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7/22/2011 http://numericalmethods.eng.usf.edu 1

Backward Divided Difference

Major: All Engineering Majors

Authors: Autar Kaw, Sri Harsha Garapati

http://numericalmethods.eng.usf.eduTransforming Numerical Methods Education for STEM

Undergraduates

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Backward DividedDifference

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Definition

ix

ix

.

x

xxfxf

x

xf

(

(

p(

!d

0

lim

Slope at

f(x)

y

x

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Backward Divided Difference

xx (x

x

x

xxfxfxf

(

($d

x

xxfxfxf iii

(

($d

)(xf

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ExampleExample:

The velocity of a rocket is given by

300,8.921001014

10

14ln2000

4

4

ee

v v!tt

ttR

where R given in m/s and t is given in seconds. Use backward difference approximation

Of the first derivative of t to calculate the acceleration at .16st! Use a step size of.2st!(

t

ttta

iii

(

1RRSolution:

16!it

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2 !t

ttt ii (!1 14216 !!

2

141616

RR !a

168.91621001014

1014ln200016

4

4

v

v!R sm /07.392!

148.91421001014

1014ln200014

4

4

v

v!R sm /24.334!

Example (contd.)

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Hence

The exact value of 16a can be calculated by differentiating

tt

t 8.921001014

1014ln2000

4

4

v

v!R

as

Example (contd.)

2/915.2

824.334

07.392

2

)14()16(16 sa !!

!

RR

? A

2/674.29)16(

3200

4.294040)()(

sma

t

tt

dt

dta

!

!! R

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The absolute relative true error is

Example (contd.)

100v

!TrueValue

eValueApproximatTrueValuet

I

%557.2

100674.29

915.28674.29

!

v

!

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Effect Of Step Sizex

exf 49)( !Value of )2.0('f Using backward Divided difference method.

h )2.0(

f aE aI % S g f ca tdigits

tE

tI %

0.05 72.61598 7.50 49 9. 65 77

0.025 76.24 76 .627777 4.758129 1 .87571 4.8 7418

0.0125 78.14946 1.905697 2.4 8529 1 1.97002 2.458849

0.00625 79.12627 0.976817 1.2 4504 1 0.99 20 1.2 9648

0.00 125 79.62081 0.4945 0.62111 1 0.49867 0.622404

0.00156 79.86962 0.248814 0. 11525 2 0.24985 0. 1185

0.000781 79.99442 0.124796 0.156006 2 0.12506 0.156087

0.000 91 80.05691 0.062496 0.078064 2 0.06256 0.078084

0.000195 80.08818 0.0 1272 0.0 9047 0.0 129 0.0 9052

9.77E-05 80.10 8 0.015642 0.019527 0.01565 0.019529

4.88E-05 80.11165 0.00782 0.009765 0.00782 0.009765

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Effect of Step Size in Backward

Divided Difference Method

75

85

95

1 3 5 7 9 11

Number of times the step size is halved, n

f'(0.2

Initial step size=0.05

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Effect of Step Size on

Approximate Error

0

1

2

3

4

1 3 5 7 9 11

Number of steps involved, n

Ea

Initial step size=0.05

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Effect of Step Size on Absolute

Relative Approximate Error

0

1

2

3

4

5

0 1 2 3 4 5 6 7 8 9 10 11 12

Number of steps involved, n

|Ea

|,

Initial step size=0.05

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Effect of Step Size on LeastNumber of Significant Digits

Correct

0

0.5

1

1.5

2

2.5

3

3.5

1 2 3 4 5 6 7 8 9 10 11

Number of steps involved, n

Leastn

umberofsignific

an

digitscorrect

Initial step size=0.05

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Effect of Step Size on True Error

0

12

3

4

5

6

7

8

0 2 4 6 8 10 12

Number of steps involved, n

Et

Initial step size=0.05

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Effect of Step Size on Absolute

Relative True Error

0

1

2

3

4

5

6

7

8

9

10

1 2 3 4 5 6 7 8 9 10 11

Number of steps involved, n

|Et|

%

Initial step size=0.05

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Additional ResourcesFor all resources on this topic such as digital audiovisuallectures, primers, textbook chapters, multiple-choice

tests, worksheets in MATLAB, MATHEMATICA, MathCadand MAPLE, blogs, related physical problems, pleasevisit

http://numericalmethods.eng.usf.edu/topics/continuous

_02dif.html

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THE END

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