2010'ff

f

millions 9.28201510

5.26201524.0

f

f